Bispectral Methods of Signal Processing: Applications in Radar, Telecommunications and Digital Image Restoration 9783110368888, 9783110374568

Table of contents :
Introduction
Contents
1 General properties of bispectrum-based digital signal processing
1.1 General properties of cumulant and moment functions
1.2 Triple correlation function and bispectrum
1.3 Bispectral density estimation techniques
1.4 Bispectrum-based algorithms in application for filtering and signal shape reconstruction
1.5 Reduction of waveform distortions in bispectrum-based signal reconstruction systems
1.6 Performance of the bispectral density estimator
1.7 Conclusions
2 Unknown noisy signal shape estimation by bispectrum-filtering techniques
2.1 Smoothing the noisy bimagnitude and biphase or the real and imaginary parts of bispectrum estimates by using nonadaptive 2-D linear and nonlinear filtering
2.2 Statistical properties of bispectrum estimate contaminated by noise
2.3 Novel techniques developed for improving noisy bispectrum estimates
2.4 Adaptive 1-D filtering applied for bispectrum-based signal reconstruction
2.5 Conclusions
3 Bispectrum-based digital image reconstruction using tapering pre-distortion
3.1 Additive predistortions in reconstruction of the images contaminated by noise and jitter
3.2 Bispectrum-based image reconstruction by using multiplicative predistortions
3.3 Search of the optimal parameters used for additive and multiplicative pre-distortion functions
3.4 Conclusions
4 Signal detection by using third-order test statistics in communications and radar applications
4.1 Detection of deterministic signals by using third-order test statistics and likelihood ratio criterion
4.2 Bispectrum-based encoding technique developed for noisy, multipath and fading radio links
4.3 Naval surface target detection and recognition by estimation of radar range profiles
4.4 Using bicoherence-based features for aerial target classification
4.5 Time-frequency analysis of backscattering in ground surveillance Doppler radar
5 Conclusions
Bibliography
Subject index

Citation preview

Alexander V. Totsky, Alexander A. Zelensky, Victor F. Kravchenko Bispectral Methods of Signal Processing

Also of Interest Compressive Sensing Applications to Sensor Systems and Image Processing Ender, 2016 ISBN 978-3-11-033531-6, e-ISBN 978-3-11-033539-2 Foundations of MIMO in Radar and Communications Dowla, Nekoogar (Eds.), 2015 ISBN 978-1-61451-732-0, e-ISBN 978-1-61451-535-7

Oscillatory Neural Networks In Problems of Parallel Information Processing Kuzmina, Manykin, Grichuk, 2013 ISBN 978-3-11-026835-5, e-ISBN 978-3-11-026920-8 Strip-Method for Image and Signal Transformation Mironovsky, Slaev, 2011 ISBN 978-3-11-025192-0, e-ISBN 978-3-11-025256-9

International Journal of Electronics and Telecommunications Ryszard Romaniuk (Editor-in-Chief) ISSN 2300-1933

Image Processing & Communications Ryszard S. Choraś (Editor-in-Chief) ISSN 2300-8709

Alexander V. Totsky, Alexander A. Zelensky, Victor F. Kravchenko

Bispectral Methods of Signal Processing | Applications in Radar, Telecommunications and Digital Image Restoration

Mathematics Subject Classification 2010 60, 60G35, 62M10, 68T10, 94A08, 60G15, 94A13 Physics and Astronomy Classification Scheme 2010 02.30, 02.30, 02.50-r, 02.50 Authors Prof. Dr. Alexander V. Totsky National Aerospace University Dept. of Receiving, Transmitting & Signal Processing Chkalova Str. 17 Kharkov 61070 Ukraine e-mail: [email protected]

Prof. Dr. Alexander A. Zelensky National Aerospace University Dept. of Receiving, Transmitting & Signal Processing Chkalova Str. 17 Kharkov 61070 Ukraine e-mail: [email protected]

Prof. Dr. Victor F. Kravchenko Russian Academy of Sciences Kotel’nikov Institute of Radio Engineering and Electronics of RAS Mokhovaya 11–7 Moscow 125009 Russia e-mail: [email protected]

ISBN 978-3-11-037456-8 e-ISBN (PDF) 978-3-11-036888-8 e-ISBN (EPUB) 978-3-11-038606-6 Set-ISBN 978-3-11-036889-5 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2015 Walter de Gruyter GmbH, Berlin/Munich/Boston Cover image: D. B. Belukha Typesetting: le-tex publishing services GmbH, Leipzig Printing and binding: CPI books GmbH, Leck ♾ Printed on acid-free paper Printed in Germany www.degruyter.com

Introduction Try extracting harmony contained in chaos

Bispectrum-based techniques of signal processing and higher-order statistical analysis [1–4] have attracted the attention of many researchers in the 1980s of the previous century as prospective addition and sometimes as an alternative tool to common second-order spectral-correlation analysis widely spread for different applications like radar, sonar, pattern recognition, digital communications, nondestructive control, biomedical engineering, and so on. A. W. Lohmann and B. Wirnitzer (see, e.g., [1]) were one of the first researchers who addressed optical and astronomical applications of triple correlation and bispectrum. Their pioneer investigations performed in astronomy stimulated other applications like sonars, biomedical engineering, nondestructive control, radars, communications, image processing, and so on, that have been generalized in [2] by C. L. Nikias and M. R. Raghuveer. Theoretical aspects of bispectrum-based signal processing were summarized by J. M. Mendel in [3] and by C. L. Nikias; A. P. Petropulu in their book [4]. Recently, the amount of publications dedicated to higher-order statistics and bispectral analysis has increased radically. For almost twenty years the researchers were basically dealing with the theoretical aspects of triple correlation and bispectrum estimation. This peculiarity can be explained by the necessity of extensive computations for processing and storage of multidimensional data required for higher-order statistical analysis. Lately, with increased computing power, the interest in practical applications of bispectrum-based signal processing has increased. This is explained with known inherent advantages of bispectrum that radically differ it from common second-order power spectrum estimation. These advantages are as follows: possibility to extract phase-coupled contributions contained in processed signals, immunity to zero mean noise with symmetric probability density function, random signal shift and jitter invariance properties, as well as preservation of phase information contained in processed data. However, many both theoretical and practical questions yet remain unclear. They are: – The statistical properties of noisy bispectrum and triple correlation estimates have not been analyzed in detail; effective ways for their improvement have not been thoroughly studied yet. – The phase wrapping problem still exists in signal processing and it influences the quality of bispectrum-based signal waveform restoration. – Recently, a lot of attention has been paid to the problem of bispecrum-based 1D signal processing; however, it is natural to expect promising results in case of bispectrum-based 2-D noisy and jittery image processing and digital image reconstruction.

vi | Introduction –

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The problem of detection of deterministic signal embedded in noise by using novel higher-order test statistics formed at the matched filter is of particular interest for digital communications and radar applications. Both sea and vegetation clutter suppression in radar applications by using bispectral-based signal processing can provide improving target recognition and classification performance. Bispectrum-based approach is able to extract the phase coupling contribution contained in nonstationary and multicomponent radar signals backscattered by moving targets observed in vegetation clutter. This will make it possible to obtain novel information features for better radar target recognition and classification. The performance of bispectrum-based signal processing is not thoroughly investigated for such real-life situations like small input signal-to-noise ratios (SNR) and a small number of observed realizations.

These problems are addressed and discussed below in this book. The goal of the book is the theoretical and experimental study of bispectrumbased techniques and algorithms developed for digital processing of signals and images. The basic application is radars of different types intended for detection and automatic recognition of aerial, ground moving and naval targets, surveillance in vegetation and sea clutter. Other applications like digital wireless communications and digital image processing are considered as well. The book contains four Chapters. Chapter 1 gives theoretical background and deals with analysis of basic properties of bispectrum and triple correlation function and accuracy of bispectrum estimation. Some particular aspects like phase unwrapping are discussed. Statistical study of bispectral estimates contaminated by interferences is performed and extreme accuracy is analytically defined by Cramér–Rao criterion. Chapter 2 is devoted to combined bispectrum-filtering techniques that exploit positive features of bispectrum and filtering, linear and nonlinear, nonadaptive and adaptive. Non-Gaussianity and nonstationarity of fluctuations observed in bispectral domain induced by leakage of input noise is demonstrated. This serves the purpose of designing novel adaptive filters suitable for this application. Reconstruction of images distorted by jitter and additive noise is considered in Chapter 3. Removal of jitter observed with influence of mixture of additive Gaussian and impulsive noise is studied. It is shown that additive predistortions provide both removal of phase ambiguity and jitter by using bispectrum-based image reconstruction. Approach based on multiplicative predistortions allows decreasing distortions in restored images as compared with additive predistortions. Optimal additive and multiplicative predistortion function parameters are evaluated and analyzed. A novel bispectral technique for signal detection and discrimination is suggested in Chapter 4 by using test detection statistics computed in the form of peak values of the third-order moment functions. A novel encoding concept using frequency diver-

Introduction |

vii

sity strategy and bispectrum-based signal processing are suggested for wireless communication systems. According to the proposed approach, binary data are transmitted by using a pair of mutually orthogonal triplet-signals contained phase-coupled frequency tones. Novel third-order test detection statistics evaluated in the form of triplet-signal bimagnitude peaks are suggested for detection and discrimination of received triplet-signals in noisy and fading communication radio links. Radar applications of bispectrum are considered in Chapter 4. It contains experimental results for coastal naval, ground surveillance and aerial target recognition and classification radars. Contributions of the authors are represented as follows. A. V. Totsky is contributed to all Chapters of the book. A. A. Zelensky is contributed to the sub-Chapters 1.1, 1.2, 2.1, 2.2, 2.4, 3.1 and 4.2. V. F. Kravchenko is contributed to the sub-Chapter 1.4.

Contents Introduction | v 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 2 2.1

2.2 2.3 2.4 2.5 3 3.1 3.2 3.3 3.4

General properties of bispectrum-based digital signal processing | 1 General properties of cumulant and moment functions | 1 Triple correlation function and bispectrum | 4 Bispectral density estimation techniques | 9 Bispectrum-based algorithms in application for filtering and signal shape reconstruction | 12 Reduction of waveform distortions in bispectrum-based signal reconstruction systems | 31 Performance of the bispectral density estimator | 42 Conclusions | 46 Unknown noisy signal shape estimation by bispectrum-filtering techniques | 48 Smoothing the noisy bimagnitude and biphase or the real and imaginary parts of bispectrum estimates by using nonadaptive 2-D linear and nonlinear filtering | 50 Statistical properties of bispectrum estimate contaminated by noise | 63 Novel techniques developed for improving noisy bispectrum estimates | 67 Adaptive 1-D filtering applied for bispectrum-based signal reconstruction | 92 Conclusions | 99 Bispectrum-based digital image reconstruction using tapering pre-distortion | 101 Additive predistortions in reconstruction of the images contaminated by noise and jitter | 101 Bispectrum-based image reconstruction by using multiplicative predistortions | 113 Search of the optimal parameters used for additive and multiplicative pre-distortion functions | 117 Conclusions | 125

x | Contents 4 4.1 4.2 4.3 4.4 4.5

5

Signal detection by using third-order test statistics in communications and radar applications | 126 Detection of deterministic signals by using third-order test statistics and likelihood ratio criterion | 126 Bispectrum-based encoding technique developed for noisy, multipath and fading radio links | 136 Naval surface target detection and recognition by estimation of radar range profiles | 148 Using bicoherence-based features for aerial target classification | 161 Time-frequency analysis of backscattering in ground surveillance Doppler radar | 168 Conclusions | 188

Bibliography | 191 Subject index | 198

1 General properties of bispectrum-based digital signal processing 1.1 General properties of cumulant and moment functions Common power or energy spectral density estimation is a well-known and widely spread tool for random signal analysis. Ensemble averaged Fourier magnitude spectrum density does not contain any information about behavior of a centered random process in the frequency domain since the spectral components are statistically independent in different observed realizations. In this case, the energy distribution of statistically independent spectral components must be estimated since the energy content does not depend on the phase relationships for separate frequencies. Indeed, for the processes containing independent spectral components, the energy spectrum estimate is the exhaustive characteristic conventionally used in spectral analysis of such processes. In several practical applications of signal processing, an analyzed process can contain phase coupled spectral contributions. Study of these spectral correlation relationships can give us very useful and important information for correct understanding, analysis and description of physical effects that cause a given process. Note that such information about phase coupling is irretrievably lost in common energy spectrum estimates. Cumulant function and cumulant spectrum estimation can serve as a very useful and promising tool for signal analysis and processing. Cumulant-based approach has several important and attractive benefits as compared with energy spectrum estimation. These benefits are listed and described below. First, consider mathematical description of cumulant spectra for a real-valued stationary and discrete-valued process given by the time series as {𝑥(𝑖), 𝑖 = 0, 1, 2, . . .}. The joint cumulants 𝑐𝑥(𝑟) (𝜏1 , 𝜏2 , . . ., 𝜏𝑟−1 ) of 𝑟th order can be defined as

𝑐𝑥(𝑟) = 𝑐𝑥(𝜏1 , 𝜏2 , . . ., 𝜏𝑟−1 ) = −𝑗𝑟 [

𝜕𝑟 ln 𝛩(𝜔1 , 𝜔2 , . . ., 𝜔𝑟 ) ] , 𝜕𝜔1 𝜕𝜔2 . . .𝜕𝜔𝑟 𝜔1 =𝜔2 =...=𝜔𝑟 =0

(1.1.1)

where 𝛩(𝜔1 , 𝜔2 , . . ., 𝜔𝑟 ) = ⟨exp[𝑗(𝜔1 𝑥1 +𝜔2 𝑥2 +. . .+𝜔𝑟 𝑥𝑟 )]⟩𝑥 is the multidimensional characteristic function; 𝜔1 , 𝜔2 , . . ., 𝜔𝑟 are the angular frequencies; 𝑗 = √−1; ⟨. . .⟩𝑥 denotes ensemble averaging procedure; and 𝜏1 , 𝜏2 , . . ., 𝜏𝑟−1 are the time shifts. The cumulants (1.1.1) serve as the characteristic of the probability distribution and they can be represented by the following coefficients in Taylor series for the function ln 𝛩(𝜔) in the neighborhood of the point of origin

𝑐𝑥(𝑟) ln 𝛩(𝜔) = ∑ (𝑗𝜔)𝑟 . 𝑟! 𝑟=1 ∞

(1.1.2)

2 | 1 General properties of bispectrum-based digital signal processing The joint moments 𝑚𝑥 (𝜏1 , 𝜏2 , . . ., 𝜏𝑟−1 ), defined for a stationary process {𝑥(𝑖), 𝑖 = 0, 1, 2, . . .} differ from the cumulants (1.1.1) as follows

𝑚(𝑟) 𝑥 = 𝑚𝑥 (𝜏1 , 𝜏2 , . . ., 𝜏𝑟−1 ) = ⟨𝑥(𝑖)𝑥(𝑖 + 𝜏1 )𝑥(𝑖 + 𝜏2 ). . .𝑥(𝑖 + 𝜏𝑟−1 )⟩𝑥 𝜕𝑟 𝛩(𝜔1 , 𝜔2 , . . ., 𝜔𝑟 ) = −𝑗𝑘 [ ] . 𝜕𝜔1 𝜕𝜔2 . . .𝜕𝜔𝑟 𝜔1 =𝜔2 =...=𝜔𝑟 =0

(1.1.3)

The joint moments (1.1.3) can be defined by the expansion coefficients of the characteristic function 𝛩(𝜔) in Taylor series in the neighborhood of the point of origin as

𝑚(𝑟) 𝑥 (𝑗𝜔)𝑟 . 𝑟! 𝑟=1 ∞

𝛩(𝜔) = 1 + ∑

(1.1.4)

The relationships between the joint cumulants (1.1.1) and the joint moments (1.1.3) in the origin under assumption that 𝜏1 = 𝜏2 = . . .𝜏𝑟−1 = 0 can be written by the following formulas

𝑐𝑥(1) = 𝑚(1) 𝑥 = ⟨𝑥(𝑖)⟩ , 2

(1) 𝑐𝑥(2) = 𝑚(2) 𝑥 − (𝑚𝑥 ) , 3

(1) (2) (1) 𝑐𝑥(3) = 𝑚(3) 𝑥 − 3𝑚𝑥 𝑚𝑥 + 2 (𝑚𝑥 ) , 2

2

4

(2) (1) (3) (1) (2) (1) 𝑐𝑥(4) = 𝑚(4) 𝑥 − 3 (𝑚𝑥 ) − 4𝑚𝑥 𝑚𝑥 + 12 (𝑚𝑥 ) 𝑚𝑥 − 6 (𝑚𝑥 ) .

(1.1.5)

For the case of a zero-mean process, that is, for 𝑚(1) 𝑥 = ⟨𝑥(𝑖)⟩ = 0, the formulas (1.1.5) transform to the following structure

𝑐𝑥(1) = 𝑚(1) 𝑥 = 0, 2

2 𝑐𝑥(2) = 𝑚(2) 𝑥 = ⟨𝑥(𝑖)⟩ = 𝜎 , 3 𝑐𝑥(3) = 𝑚(3) 𝑥 = ⟨𝑥 (𝑖)⟩ , 2

2

(2) 4 2 𝑐𝑥(4) = 𝑚(4) 𝑥 − 3(𝑚𝑥 ) = ⟨𝑥 (𝑖)⟩ − 3(𝜎 ) ,

(1.1.6)

where 𝜎2 is the variance of a process under consideration. Let us consider a real-valued discrete and zero-mean process {𝑥(𝑖), 𝑖 = 0, 1, 2, . . ., 𝐼 − 1}, ⟨𝑥(𝑖)⟩ = 0. The relationships between the moment and cumulant functions for this zero-mean process can be described by the following formulas (2) (2) ⟨𝑥(𝑖)𝑥(𝑖 + 𝑘)⟩ = 𝑚𝑥 (𝑘) = 𝑐𝑥 (𝑘) ,

⟨𝑥(𝑖)𝑥(𝑖 + 𝑘)𝑥(𝑖 + 𝑙)⟩ = ⟨𝑥(𝑖)𝑥(𝑖 + 𝑘)𝑥(𝑖 + 𝑙)𝑥(𝑖 + 𝑚)⟩ =

(3) 𝑚(3) 𝑥 (𝑘, 𝑙) = 𝑐𝑥 (𝑘, 𝑙) , (4) (2) (2) 𝑚(4) 𝑥 (𝑘, 𝑙, 𝑚) = 𝑐𝑥 (𝑘, 𝑙, 𝑚) + 𝑐𝑥 (𝑘)𝑐𝑥 (𝑚 + 𝑐𝑥(2) (𝑘)𝑐𝑥(2) (𝑚 − 𝑘) + 𝑐𝑥(2) (𝑚)𝑐𝑥(2) (𝑙 − 𝑘) ,

where 𝑘, 𝑙 and 𝑚 are the shift indices.

(1.1.7a) (1.1.7b)

− 𝑙) (1.1.7c)

1.1 General properties of cumulant and moment functions |

3

The formula (1.1.7a) describes the relationship between the second-order statistics and it defines conventional autocorrelation function. Note that second-order moment and cumulant functions are equal to each other in this case. The formula (1.1.7b) describes the relationship between the third-order statistics and it defines triple or third-order autocorrelation function. It should be noted that the third-order moments and cumulants are equal to each other in this case. According to the formula (1.1.7c) defining the relationship existing between the fourth-order statistics, the fourth-order moment function is not equal to the fourthorder cumulant function. Spectral density of the 𝑟th order called as polispectrum or cumulant spectrum 𝐶𝑥 (𝜔1 , 𝜔2 ,. . . , 𝜔𝑟−1 ) of a process {𝑥(𝑖), 𝑖 = 0, 1, 2, . . ., 𝐼 − 1} can be defined by the following multidimensional Fourier transform of the 𝑟th order cumulant 𝑐𝑥(𝑟) (𝜏1 , 𝜏2 , . . ., 𝜏𝑟−1 ) as

𝐶(𝑟) 𝑥 (𝜔1 , 𝜔2 , . . ., 𝜔𝑟−1 , ) +∞

+∞

𝜏1 =−∞

𝜏𝑟−1 =−∞

= ∑ . . . ∑ 𝑐𝑥(𝑟) (𝜏1 , 𝜏2 , . . ., 𝜏𝑟−1 ) exp[−𝑗(𝜔1 𝜏1 + 𝜔2 𝜏2 + . . . + 𝜔𝑟−1 𝜏𝑟−1 )]. (1.1.8) The generalized formula (1.1.8) allows to define the energy spectrum 𝑃𝑥 (𝜔) (for 𝑟 = 2), the bispectrum 𝐵𝑥 (𝜔1 , 𝜔2 ) (for 𝑟 = 3) and the trispectrum 𝑇𝑥 (𝜔1 , 𝜔2 , 𝜔3 ) (for 𝑟 = 4), respectively, in the forms: +∞

𝑃𝑥(𝜔) = ∑ 𝑐𝑥(2) (𝑙) exp[−𝑗(𝜔𝑙)] , 𝑙=−∞ +∞

𝐵𝑥 (𝜔1 , 𝜔2 ) = ∑

(1.1.9a)

+∞

∑ 𝑐𝑥(3) (𝑙1 , 𝑙2 ) exp[−𝑗(𝜔1 𝑙1 + 𝜔2 𝑙2 )] ,

(1.1.9b)

𝑙1 =−∞ 𝑙2 =−∞ +∞

+∞

𝑇𝑥 (𝜔1 , 𝜔2 , 𝜔3 ) = ∑

∑

+∞

∑ 𝑐𝑥(4) (𝑙1 , 𝑙2 , 𝑙3 ) exp[−𝑗(𝜔1 𝑙1 + 𝜔2 𝑙2 + 𝜔3 𝑙3 )]. (1.1.9c)

𝑙1 =−∞ 𝑙2 =−∞ 𝑙3 =−∞

The expressions (1.1.9a–b) contain the cumulant functions whose properties are interesting and worth considering in detail. (1) If 𝑥𝑖 , 𝑖 = 1, 2, . . ., 𝐾 is a sequence of random variables and 𝛼𝑖 = 1, 2, . . ., 𝐾 are some constant values, then 𝐾

𝑐(𝛼1 𝑥1 , 𝛼2 𝑥2 , . . ., 𝛼𝐾 𝑥𝐾 ) = (∏ 𝛼𝑖 ) 𝑐(𝑥1 , 𝑥2 , . . ., 𝑥𝐾 ) .

(1.1.10)

𝑖=1

(2) Permutation property for random variables

𝑐(𝑥1 , 𝑥2 , . . .𝑥𝐾 ) = 𝑐(𝑥𝑖1 , 𝑥𝑖2 , . . ., 𝑥𝑖𝐾 ) , where (𝑖1 , 𝑖2 , . . ., 𝑖𝐾 ) is the permutation index (1, 2, . . ., 𝐾).

(1.1.11)

4 | 1 General properties of bispectrum-based digital signal processing (3) Additivity property of the cumulants for their arguments

𝑐(𝑥 + 𝑦, 𝑧1 , 𝑧2 , . . ., 𝑧𝐾 ) = 𝑐(𝑥, 𝑧1 , 𝑧2 , . . ., 𝑧𝐾 ) + 𝑐(𝑦, 𝑧1 , 𝑧2 , . . ., 𝑧𝐾 ) ,

(1.1.12)

which signifies that the cumulant of the sum of arguments is equal to the sum of the cumulants of the separate arguments. (4) If 𝛼 is a constant value then

𝑐(𝛼 + 𝑥1 , 𝑥2 , . . ., 𝑥𝐾 ) = 𝑐(𝑥1 , 𝑥2 , . . ., 𝑥𝐾 ) .

(1.1.13)

(5) In the case when the random variables 𝑥𝑖 , 𝑖 = 1, 2, . . ., 𝐾 and 𝑦𝑖 , 𝑖 = 1, 2, . . ., 𝐾 are independent, we have

𝑐(𝑥1 + 𝑦1 , 𝑥2 + 𝑦2 , . . ., 𝑥𝐾 + 𝑦𝐾 ) = 𝑐(𝑥1 , 𝑥2 , . . ., 𝑥𝐾 ) + 𝑐(𝑦1 , 𝑦2 , . . ., 𝑦𝐾 ) . (1.1.14) Assume that an observed process is 𝑧(𝑖) = 𝑥(𝑖) + 𝑛(𝑖), 𝑖 = 1, 2, . . ., 𝐾, and 𝑥(𝑖) and 𝑛(𝑖) are the independent processes. According to the property (1.1.14), one can obtain

𝑐𝑧(𝐾) (𝑙1 , 𝑙2 , . . ., 𝑙𝐾−1 ) = 𝑐𝑥(𝐾) (𝑙1 , 𝑙2 , . . ., 𝑙𝐾−1 ) + 𝑐𝑛(𝐾) (𝑙1 , 𝑙2 , . . ., 𝑙𝐾−1 ) .

(1.1.15)

If one of the process, for example, 𝑛(𝑖) is Gaussian, then under condition of 𝐾 ≥ 3,

𝑐𝑛(𝐾) (𝑙1 , 𝑙2 , . . ., 𝑙𝐾−1 ) = 0 we obtain

𝑐𝑧(𝐾) (𝑙1 , 𝑙2 , . . ., 𝑙𝐾−1 ) = 𝑐𝑥(𝐾) (𝑙1 , 𝑙2 , . . ., 𝑙𝐾−1 ) .

(1.1.16)

The latter expression (1.1.16) demonstrates important insensitivity property to the additive Gaussian noise valid for the cumulants the order of which is equal or more than three. From the practical point of view of signal processing in additive Gaussian noise environment, the cumulant estimates permit to separate non-Gaussian signal from additive Gaussian noise and, hence, to increase signal-to-noise (SNR) ratio. Below, we will pay attention to the third-order statistics from the point of view of their application in digital signal and image processing. For this purpose, first, we will consider the general properties of triple correlation and bispectrum and the techniques used for their estimation.

1.2 Triple correlation function and bispectrum One of the main motivations referred to using bispectrum-based signal processing is the following. Bispectrum density estimate or third-order cumulant spectrum estimate, opposite to the energy spectrum estimate, not only allows to describe the statistical properties of an observed process more correctly and completely, but also to extract novel information features such as spectral component correlation relationships. Moreover, bispectrum estimate allows extracting the phase relationships existing between the spectral components contained in the process under study. Therefore, the

1.2 Triple correlation function and bispectrum |

5

main difference of bispectrum from energy spectrum is in preservation of phase information contained in a process and the possibility to recover this important information. Already, this promising peculiarity of bispectrum has contributed to wide usage of the bispectrum analysis and bispectral estimation techniques for digital signal processing. Permanent growth in the interest in bispectrum analysis is accompanied by appearance of a great number of publications. Consider the benefits of bispectrum analysis more in detail. One of the most promising bispectrum property usually used for recovering a signal embedded in Gaussian noise, is the tendency to zero the bispectrum of an interference having a symmetrical probability density function (PDF) . This property provides robustness of the bispectrum-based filtering techniques regarding additive Gaussian interference in radar [5–8], astronomy [9–11], underwater acoustics [12–14], and biomedical [15, 16] signal processing systems, as well as in digital image processing systems [17–20]. Bispectral analysis can serve as quite a sensitive and precise tool permitting to define and measure the deviation of the observed process from Gaussian distribution, that is, to estimate non-Gaussianity. This property seems to be very useful in noisylike processes in machine diagnostics systems [21], underwater acoustic systems [12], nondestructive monitoring [22], and biomedical diagnostics [16]. Let us consider the properties of bispectrum for a real-valued stationary discrete process {𝑥(𝑚) (𝑖)} given by finite number of samples 𝑖 = 0, 1, 2, . . ., 𝐼 − 1 and observed with a finite sequence of 𝑚 = 1, 2, . . ., 𝑀 realizations. Common autocorrelation discrete function 𝑅𝑥 (𝑘) belonging to the class of secondorder statistics can be written as a function of a single variable 𝐼−1

𝑅𝑥 (𝑘) = ⟨ ∑ [𝑥(𝑚) (𝑖) − 𝐸] [𝑥(𝑚) (𝑖 + 𝑘) − 𝐸]⟩ ,

(1.2.1)

∞

𝑖=0

where 𝑘 = −𝐼 + 1, . . ., 𝐼 − 1 is the temporal or spatial shift index; ⟨. . . ⟩∞ denotes ensemble averaging assuming that number of accumulated realizations tends to in𝐼−1 finity, that is, 𝑀 → ∞; 𝐸 = ⟨(1/𝐼) ∑𝑖=0 𝑥(𝑚) (𝑖)⟩∞ is the mean value; 𝑅𝑥 (0) = 𝜎𝑥2 = 𝐼−1 (𝑚) 2 ⟨∑𝑖=0 [𝑥 (𝑖) − 𝐸] ⟩∞ is the variance. Autocorrelation function 𝑅𝑥 (𝑘) (1.2.1) has the following symmetry property

𝑅𝑥(𝑘) = 𝑅𝑥 (−𝑘) .

(1.2.2)

According to the Wiener–Khinchin theorem, the spectral density 𝑃𝑥 (𝑝) can be defined in the form of the following discrete direct Fourier transform (DFT) 𝑘=+∞

𝑃𝑥 (𝑝) = ∑ 𝑅𝑥 (𝑘) exp(−𝑗2𝜋𝑘𝑝) , 𝑘=−∞

(1.2.3)

6 | 1 General properties of bispectrum-based digital signal processing or by

𝑃𝑥 (𝑝) = ⟨𝑋̇ (𝑚) (𝑝)𝑋̇ ∗(𝑚) (𝑝)⟩∞ ,

(1.2.4) 𝐼−1

where 𝑝 = −𝐼 + 1, . . ., 𝐼 + 1 is the frequency sample index; 𝑋̇ (𝑚) (𝑝) = ∑𝑖=0 𝑥(𝑚) (𝑖) exp(−𝑗2𝜋𝑖𝑝) is the complex-valued DFT computed for 𝑚th arbitrary realization; ∗ denotes complex conjugation. It should be taken into account one more time that Fourier phase spectrum information is irretrievably lost in the spectral density (1.2.4). Own autocorrelation function corresponds to each concrete signal, but not inversely. It is impossible to restore signal shape by autocorrelation function as it is impossible to restore the shape of some plane figure by its known square. Triple autocorrelation function (TAF) 𝑅𝑥 (𝑘, 𝑙) represents the third-order statistic. TAF is a function of two variables and it can be represented in the discrete form as 𝐼−1

𝑅𝑥 (𝑘, 𝑙) = ⟨ ∑ [𝑥(𝑚) (𝑖) − 𝐸] [𝑥(𝑚) (𝑖 + 𝑘) − 𝐸] [𝑥(𝑚) (𝑖 + 𝑙) − 𝐸]⟩ ,

(1.2.5)

∞

𝑖=0

where 𝑘 = −𝐼 + 1, . . ., 𝐼 − 1 and 𝑙 = −𝐼 + 1, . . ., 𝐼 − 1 are the independent shift indices. Note that the TAF (1.2.5) possesses the following symmetry property [2]

𝑅𝑥 (𝑘, 𝑙) = 𝑅𝑥 (𝑙, 𝑘) = 𝑅𝑥(𝑙 − 𝑘, −𝑘) = 𝑅𝑥 (𝑘 − 𝑙, −𝑙) = 𝑅𝑥 (−𝑘, 𝑙 − 𝑘) .

(1.2.6)

According to the definitions given in [1, 2], bispectrum is the 2-D DFT of TAF. Unlike the real-valued spectral density (1.2.3) and (1.2.4), bispectrum (or bispectral density) is the complex-valued function 𝐵̇ 𝑥 (𝑝, 𝑞) of two independent frequencies 𝑝 and 𝑞 that can be written as the following 2-D discrete DFT of TAF (1.2.5) 𝐼−1

𝐵̇ 𝑥 (𝑝, 𝑞) = ∑

𝐼−1

∑ 𝑅𝑥(𝑘, 𝑙) exp[−𝑗2𝜋(𝑘𝑝 + 𝑙𝑞)] ,

(1.2.7a)

𝑘=−𝐼+1 𝑙=−𝐼+1

or as

𝐵̇ 𝑥 (𝑝, 𝑞) = ⟨𝑋̇ (𝑚) (𝑝)𝑋̇ (𝑚) (𝑞)𝑋̇ ∗(𝑚) (𝑝 + 𝑞)⟩∞ = ⟨𝑋̇ (𝑚) (𝑝)𝑋̇ (𝑚) (𝑞)𝑋̇ (𝑚) (−𝑝 − 𝑞)⟩∞ ,

(1.2.7b) where 𝐵̇ 𝑥 (𝑝, 𝑞) = |𝐵̇ 𝑥 (𝑝, 𝑞)| exp[𝑗𝛾𝑥 (𝑝, 𝑞)]; |𝐵̇ 𝑥 (𝑝, 𝑞)| and 𝛾𝑥 (𝑝, 𝑞) are the magnitude bispectrum (bimagnitude) and phase bispectrum (biphase), respectively; 𝑝 = −𝐼 + 1, . . ., 𝐼 − 1 and 𝑞 = −𝐼 + 1, . . ., 𝐼 − 1 are the frequency indices. Comparing the spectral (1.2.4) and bispectral (1.2.7b) densities allows noting that spectral density 𝑃𝑥 (𝑝) is the ensemble averaging performed for the multiplication of two complex conjugated functions corresponding to the same frequency 𝑝 and bispectral density 𝐵̇ 𝑥 (𝑝, 𝑞) is the ensemble averaging of triple product of three complexvalued functions related to three different frequencies: 𝑝, 𝑞 and 𝑝 + 𝑞.

7

1.2 Triple correlation function and bispectrum |

Abs sig1 (Gauss)

Abs sig4 (exp) 101

Px(f)

Px(f)

101

100

10–1

0

0.2

0.4 0.6 Frequency

0.8

10–1

1

(a)

100

0

0.2

0.4 0.6 Frequency

0.8

1

(b)

Fig. 1.2.1. Spectral density as a function of normalized frequency: (a) Gaussian process and (b) exponentially distributed process.

Illustrative examples and comments referred to the main properties of bispectrum are listed below. (1) TAF and bispectrum tend to zero for stationary zero-mean Gaussian process [2]

𝑅𝑥 (𝑚, 𝑛) = 0,

𝐵̇ 𝑥 (𝑝, 𝑞) = 0 .

(1.2.8)

However, for a process having nonsymmetrical PDF its bispectrum differs from zero. This property permits to exploit bispectral analysis for detecting of nonGaussianity. It can be demonstrated by using the following comparative example. The results of computations of spectral and bispectral densities of a Gaussian and exponentially distributed processes are shown in Figures 1.2.1 and 1.2.2. It is clearly seen from Figure 1.2.1 that the spectral densities are practically of the same shape. At the same time, bimagnitude function computed for exponentially distributed process contains pronounced parts where |𝐵̇ 𝑥 (𝑓1 , 𝑓2 )| ≠ 0 (see Figure 1.2.2 (b)) as opposed to Gaussian process when |𝐵̇ 𝑥 (𝑓1 , 𝑓2 )| ≈ 0 (see Figure 1.2.2 (a)). (2) TAF and bispectrum are equal to zero for deterministic signals having zero asymmetry. For example, TAF and bispectrum are of zero values for a simple harmonic oscillation 𝑥(𝑖) = 𝐴 0 cos(2𝜋𝑓𝑖) (here, symbol 𝑓 denotes the frequency). However, when little signal nonlinear distortions appear or when a constant component is contained in harmonic signal, TAF and bispectrum become of nonzero-valued functions. The latter property can serve as a very sensitive tool for detection and assessment of nonlinear distortions contained in a signal under study. (3) Bispectrum is the periodical function [2] with period equal to 2𝜋

̇ + 2𝜋, 𝑞 + 2𝜋) . 𝐵̇ 𝑥 (𝑝, 𝑞) = 𝐵(𝑝

(1.2.9)

8 | 1 General properties of bispectrum-based digital signal processing Abs sig4 (exp) 10

3

10

2

0

2

0

1

–10

1

–10

0

–20

0

–20

–1

–30

–1

–30

–2

–40

–2

–40

–50

–3

–3 –3

–2

–1 0 1 2 Frequency [rad’s]

3

(a)

Frequency [rad’s]

Frequency [rad’s]

Abs sig1 (Gauss) 3

–3

–2

–1 0 1 2 Frequency [rad’s]

–50

3

(b)

Fig. 1.2.2. Bimagnitude as a function of two normalized frequencies: (a) Gaussian process and (b) exponentially distributed process.

(4) Bispectral density has the following symmetry property [2]

̇ 𝑞) = 𝐵(𝑞, ̇ 𝑝) = 𝐵̇ ∗ (−𝑝, −𝑞) = 𝐵̇ ∗ (−𝑞, −𝑝) 𝐵(𝑝, ̇ ̇ −𝑝 − 𝑞) = 𝐵(−𝑝 ̇ ̇ −𝑝 − 𝑞) . (1.2.10) = 𝐵(−𝑝 − 𝑞, 𝑞) = 𝐵(𝑝, − 𝑞, 𝑝) = 𝐵(𝑞, Symmetry relationships (1.2.10) define bispectrum within the limits given by a hexagon in the bifrequency domain. Analysis of the expressions (1.2.10) shows that bispectrum of a real-valued process can be defined completely just only within the area of the main triangular domain limited by the inequalities

𝑞 ≥ 0,

𝑝 ≥ 𝑞,

𝑝+𝑞≤𝐼−1.

(1.2.11)

It is sufficient to use the symmetry relationships (1.2.10) taking into account (1.2.11) for computation of bispectrum function within all other parts of hexagonal domain. Note that the conditions (1.2.11) limiting the total number of bispectrum samples allow essential decreasing of the requirements to the PC memory and reducing the data processing time in real-life computations. (5) Unlike power spectral density containing only information about signal magnitude Fourier spectrum, the bispectral density permits preserving and recovering the data related to both magnitude and phase signal Fourier spectra. This property follows from the equations (1.2.7a) and (1.2.7b). (6) Important invariance property of bispectrum to a signal time delay or signal spatial shift follows from (1.2.7b). This property can be demonstrated by the following simple formula

𝐵̇ 𝑥𝜏 (𝑝, 𝑞) = 𝑋̇ 𝜏 (𝑝)𝑋̇ 𝜏 (𝑞)𝑋̇ 𝜏 (−𝑝 − 𝑞) ̇ 𝑋(𝑞) ̇ 𝑋(−𝑝 ̇ = 𝑋(𝑝) − 𝑞) exp(−𝑗2𝜋𝜏𝑝) exp(−𝑗2𝜋𝜏𝑞) exp[−𝑗2𝜋𝜏(−𝑝 − 𝑞)] = 𝐵̇ 𝑥 (𝑝, 𝑞) ,

(1.2.12)

1.3 Bispectral density estimation techniques | 9

̇ exp(−𝑗2𝜋𝜏) is the Fourier transform of a process shifted by where 𝑋̇ 𝜏 (𝑝) = 𝑋(𝑝) a 𝜏 value in temporal domain. It follows from the expression (1.2.12) that both for the original process 𝑥(𝑚) (𝑖) and for its replica 𝑥(𝑚) (𝑖−𝜏) shifted by 𝜏, the bispectra are congruent. This important property seems to be useful for solving the jittered image reconstruction problems. However, at the same time, this bispectrum invariance property (1.2.12) can cause the problems in some bispectrum-based signal processing algorithms. Particularly, undesirable image line shifts can arise with using bispectrum-based image reconstruction techniques. These techniques will be considered below in detail. (7) Very often in practice, one can study behavior of a wideband process passing through a device having square-law characteristic. In this case, sum or difference frequency components can appear at the nonlinear device output. Quadratic phase coupling phenomenon can be observed at the nonlinear device output. In many practically-important cases, it is necessary to detect and estimate just those spectral components that contain quadratic coupling. Since the phase relationships are lost in power spectral density, it is impossible to extract information about phase coupling from power spectral density. Meanwhile, it is possible to do that by using bispectrum. This property can be explained by using the following demonstrative example. Let us consider the following two processes 𝑥1 (𝑖) = cos(2𝜋𝑓1 𝑖 + 𝜑1 ) + cos(2𝜋𝑓2 𝑖 + 𝜑2 ) + cos(2𝜋𝑓3 𝑖 + 𝜑3 ) ,

(1.2.13a)

𝑥2 (𝑖) = cos(2𝜋𝑓1 𝑖 + 𝜑1 ) + cos(2𝜋𝑓2 𝑖 + 𝜑2 ) + cos[2𝜋𝑓3 𝑖 + (𝜑1 + 𝜑2 )] (1.2.13b) where 𝑓3 = 𝑓1 + 𝑓2 are the frequencies and 𝜙1 , 𝜙2 , 𝜙3 are the initial phases that are supposed to be independent random values having uniform distribution law and they vary within the limits of [0, 2𝜋]. Assume that the frequency 𝑓3 in (1.2.13a) is an independent component and an initial phase 𝜙3 corresponding to this frequency is an independent random value. Assume also that the frequency 𝑓3 in (1.2.13b) is the result of quadratic phase coupling. It is evident that the power spectrum densities for these two considered processes are of the same shape, that is, 𝑃𝑥1 (𝑝) = 𝑃𝑥2 (𝑝). At the same time, magnitude bispectrum of the process (1.2.13a) tends to be zero, that is, |𝐵̇ 𝑥1 (𝑝, 𝑞)| = 0, and the magnitude bispectrum of the process (1.2.13b) is of nonzero value, that is, |𝐵̇ 𝑥2 (𝑝, 𝑞)| ≠ 0. Therefore, bispectrum can serve as a very sensitive indicator for detection of phase coupling.

1.3 Bispectral density estimation techniques Note that the existing bispectrum-based approaches used in digital signal processing can be divided into nonparametric and parametric, as well as into direct and indirect

10 | 1 General properties of bispectrum-based digital signal processing techniques [2, 23]. The main attention in this book will be paid to the nonparametric bispectrum estimation. First, we start with consideration of nonparametric indirect bispectral density estimation technique [2] containing the following sequence of data processing procedures. (1) Computation of a number of 𝑀 TAF estimates 𝑅̂ (𝑚) (𝑘, 𝑙) for each 𝑚th realization given by a sequence of real-valued samples {𝑥(𝑚) (0), 𝑥(𝑚) (1), 𝑥(𝑚) (2), . . ., 𝑥(𝑚) (𝐼 − 1)}, 𝑚 = 1, 2, . . ., 𝑀 observed at the input of signal processing system as 𝐼−1

(𝑚) (𝑖)𝑥(𝑚) (𝑖 + 𝑘)𝑥(𝑚) (𝑖 + 𝑙) . 𝑅̂ (𝑚) 𝑥 (𝑘, 𝑙) = ∑ 𝑥

(1.3.1)

𝑖=0

Assume that a considered process in (1.3.1) {𝑥(𝑚) (𝑖)} is of zero-mean-valued, that is, 𝐸𝑥 = 0. (2) Averaging the separate estimates (1.3.1) over the ensemble of 𝑀 realizations to obtain the smoothed TAF estimate 𝑅̂ 𝑀 (𝑘, 𝑙) in the form of

1 𝑀 ̂ (𝑚) ∑ 𝑅 (𝑘, 𝑙) . 𝑅̂ 𝑀 (𝑘, 𝑙) = 𝑀 𝑚=1 𝑥

(1.3.2)

Note that for an ergodic process the ensemble averaging (1.3.2) can be replaced by the averaging procedure performed in temporal domain, that is, by dividing the observed sequence {𝑥(𝑚) (𝑖)} into 𝑀 separate segments. (3) Computation of bispectral density in the form of a complex-valued function ̂ (𝑝,𝑞) by using direct Fourier transform of 𝑅̂ 𝑀 (𝑘, 𝑙) 𝐵̂̇ 𝑥ind (𝑝, 𝑞) = |𝐵̂̇ 𝑥ind (𝑝, 𝑞)|𝑒𝑗𝛾𝑥ind

(1.3.2) as 𝐼−1 𝐼−1

𝐵̂̇ 𝑥ind (𝑝, 𝑞) = ∑ ∑ 𝑅̂ 𝑀 (𝑘, 𝑙)𝑊(𝑘, 𝑙) exp[−𝑗2𝜋(𝑘𝑝 + 𝑙𝑞)] ,

(1.3.3)

𝑘=0 𝑙=0

̂ (𝑝, 𝑞) are the magnitude and phase bispectral estiwhere |𝐵̂̇ 𝑥ind (𝑝, 𝑞)| and 𝛾𝑥ind mates, respectively; 𝑊(𝑘, 𝑙) is the weighting window function that is desirable for improving the accuracy in bispectral estimate by decreasing spectral leakage. A weighting window function optimization is the separate problem that is considered, for example, in [24, 25] and the readers interested in more detailed information can be directed to these papers. Direct bispectral density estimation technique comparing to the indirect technique has better processing speed, first, due to using FFT algorithm and, second, due to excluding the computationally-consuming procedure necessary for evaluation of the TAF (1.3.1). The direct technique contains the following sequence of signal processing procedures.

1.3 Bispectral density estimation techniques |

11

(1) Direct Fourier transform performed for each realization {𝑥(𝑚) (𝑖)} as 𝐼−1

𝑋̇ (𝑚) (𝑝) = ∑ 𝑥(𝑚) (𝑖) exp(−𝑗2𝜋𝑖𝑝) ,

𝑝 = 0, 1, 2, . . ., 𝐼 − 1 .

(1.3.4)

𝑖=0

(2) Computation of 𝑚th bispectrum estimate by using the following triple product of complex-valued functions (1.3.4) as

̇ (𝑚) (𝑝)𝑋̇ (𝑚) (𝑞)𝑋̇ ∗(𝑚) (𝑝 + 𝑞) , 𝐵̂̇ (𝑚) 𝑥dir (𝑝, 𝑞) = 𝑋

𝑚 = 1, 2, . . ., 𝑀 .

(1.3.5)

(3) Averaging the sample estimates (1.3.5) performed over the ensemble of 𝑀 realizations to obtain smoothed estimate 𝐵̂̇ 𝑥dir (𝑝, 𝑞) in the form of

1 𝑀 ̂̇ (𝑚) ∑ 𝐵 (𝑝, 𝑞) . 𝐵̂̇ 𝑥dir (𝑝, 𝑞) = 𝑀 𝑚=1 𝑥dir

(1.3.6)

It should be stressed that bispectral density estimates formed by indirect and direct techniques differ from each other. These estimates are congruent in the case of 𝑊(𝑘, 𝑙) = 1 in (1.3.3). It was shown by Brillinger in [26] that for unlimited sample number 𝐼 of an input sequence or for an unlimited realization number 𝑀, bispectrum estimates obtained by indirect and direct techniques converge in average to the true bispectral density, that is, these estimates are asymptotically unbiased and consistent

⟨𝐵̂̇ 𝑥dir (𝑝, 𝑞)⟩ = ⟨𝐵̂̇ 𝑥ind (𝑝, 𝑞)⟩ ≅ 𝐵̇ 𝑥 (𝑝, 𝑞) ,

𝐼, 𝑀 → ∞ .

(1.3.7)

The following asymptotical expressions are represented in [2] for bispectrum density estimate variances for indirect and direct techniques, respectively

var{Re𝐵̂̇ 𝑥ind (𝑝, 𝑞)} = var{Im𝐵̂̇ 𝑥ind (𝑝, 𝑞)} ≅

𝑉 𝑃(𝑝)𝑃(𝑞)𝑃(𝑝 + 𝑞) (1.3.8a) (2𝐿 + 1)2 𝑀

and

1 𝑃(𝑝)𝑃(𝑞)𝑃(𝑝 + 𝑞) , var{Re𝐵̂̇ 𝑥dir (𝑝, 𝑞)} = var{Im𝐵̂̇ 𝑥dir (𝑝, 𝑞)} ≅ 𝑀

(1.3.8b)

where real and imaginary parts of bispectrum density estimate are denoted by Re and Im, respectively; 𝑉 = ∑𝐿𝑘=−𝐿 ∑𝐿𝑙=−𝐿 |𝑊(𝑘, 𝑙)|2 is the energy parameter corresponding to the weighting window; 𝐿 = 𝐼−1 is the weighting window length; 𝑃(. . . ) is the power density; var{. . . } is the variance. Note that for window of uniform shape 𝑊(𝑘, 𝑙) = 1, and 𝑉/(2𝐿 + 1)2 = 1 in (1.3.8a). Because of this, the variances (1.3.8a) and (1.3.8b) become the same. It should especially be noted that according to the asymptotic expressions (1.3.8a) and (1.3.8b) it is necessary to observe quite large numbers of realizations 𝑀 for obtaining an unbiased bispectrum density estimate with small variance. However, in signal processing practice, a number 𝑀 is not too large. Therefore, one of the most important problems is improving the bispectrum density estimates obtained for a limited realization number. Some new approaches to solving this problem will be considered and the obtained results will be discussed below.

12 | 1 General properties of bispectrum-based digital signal processing

1.4 Bispectrum-based algorithms in application for filtering and signal shape reconstruction A typical problem arising in aforementioned signal processing systems is the process estimation problem or filtering problem that comes to the reconstruction of unknown signal shape (waveform) at the filter output in interference environment with high accuracy in statistical sense. Attraction of the bispectrum-based techniques in application to filtering and signal reconstruction problems is, first of all, in the high accuracy of non-Gaussian signal shape reconstruction in additive Gaussian and, in general, symmetric PDF noise environment for low input SNR. In this Chapter, we pay attention to consideration of different bispectrum-based algorithms used for signal reconstruction from additive mixture of signal and noise. We will study the following typical signal and noise model observed at the digital filtering system input. Let us consider a real-valued deterministic signal 𝑠(𝑖) given by a uniform array of samples, 𝑖 = 0, 1, . . ., 𝐼 − 1. Assume that 𝑀 realizations of the randomly shifted signal are observed. Thus, each 𝑚th (𝑚 = 1, 2, . . ., 𝑀) realization 𝑥(𝑚) (𝑖) could be presented as

𝑥(𝑚) (𝑖) = 𝑠(𝑖 − 𝜏(𝑚) ) + 𝑛(𝑚) (𝑖),

(1.4.1)

where 𝑛(𝑚) (𝑖) is the 𝑚th realization of the stationary additive white Gaussian noise (AWGN) with zero mean and sampling variance 𝜎(𝑚)2 ; 𝜏(𝑚) is a random integer shift of the deterministic signal 𝑠(𝑖). AWGN is assumed to be uncorrelated with the original signal 𝑠(𝑖). The signal waveform is supposed to be invariable for all realizations. The signal’s TAF is assumed to be a priori nonzero. The observation model (1.4.1) is rather typical for many applications [5, 14, 27–29]. In particular, 𝑥(𝑚) (𝑖) can describe the noisy output of a high resolution radar system that forms a target range profile 𝑠(𝑖). The temporal shifts 𝜏(𝑚) can arise in practice due to the target’s random motion from one scan to another or random vibration of radar platform. Statistical characteristics of 𝜏(𝑚) (PDF, variance, etc.) depend upon many physical phenomena. They are a priori unknown and different for various practical applications. However, it is realistic to observe deviations of 𝜏(𝑚) comparable to the total available signal length. Similar properties of signal and noise can be observed for an active sonar operating in the pulse mode. They can also relate to a set of preliminary outputs of a passive sonar system. According to the observation equation (1.4.1), filtering problem is formulated as follows: to estimate a priori unknown signal shape 𝑠(𝑖) in AWGN environment. The conventional bispectrum-based approach to signal shape reconstruction from noisy mutually shifted realizations [9, 10] includes the following three main stages: (1) obtaining an estimate of signal bispectrum; (2) recovery of the signal Fourier spectrum (magnitude and phase spectra) from the signal bispectrum estimate;

1.4 Bispectrum-based filtering and signal shape reconstruction

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13

(3) signal shape estimate reconstruction from the obtained estimate of the signal Fourier spectrum by inverse Fourier transform. Existing approaches [9, 10] are based on the following pair of equations that connect ̇ the signal bispectrum 𝐵̇ 𝑠 (𝑝, 𝑞) and the signal Fourier spectrum 𝑆(𝑝)

󵄨󵄨 󵄨 󵄨 ̇ 󵄨󵄨󵄨 󵄨󵄨󵄨 ̇ 󵄨󵄨󵄨 󵄨󵄨󵄨 ̇∗ 󵄨󵄨 ̇ 󵄨󵄨𝐵𝑠 (𝑝, 𝑞)󵄨󵄨󵄨 = 󵄨󵄨󵄨󵄨𝑆(𝑝) 󵄨󵄨 󵄨󵄨𝑆(𝑞)󵄨󵄨 󵄨󵄨𝑆 (𝑝 + 𝑞)󵄨󵄨󵄨 , 𝛾𝑠 (𝑝, 𝑞) = 𝜑(𝑝) + 𝜑(𝑞) − 𝜑(𝑝 + 𝑞) , [−𝜋, 𝜋]

(1.4.2) (1.4.3)

where 𝜑(. . . ) denotes signal phase Fourier spectrum. Note that equations (1.4.2) and (1.4.3) are valid for Hermitian conjugated real̇ valued signal Fourier spectrum, that is, for 𝑆(𝑝) = 𝑆̇∗ (−𝑝) and 𝜑(𝑝) = −𝜑(−𝑝). A recursive algorithm for signal Fourier spectrum reconstruction by using bispectral density estimation [10] is based on the equations (1.4.2) and (1.4.3) and it can be represented in the form of

̂ + 𝑞) = 𝜑(𝑝) ̂ ̂ − 𝛾𝑠̂ (𝑝, 𝑞) , 𝜑(𝑝 + 𝜑(𝑞)

𝑝 = 0, . . ., 𝐼 − 1 ; 0 ≤ 𝑞 ≤ 𝑝 ; 𝑝 + 𝑞 ≤ 𝐼 − 1 , (1.4.4)

󵄨󵄨 ̂ 󵄨 󵄨󵄨𝐵̇ 𝑠 (𝑝, 𝑞)󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 ̂ 󵄨 󵄨󵄨 󵄨 󵄨󵄨𝑆(𝑝 ̇ + 𝑞)󵄨󵄨 = 󵄨󵄨 󵄨󵄨 ̂ 󵄨󵄨 󵄨󵄨 ̂ 󵄨󵄨 , 𝑝 = 0, . . ., 𝐼 − 1 ; 0 ≤ 𝑞 ≤ 𝑝 ; 𝑝 + 𝑞 ≤ 𝐼 − 1 , (1.4.5) 󵄨󵄨 󵄨󵄨𝑆(𝑝) ̇ 󵄨󵄨 ̇ 󵄨󵄨 󵄨󵄨𝑆(𝑞) 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 where |𝐵̂̇ (𝑝, 𝑞)| and 𝛾̂ (𝑝, 𝑞) are the sampling estimates of the magnitude |𝐵̇ (𝑝, 𝑞)| 𝑠

𝑠

𝑠

and phase 𝛾𝑠 (𝑝, 𝑞) bispectra, respectively, computed for a finite number of realiza-

̂̇ . .)| and 𝜑(. ̂ . .) are the signal magnitude and phase Fourier spectrum estions; |𝑆(. timates reconstructed from the magnitude and phase bispectrum estimates, respectively. The following sequence of data processing steps is exploited in [10] for recursive evaluation of signal phase and magnitude Fourier spectrum estimates ̂ 𝜑(0) = 0, ̂ ̂ + 𝜑(0) ̂ − 𝛾𝑠̂ (1, 0) , 𝜑(1) = 𝜑(1) ̂ ̂ − 𝛾𝑠̂ (1, 1) , 𝜑(2) = 2𝜑(1) ̂ ̂ + 𝜑(1) ̂ − 𝛾𝑠̂ (2, 1) , 𝜑(3) = 𝜑(2) ̂ ̂ − 𝛾𝑠̂ (2, 2) , 𝜑(4) = 2𝜑(2) ̂ ̂ + 𝜑(1) ̂ − 𝛾𝑠̂ (3, 1) , 𝜑(4) = 𝜑(3) ̂ ̂ + 𝜑(2) ̂ − 𝛾𝑠̂ (3, 2) , 𝜑(5) = 𝜑(3) ̂ ̂ + 𝜑(3) ̂ − 𝛾𝑠̂ (3, 3) , 𝜑(6) = 𝜑(3) ̂ ̂ + 𝜑(1) ̂ − 𝛾𝑠̂ (4, 1) , 𝜑(5) = 𝜑(4) ..., ̂ − 1) = 𝜑(𝐼 ̂ − 1) + 𝜑(0) ̂ − 𝛾𝑠̂ (𝐼 − 1, 0) , 𝜑(𝐼

(1.4.6)

14 | 1 General properties of bispectrum-based digital signal processing 3 ̂̇ |𝑆(0)| = √|𝐵̂̇ 𝑠 (0, 0)|, ̂̇ ̂̇ ̂̇ |𝑆(1)| = |𝐵̂̇ (1, 0)|/(|𝑆(1)|| 𝑆(0)|),

𝑠

2 ̂̇ ̂̇ |𝑆(2)| = |𝐵̂̇ 𝑠 (1, 1)|/(|𝑆(1)|) ,

̂̇ ̂̇ ̂̇ |𝑆(3)| = |𝐵̂̇ 𝑠 (2, 1)|/(|𝑆(2)|| 𝑆(1)|), 2 ̂̇ ̂̇ |𝑆(4)| = |𝐵̂̇ (2, 2)|/(|𝑆(2)|) , 𝑠

̂̇ ̂̇ ̇̂ |𝑆(3)| = |𝐵̂̇ 𝑠 (3, 0)|/(|𝑆(3)|| 𝑆(0)|), ̂̇ ̂̇ ̇̂ |𝑆(4)| = |𝐵̇̂ (3, 1)|/(|𝑆(3)|| 𝑆(1)|), 𝑠

̂̇ ̂̇ ̇̂ |𝑆(5)| = |𝐵̂̇ 𝑠 (3, 2)|/(|𝑆(3)|| 𝑆(2)|), 2 ̂̇ ̇̂ |𝑆(6)| = |𝐵̇̂ (3, 3)|/(|𝑆(3)|) , 𝑠

..., ̂̇ − 1)| = |𝐵̂̇ (𝐼 − 1, 0)|/(|𝑆(𝐼 ̂̇ − 1)||𝑆(0)|) ̂̇ |𝑆(𝐼 . 𝑠

(1.4.7)

We would like to emphasize the following important peculiarities of the recursive algorithms (1.4.6) and (1.4.7). ̂ is ambiguous. This peculiar(1) It is seen from (1.4.6) that the first phase value 𝜑(1) ity results from the above-considered bispectrum invariance property to signal translation (1.2.12). Due to this ambiguity, the first phase sampling value is usû = 0. Because of this, neglecting the linear phase ally supposed to be equal to 𝜑(1) that corresponds to original signal shift leads to phase error just in the first step of recursive processing. This phase error can be substantial or nonsubstantial in different signal processing applications and this peculiarity will be discussed later. ̂ Now we only note that if the value 𝜑(1) differs by 2𝜋, phase distortions do not appear in the reconstructed signal. Otherwise, systematic phase error caused by ̂ = 0 is accumulated step-by-step in recursive procedure. arbitrary choice of 𝜑(1) (2) The sampling values of 𝛾𝑠̂ (𝑝, 𝑞) computed usually by using standard softwares within the limits of the principal value in arc tangent function as 𝛾𝑠̂ (𝑝, 𝑞) =

Arc tan{Im𝐵̂̇ 𝑠 (𝑝, 𝑞)/Re𝐵̂̇ 𝑠 (𝑝, 𝑞)} are of ambiguous values in the sense that any quantity multiplied by 2𝜋 added to 𝛾𝑠̂ (𝑝, 𝑞) in an arbitrary sample does not change

its value. Therefore, bispectrum phase computations are accompanied by phase wrapping (or phase function discontinuity) and, hence, computed phase bispectrum differs from true function that has continuous behavior. (3) Due to the bispectrum symmetry property (1.2.10), only the samples located within the limits of the principal triangular domain (see inequalities (1.2.11)) are necessary for signal Fourier spectrum reconstruction.

1.4 Bispectrum-based filtering and signal shape reconstruction

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(4) Both each signal Fourier phase (1.4.6) and magnitude (1.4.7) sample value is contained in the recursive steps by (𝑝 − 1)/2 times for odd 𝑝 and 𝑝/2 times for even 𝑝. This redundancy in some cases can provide improving bispectrum estimate by exploiting the averaging of the corresponding redundant samples. Note that in order to avoid phase wrapping, it is reasonable to perform averaging of the values of exp[𝜙(𝑝)]. After performing the averaging, the phase and magnitude values have to be placed in the consequent recursive step. (5) Since correlation factors between the mentioned redundant samples are, in general, not equal to zeros and in each concrete case their values depend upon signal and noise characteristics, the efficiency of noise smoothing by using redundant data averaging depends upon their correlatedness. (6) Improving of bispectrum estimate by averaging the redundant data differs in different parts of principal bispectrum triangular domain. For low frequencies, that is, for small 𝑝 and 𝑞 values, noise smoothing is worse comparing to possible noise smoothing in high frequency domain because for larger 𝑝 and 𝑞 more averaged values participate in averaging procedure. In order to be able to recover a signal waveform from its bispectrum with maximal accuracy, it is necessary to obtain such bispectrum estimate that is as close as possible to the true bispectrum. In other words, the accuracy of bispectrum-based signal waveform reconstruction largely depends on the accuracy of the bispectrum estimate. Because efficiency of bispectrum-based signal shape reconstruction algorithms directly depends upon quality of bispectrum estimates, it is a very important problem to analyze bispectrum estimation accuracy. Theoretically, it has been demonstrated in [2] that under the assumption that appropriate windowing is accomplished, the conventional direct bispectrum estimate (1.3.6) is unbiased and asymptotically consistent for 𝑀 → ∞. However, in the majority of practical applications we can only operate with short data blocks and the number of available realizations 𝑀 is essentially limited by the restricted time interval given for data measurements. It should be especially noted that in many important real-life cases, for example, for bispectrum-based signal processing in radar applications, low input SNRs could cause an intolerably large variance in bispectrum estimate with respect to what is required for providing the satisfactory performance of a signal reconstruction system. Thus, one of the aims of this Subsection is the study of the performance of bispectral estimators in typical practical signal processing situations: with rather short data blocks, limited sample volume and low input SNR. According to the observation equation (1.4.1) and direct bispectrum estimation technique (see the formulas (1.3.4–1.3.6)), noisy bispectrum estimates can be written

16 | 1 General properties of bispectrum-based digital signal processing as [30]

̇ 𝑆(𝑞) ̇ 𝑆̇∗ (𝑝 + 𝑞) + 𝑆(𝑝) ̇ 𝑆(𝑞)𝐸[ ̇ 𝑁̇ 𝑚∗ (𝑝 + 𝑞)𝑒−𝑗2𝜋𝜏𝑚 (𝑝+𝑞) ] 𝐵̂̇ 𝑥 (𝑝, 𝑞) = 𝑆(𝑝) ̇ 𝑆̇∗ (𝑝 + 𝑞)𝐸[𝑁̇ 𝑚 (𝑝)𝑒𝑗2𝜋𝜏𝑚 𝑝 ] ̇ 𝑆̇∗ (𝑝 + 𝑞)𝐸[𝑁̇ 𝑚 (𝑞)𝑒𝑗2𝜋𝜏𝑚 𝑞 ] + 𝑆(𝑞) + 𝑆(𝑝) ̇ ̇ + 𝑆(𝑝)𝐸[ 𝑁̇ 𝑚 (𝑞)𝑁̇ 𝑚∗ (𝑝 + 𝑞)𝑒−𝑗2𝜋𝜏𝑚 𝑝 ] + 𝑆(𝑞)𝐸[ 𝑁̇ 𝑚 (𝑝)𝑁̇ 𝑚∗ (𝑝 + 𝑞)𝑒−𝑗2𝜋𝜏𝑚 𝑞 ] + 𝑆̇∗ (𝑝 + 𝑞)𝐸[𝑁̇ 𝑚 (𝑝)𝑁̇ 𝑚 (𝑞)𝑒𝑗2𝜋𝜏𝑚 (𝑝+𝑞) ] + 𝐸[𝑁̇ 𝑚 (𝑝)𝑁̇ 𝑚 (𝑞)𝑁̇ ∗ (𝑝 + 𝑞)] 𝑚

= 𝐵̇ 𝑠 (𝑝, 𝑞) + 𝐵̇ err (𝑝, 𝑞) , (1.4.8) where 𝑁̇ 𝑚 (𝑝) = ∑𝑖=0 𝑛𝑚 (𝑖)𝑒−𝑗(2𝜋𝑖𝑝/𝐼) ) is DFT of the 𝑚th noise realization; 𝐵̂̇ 𝑠 (𝑝, 𝑞) = 𝐼−1

̇ 𝑆(𝑞) ̇ 𝑆̇∗ (𝑝 + 𝑞) and 𝐵̇̂ err (𝑝, 𝑞) are the true signal bispectrum and error, that is, con𝑆(𝑝) ̇ taminated by noise and random signal delay components, respectively; 𝑆(𝑝) is the complex-valued Fourier spectrum of true signal 𝑠(𝑖). Analyzing the formula (1.4.8), we see that: –

–

–

the first three terms of the error component 𝐵̂̇ err (𝑝, 𝑞), in fact, tend to zero value asymptotically, that is, for sample volume 𝑀 → ∞, and under assumption that AWGN {𝑛𝑚 (𝑖)} in (1.4.1) has zero mean;

the last term of the error component 𝐵̂̇ err (𝑝, 𝑞) is the third-order statistic that also asymptotically, that is, for 𝑀 → ∞ tends to zero because AWGN is assumed to have a symmetrical PDF; the fourth, fifth, and sixth terms of the error component 𝐵̂̇ err (𝑝, 𝑞) depend multi̇ and these terms mainly cause plicatively on the true signal Fourier spectrum 𝑆(𝑝) the bias and contribute considerably to the total variance of the bispectrum estimator.

TAF estimate evaluated by indirect technique (see the formulas (1.3.1–1.3.3)) can be represented in the form of

𝑅̂ 𝑀 (𝑘, 𝑙) = ⟨𝑅̂ (𝑚) (𝑘, 𝑙)⟩ = 𝑅𝑠 (𝑘, 𝑙) + ⟨𝑛(𝑚) (𝑖)⟩ [𝑅𝑠 (𝑘) + 𝑅𝑠 (𝑙) + 𝑅𝑠 (𝑘 + 𝑙)] ̂ (2) ̂ (2) ̂ ̄ (𝑖)[𝑅̂ (2) + 𝑠𝑀 𝑛𝑀 (𝑘) + 𝑅𝑛𝑀 (𝑙) + 𝑅𝑛𝑀 (𝑘 + 𝑙)] + 𝑅𝑛𝑀 (𝑘, 𝑙) , (1.4.9) where 𝑅𝑠 (𝑘, 𝑙) is the TAF related to original signal 𝑠(𝑖); 𝑅𝑠 (. . .) is the signal autocorre(2) ̄ (𝑖) is the lation function; 𝑅̂ 𝑛 𝑀 (. . .) is the noise autocorrelation function estimate; 𝑠𝑀 (𝑚) signal mean value; 𝑅̂ 𝑛 𝑀 (𝑘, 𝑙) is the noise TAF estimate. Analysis of (1.4.9) indicates that the estimate 𝑅̂ 𝑀 (𝑘, 𝑙) is equal to signal TAF 𝑅𝑠 (𝑘, 𝑙) if and only if the three following conditions are satisfied simultaneously

⟨𝑛(𝑚) (𝑖)⟩ = 0 ;

̄ =0; 𝑠(𝑖)

𝑅̂ 𝑛𝑀 (𝑘, 𝑙) = 0 .

(1.4.10)

1.4 Bispectrum-based filtering and signal shape reconstruction

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It has been demonstrated in [29] that the estimates (1.4.8) and (1.4.9) are asymptotically unbiased and consistent within the principal triangular symmetry domain (1.2.11) if and only if 󵄨󵄨 ̇ 󵄨󵄨 󵄨󵄨𝑆(0)󵄨󵄨 = 𝑠(𝑖) ̄ = 0. (1.4.11)

󵄨

󵄨

Therefore, the necessary condition for obtaining unbiased bispectrum estimate is that a signal mean is zero according to (1.4.11). It means that the mean value of the true signal 𝑠(𝑖) in (1.4.1) should be zero. Moreover, according to [29], even if the signal has a nonzero mean, the bispectrum estimate can be asymptotically unbiased for all frequencies except the bispectral samples belonging to the frequency axes and diagonal in the principal triangular domain:

𝑝 = 0 or 𝑞 = 0 or 𝑝 + 𝑞 = 0 or 𝑝 = 𝑞 or 𝑝 + 𝑞 = 𝐼/2 .

(1.4.12)

However, this statement is correct only asymptotically, that is, for 𝑀 → ∞ and, in addition, for input SNR ≫ 1. Indeed, in the majority of the papers dedicated to the problems of bispectrum-based deterministic signal reconstruction, the latter assumption SNR ≫ 1 is usually supposed. Nevertheless, in practice this assumption is not always valid. Thus, the task of studying the accuracy of the bispectrum estimator in case of processing of zero-mean and nonzero-mean deterministic signals under low input SNRs, limited sample volume 𝑀, and short data blocks (𝐼 is less than hundred samples) is of paramount interest. By using the definition of the variance var(𝑧)̇ of complex random variables [31]

̇ 2) , var(𝑧)̇ = 𝐸(|𝑧̇ − 𝐸(𝑧)|

(1.4.13a)

where 𝑧̇ is an arbitrary complex random variable, a formula for the variance of the bispectrum estimate (1.4.8) has been obtained in [30] as

var{𝐵̂̇ 𝑥 (𝑝, 𝑞)} = 𝐸{|𝐵̂̇ 𝑥 (𝑝, 𝑞) − 𝐸[𝐵̂̇ 𝑥 (𝑝, 𝑞)]|2 } = 𝐸{|Re𝐵̂̇ 𝑥 (𝑝, 𝑞) − 𝐸{Re𝐵̂̇ 𝑥 (𝑝, 𝑞)} + 𝑗{Im𝐵̂̇ 𝑥 (𝑝, 𝑞) − 𝐸{Im𝐵̂̇ 𝑥 (𝑝, 𝑞)}}|2 } = 𝐸{Re𝐵̇̂ (𝑝, 𝑞) − 𝐸{Re𝐵̇̂ (𝑝, 𝑞)}}2 + 𝐸{Im𝐵̇̂ (𝑝, 𝑞) − 𝐸{Im𝐵̂̇ (𝑝, 𝑞)}}2 , 𝑥

𝑥

𝑥

𝑥

(1.4.13b) where Re{. . .} and Im{. . .} are the real and imaginary parts of bispectrum estimate, respectively. It is seen from (1.4.13) that overall variance of bispectrum estimate contains the following contributions:

var{𝐵̂̇ 𝑥 (𝑝, 𝑞)} = var{Re{𝐵̂̇ 𝑥 (𝑝, 𝑞)}} + var{Im{𝐵̂̇ 𝑥 (𝑝, 𝑞)}} .

(1.4.14)

Thus, the total variance of bispectrum estimates of an arbitrary signal corrupted by AWGN can be written as the sum of variances of real and imaginary parts of the bispectrum estimate.

18 | 1 General properties of bispectrum-based digital signal processing After substitution of the function 𝐵̂̇ err (𝑝, 𝑞) from (1.4.8) to (1.4.14), each of two terms in (1.4.14) will contain the sum of 55 terms. It is difficult to analyze them analytically. Below, the statistical analysis of bispectrum estimator performance will be given using computer simulations results performed in [30]. In the paper [29], the problem of bispectral reconstruction of deterministic signals embedded in AWGN was considered. Unlike the algorithm (1.4.7) considered above, the algorithm described in [29] has the following form 𝑀 𝐼−1 󵄨󵄨 ̂ 󵄨󵄨 ̇ 󵄨󵄨 = 1 ∑ ∑ 𝑥(𝑚) (𝑖) , 𝑝 = 0 , 𝑞 = 0 , 󵄨󵄨𝑆(0) 󵄨󵄨 𝑀 󵄨󵄨 𝑚=1 𝑖=0 1 󵄨󵄨 ̂ 󵄨󵄨3 󵄨󵄨 ̂ 󵄨󵄨 6 󵄨 󵄨 󵄨 󵄨 ̇ ̇ 𝐵 𝐵 (1, 1) (3, 1) 󵄨󵄨 󵄨󵄨 𝑠 󵄨󵄨 ] 󵄨󵄨 ̂ 󵄨󵄨 [ 󵄨󵄨󵄨 𝑠 󵄨 󵄨 󵄨 󵄨󵄨𝑆(1) ̇ 󵄨󵄨 = [ 󵄨󵄨 ̂ 󵄨󵄨 󵄨󵄨 ̂ 󵄨󵄨 ] , 𝑝 = 1, . . ., 𝐼 − 1; 1 ≤ 𝑞 ≤ 𝑝; 𝑝 + 𝑞 ≤ 𝐼 − 1 , 󵄨󵄨 󵄨󵄨 󵄨󵄨𝐵̇ 𝑠 (2, 1)󵄨󵄨 󵄨󵄨𝐵̇ 𝑠 (2, 2)󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 ] [ 󵄨󵄨 󵄨󵄨 󵄨󵄨 ̂ 󵄨󵄨2 󵄨󵄨 ̂ 󵄨󵄨 󵄨󵄨 ̂ ̇ 󵄨󵄨 , ̇ 󵄨󵄨 = 󵄨󵄨𝐵̇ 𝑠 (1, 1)󵄨󵄨/󵄨󵄨𝑆(1) 󵄨󵄨󵄨𝑆(2) 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 ... , 󵄨󵄨 󵄨󵄨 ̂ 󵄨󵄨 󵄨󵄨 󵄨󵄨 ̂ 󵄨󵄨 󵄨󵄨 ̂ 󵄨󵄨 ̂ 󵄨󵄨 ̇ 󵄨 󵄨 󵄨 ̇ 󵄨 󵄨 ̇ 󵄨󵄨𝑆(𝐼 ̇ (1.4.15) 󵄨󵄨 − 1)󵄨󵄨󵄨 = 󵄨󵄨󵄨𝐵𝑠 (𝐼 − 2, 1)󵄨󵄨󵄨/󵄨󵄨󵄨𝑆(𝐼 − 2)󵄨󵄨󵄨 󵄨󵄨󵄨𝑆(1)󵄨󵄨󵄨 .

The authors of the cited paper [29], guided by results of theoretical analysis, affirm that bispectrum estimates for signals with nonzero mean values become asymptotically unbiased by removing the bispectrum samples belonging to the diagonal axis and frequency axes in the principal triangular region of the support of the bispectrum. Unfortunately, one can reach this ideal result only if the data length used for the bispectrum estimation is infinite. The second approach dedicated to improving a bispectrum estimator performance and commonly employed in bispectral signal processing is based on the subtraction of a mean value (DC component) from an observed input signal sequence (see, for example, [2, 15, 29]). This way of improving the bispectrum estimate theoretically permits to exclude the distorting influence of the signal-dependent terms in bispectrum estimates. However, spectral leakage does not allow completely removing the influence of these signal-dependent terms. Windowing, well-known in spectral analysis to decrease spectral leakage, has also been theoretically studied in the estimation of third-order statistics [2, 25, 26] because it seems to be an effective way for improvement of bispectrum estimates. However, for many practical applications, particularly, for bispectrum-based reconstruction of deterministic signals embedded in AWGN of large intensity, the performance of those techniques has not been investigated yet. Thus, several important peculiarities of bispectrum estimation have not been studied yet. Therefore, statistical investigation of three aforementioned methods will permit to assess bispectrum estimator performance achievable in realistic cases.

1.4 Bispectrum-based filtering and signal shape reconstruction

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19

Simple test signals 𝑠(𝑖) in the form of single pulse with three different shapes typical in signal processing: rectangular, triangular and Gaussian have been used in computer simulations [30]. The pulse durations have been varied and the maximal amplitude for all types of the pulses has been set to unity. AWGN with different variances and random signal delays 𝜏𝑚 of different deviations have been studied. Total (TOSD) and truncated (TRSD) standard deviations of bispectrum estimator evaluated within the limits of the main bispectrum triangular domain (see (1.2.11)) have been computed and analyzed. TOSD is defined in the form of TOSD = √∑ ∑ var{𝐵̂̇ 𝑥 (𝑝, 𝑞)} , 𝑝

𝑝, 𝑞 ∈ 𝐺 .

(1.4.16)

𝑞

In order to estimate the contribution to the TOSD (1.4.16) of the bispectrum samples belonging to the axes and diagonal, TRSD has been introduced. The TRSD is involved to assess influence of the bispectrum samples belonging to the frequency axes and diagonal (see conditions (1.4.12)) on bispectrum estimator performance. We define TRSD in the form similar to (1.4.16) but excluding the bispectral samples that are located on the aforementioned axes 𝑝 = 0 and 𝑞 = 0 and on the diagonal strips 𝑝 = 𝑞 and 𝑝 + 𝑞 = 𝐼/2. The test signal 𝑠(𝑖), 𝑖 = 0, 1, . . ., 64 selected for studying the bispectrum estimator performance is simulated as the sequence of nonnegative and real-valued samples. The random signal shift 𝜏𝑚 is supposed to be of uniform distribution. Maximum deviation in our computer simulations was {𝜏𝑚 }max = 10 samples. AWGN has been modeled as a zero mean WGN with the specified standard deviation 𝜎𝑛 . It was of constant value within one set of experiments but varied from one set to another to get the dependence on noise standard deviation. First, we analyze the behavior of the function 𝜎(𝑝, 𝑞) = √var{𝐵̂̇ 𝑥 (𝑝, 𝑞)} for different number of realizations 𝑀 for the signals 𝑠(𝑖) of rectangular and triangular waveforms and for different standard deviations of AWGN. Statistical stability has been observed starting with 𝑀 = 50 Monte Carlo runs. The illustrative examples of behavior of function 𝜎(𝑝, 𝑞) for the test pulse signal of rectangular waveform and nonzero DC component, different lengths 𝑇 and for 𝑀 = 100 Monte Carlo runs are represented in Figures 1.4.1–1.4.4 by 3-D graphs. For comparison, the graphs computed for the test signal with subtracted DC component are shown in Figures 1.4.5–1.4.8 (the rest of conditions employed in derivations are the same as in Figures 1.4.1–1.4.4). Note that the graphs in Figures 1.4.1–1.4.8 are plotted only for the bispectral samples belonging to the principal triangular domain of symmetry given by inequalities (1.2.11).

20 | 1 General properties of bispectrum-based digital signal processing

10

SD

8 6 4 2 0 40 30 20 10

p

0

10

5

0

20

15

25

30

35

q

Fig. 1.4.1. Plot of SD = 𝜎(𝑝, 𝑞) computed for 𝑇 = 2, 𝜎𝑛 = 0.1, 𝑀 = 100 and nonzero DC component.

250 200

SD

150 100 50 0 40 30 20 p

10 0

0

5

10

15

20

25

30

35

q

Fig. 1.4.2. Plot of SD = 𝜎(𝑝, 𝑞) computed for 𝑇 = 2, 𝜎𝑛 = 0.5, 𝑀 = 100 and nonzero DC component.

Comparative analysis of these 3-D graphs allows the following conclusion. (1) According to the formula (1.4.8), the largest values of the functions 𝜎(𝑝, 𝑞) are concentrated on the frequency axis 𝑞 = 0 and on two strips that limit the considered principal triangular domain on the bifrequency plane (𝑝, 𝑞). (2) Behavior of the function 𝜎(𝑝, 𝑞) is definitely determined by the form of the signal magnitude Fourier spectrum. This is clearly visible for low standard deviations 𝜎𝑛 of AWGN, when the values of 𝜎(𝑝, 𝑞) are in strict correlation with the magnitude of the Fourier spectrum:

1.4 Bispectrum-based filtering and signal shape reconstruction

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21

200

SD

150 100 50 0 40 30 20 p

10 0

0

10

5

15

20

25

30

35

q

Fig. 1.4.3. Plot of SD = 𝜎(𝑝, 𝑞) computed for 𝑇 = 10, 𝜎𝑛 = 0.1, 𝑀 = 100 and nonzero DC component.

1400 1200

SD

1000 800 600 400 200 0 40 30 20 p

10 0 0

5

10

15 20 q

25

30

35

Fig. 1.4.4. Plot of SD = 𝜎(𝑝, 𝑞) computed for 𝑇 = 10, 𝜎𝑛 = 0.5, 𝑀 = 100 and nonzero DC component.

–

–

the functions 𝜎(𝑝, 𝑞) decay slowly from their maximums that are concentrated at low frequencies for short pulse length of 𝑇 = 2 samples that corresponds to spread signal magnitude Fourier spectrum 𝑆(𝑝) = 𝑇sinc (2𝜋𝑝𝑇/𝐼) (see Figures 1.4.1 and 1.4.5); decaying of the functions 𝜎(𝑝, 𝑞) becomes more rapid for the wider pulse length of 𝑇 = 10 samples and side lobes of the magnitude Fourier spectrum become apparent in the graphs (see Figures 1.4.3 and 1.4.7).

22 | 1 General properties of bispectrum-based digital signal processing

10

SD

8 6 4 2 0 40 40

30 30

20 10

p

10 0

0

20 q

Fig. 1.4.5. Plot of SD = 𝜎(𝑝, 𝑞) computed for 𝑇 = 2, 𝜎𝑛 = 0.1, 𝑀 = 100 and subtracted DC component.

200

SD

150 100 50 0 35 30

40

25

30

20 15 p

20

10

10

5

q

0 0 Fig. 1.4.6. Plot of SD = 𝜎(𝑝, 𝑞) computed for 𝑇 = 2, 𝜎𝑛 = 0.5, 𝑀 = 100 and subtracted DC component.

(3) Spectral leakage effect is evidently observed in the graphs of the functions 𝜎(𝑝, 𝑞) and it becomes more visible for large standard deviations 𝜎𝑛 of AWGN. Increasing of the leakage for larger 𝜎𝑛 is clearly seen from the corresponding comparison of the pairs of the plots in Figure 1.4.1 and Figure 1.4.2; Figure 1.4.3 and Figure 1.4.4; Figure 1.4.5 and Figure 1.4.6 and Figure 1.4.7 and Figure 1.4.8, respectively. (4) Subtraction of DC component from the original signal leads to decreasing the maximum of the function 𝜎(𝑝, 𝑞). It is clearly seen from the corresponding compara-

1.4 Bispectrum-based filtering and signal shape reconstruction

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23

140 120

SD

100 80 60 40 20 0 40 40

30 30

20 p

10

10 0

0

20 q

Fig. 1.4.7. Plot of SD = 𝜎(𝑝, 𝑞) computed for 𝑇 = 10, 𝜎𝑛 = 0.1, 𝑀 = 100 and subtracted DC component.

140 120

SD

100 80 60 40 20 0 40 40

30 30

20 p

10

10 0

0

20 q

Fig. 1.4.8. Plot of SD = 𝜎(𝑝, 𝑞) computed for 𝑇 = 10, 𝜎𝑛 = 0.5, 𝑀 = 100 and subtracted DC component.

tive analysis of the pairs of the graphs in Figure 1.4.1 and Figure 1.4.5; Figure 1.4.2 and Figure 1.4.6; Figure 1.4.3 and Figure 1.4.7 and Figure 1.4.4 and Figure 1.4.8. Though subtraction of DC component from the original signal allows slightly improving bispectrum estimate, the latter still remains quite distorted in the practical case of limited data blocks length 𝐼 and sample volume 𝑀 that are available for averaging. Hence, in practice, the abovementioned DC component subtraction proposed in [2, 15, 29] does not lead to considerable improvement of bispectrum estimator performance.

24 | 1 General properties of bispectrum-based digital signal processing (5) It should be stressed, that since the main part of energy of the spectrum of the test signal 𝑇sinc (2𝜋𝑝𝑇/𝐼) is concentrated within the frequency interval of [−1/𝑇, 1/𝑇], the SNR in this frequency band is also maximal. Hence, the contribution of the errors caused by the fifth, sixth, seventh, and eighth term in (1.4.8) is not too crucial for large SNRs from the point of view of distortions in bispectrumbased signal waveform estimates reconstruction. This statement is confirmed by the results obtained in [32]. (6) One more important feature of bispectrum estimators should be noted. It concerns the invariance of a bispectrum estimate with respect to signal translations. The terms of the form of 𝑒−𝑗2𝜋𝜏𝑚 that appear in (1.4.8) due to the random shifts 𝜏𝑚 of the true signal, fortunately, do not cause a decrease in the bispectrum estimator accuracy. The results obtained for the signals of rectangular, triangular and Gaussian waveforms, maximum signal shift deviation equal to 𝜏𝑚 = 10 samples and nonzero DC component are demonstrated by Tables 1.4.1–1.4.3, respectively. The lengths of pulse of triangular and Gaussian waveforms were selected according to the length of the Table 1.4.1. The results obtained for rectangular signal waveform (𝑀 = 100 Monte Carlo runs, nonzero DC component).

𝑇 TOSD

0.2

0.3

0.4

0.5

2

896

2.51 ⋅ 103

5.87 ⋅ 103

1.09 ⋅ 104

1.85 ⋅ 104

6

1.84 ⋅ 10

3

5.03 ⋅ 10

3

1.02 ⋅ 10

4

1.71 ⋅ 10

4

2.60 ⋅ 104

2.63 ⋅ 10

3

6.39 ⋅ 10

3

1.26 ⋅ 10

4

2.05 ⋅ 10

4

3.33 ⋅ 104

10 TRSD

𝜎𝑛 0.1

2

760

2.17 ⋅ 103

5.04 ⋅ 103

9.42 ⋅ 103

1.60 ⋅ 104

6

1.34 ⋅ 10

3

3.79 ⋅ 10

3

7.89 ⋅ 10

3

1.36 ⋅ 10

4

2.13 ⋅ 104

1.66 ⋅ 10

3

4.22 ⋅ 10

3

8.62 ⋅ 10

3

1.49 ⋅ 10

4

2.54 ⋅ 104

10

Table 1.4.2. The results obtained for triangular signal waveform (𝑀 = 100 Monte Carlo runs, nonzero DC component).

𝑇 TOSD

0.2

4

694

12

1.06 ⋅ 10

3

1.30 ⋅ 10

3

20 TRSD

𝜎𝑛 0.1

4 12 20

0.3

2.15 ⋅ 10

3

3.42 ⋅ 10

3

4.07 ⋅ 10

3

0.4

5.39 ⋅ 10

3

7.33 ⋅ 10

3

9.26 ⋅ 10

3

0.5

1.09 ⋅ 10

4

1.85 ⋅ 104

1.39 ⋅ 10

4

2.41 ⋅ 104

1.59 ⋅ 10

4

2.60 ⋅ 104

578

1.84 ⋅ 103

4.6 ⋅ 103

9.40 ⋅ 103

1.60 ⋅ 104

685

2.38 ⋅ 10

3

5.35 ⋅ 10

3

1.10 ⋅ 10

4

1.93 ⋅ 104

2.37 ⋅ 10

3

5.86 ⋅ 10

3

1.09 ⋅ 10

4

1.92 ⋅ 104

680

1.4 Bispectrum-based filtering and signal shape reconstruction

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25

2 2 Table 1.4.3. The results obtained for Gaussian signal waveform of 𝑠(𝑖) = 𝑒−𝛼 (𝑖−𝐼/2−1) (𝑀 = 100 Monte Carlo runs, nonzero DC component).

𝛼 TOSD

TRSD

𝜎𝑛 0.1

0.2

0.3

0.4

0.5

0.78

665

2.09 ⋅ 103

5.3 ⋅ 103

1.08 ⋅ 104

1.84 ⋅ 104

0.09 0.03

3

1.12 ⋅ 10 1.43 ⋅ 103

3

3.54 ⋅ 10 4.39 ⋅ 103

3

7.49 ⋅ 10 9.44 ⋅ 103

4

1.41 ⋅ 10 1.73 ⋅ 104

2.43 ⋅ 104 2.89 ⋅ 104

0.78

552

1.78 ⋅ 103

4.52 ⋅ 103

9.30 ⋅ 103

1.58 ⋅ 104

728 737

3

3

4

1.95 ⋅ 104 2.08 ⋅ 104

0.09 0.03

2.50 ⋅ 10 2.63 ⋅ 103

5.70 ⋅ 10 6.03 ⋅ 103

1.13 ⋅ 10 1.20 ⋅ 104

Table 1.4.4. The results obtained for rectangular signal waveform (𝑀 = 100 Monte Carlo runs; subtracted DC component).

𝑇 TOSD

0.2

2

810

6

1.57 ⋅ 10

3

1.97 ⋅ 10

3

10 TRSD

𝜎𝑛 0.1

0.3

2.27 ⋅ 10

3

4.18 ⋅ 10

3

5.1 ⋅ 10

3

0.4

5.47 ⋅ 10

3

8.61 ⋅ 10

3

9.83 ⋅ 10

3

0.5

1.03 ⋅ 10

4

1.98 ⋅ 104

1.45 ⋅ 10

4

2.54 ⋅ 104

1.76 ⋅ 10

4

2.84 ⋅ 104

2

738

2.05 ⋅ 103

4.81 ⋅ 103

8.97 ⋅ 103

1.74 ⋅ 104

6

1.34 ⋅ 10

3

3.68 ⋅ 10

3

7.54 ⋅ 10

3

1.28 ⋅ 10

4

2.20 ⋅ 104

1.66 ⋅ 10

3

4.31 ⋅ 10

3

8.35 ⋅ 10

3

1.50 ⋅ 10

4

2.50 ⋅ 104

10

pulse of rectangular waveform to preserve the same signal power for correct comparative analysis. The influence of the signal DC contribution on the bispectrum estimator performance has also been studied for the three abovementioned test signal waveforms with subtracted DC component. Computer simulation results for this case are presented below in Tables 1.4.4–1.4.6 for rectangular, triangular and Gaussian signal waveforms, respectively. Comparative analysis of the data represented in Tables 1.4.1–1.4.6 demonstrates the following: (1) Removing the bispectrum samples located on the boundaries of abovementioned triangular domain allows decreasing TRSD in comparison to TOSD. This coincides with conclusions in [29]. We have analyzed the ratio of TOSD to TRSD by RTT value. RTT grows with increasing of signal length and it becomes considerably larger than unity for small standard deviations 𝜎𝑛 of AWGN. Maximum values of RTT reached approximately 90% as seen in Tables 1.4.2 and 1.4.3 for triangular and Gaussian signal waveforms of maximal length and for 𝜎𝑛 = 0.1. But for low input SNR the expedience of removal data on aforementioned borders in bispectrum-

26 | 1 General properties of bispectrum-based digital signal processing Table 1.4.5. The results obtained for triangular signal waveform (𝑀 = 100 Monte Carlo runs; subtracted DC component).

𝜎𝑛

𝑇 TOSD

TRSD

0.1

4

623

0.2 2.00 ⋅ 10

0.3 3

0.4

4.76 ⋅ 10

3

1.04 ⋅ 10

0.5 4

1.83 ⋅ 104

12 20

854 861

2.78 ⋅ 10 2.87 ⋅ 103

6.53 ⋅ 10 7.04 ⋅ 103

1.29 ⋅ 10 1.31 ⋅ 104

2.17 ⋅ 104 2.34 ⋅ 104

4

566

1.78 ⋅ 103

4.25 ⋅ 103

9.10 ⋅ 103

1.60 ⋅ 104

710 663

3

3

4

1.90 ⋅ 104 1.92 ⋅ 104

12 20

3

3

4

2.34 ⋅ 10 2.34 ⋅ 103

5.60 ⋅ 10 5.82 ⋅ 103

1.10 ⋅ 10 1.10 ⋅ 104

Table 1.4.6. The results obtained for Gaussian signal waveform of (𝑀 = 100 Monte Carlo runs, subtracted DC component).

𝛼 TOSD

0.78 0.09

TRSD

𝜎𝑛 0.1 610 887

0.2

0.3

1.99 ⋅ 10

3

2.97 ⋅ 10

3 3

0.4

5.11 ⋅ 10

3

6.47 ⋅ 10

3

7.25 ⋅ 10

3

0.5

1.05 ⋅ 10

4

1.80 ⋅ 104

1.27 ⋅ 10

4

2.24 ⋅ 104

1.39 ⋅ 10

4

2.47 ⋅ 104

0.03

938

3.18 ⋅ 10

0.78

552

1.78 ⋅ 103

4.52 ⋅ 103

9.30 ⋅ 103

1.58 ⋅ 104

728

2.50 ⋅ 10

3

5.70 ⋅ 10

3

1.12 ⋅ 10

4

2.08 ⋅ 104

2.63 ⋅ 10

3

6.03 ⋅ 10

3

1.20 ⋅ 10

4

2.08 ⋅ 104

0.09 0.03

737

based signal waveform reconstruction is doubtful since significant part of signal Fourier spectrum is rejected. (2) RTT decreases with increasing of 𝜎𝑛 since the fluctuation errors in bispectrum estimates distributed on the whole triangular bispectral domain become prevailing. (3) The method of improving bispectrum estimate by subtraction of signal DC component, in general, leads to decreasing TOSD and TRSD as seen from the corresponding comparison of Tables 1.4.1 and 1.4.4 (rectangular signal waveform), Tables 1.4.2 and 1.4.5 (triangular signal waveform), and Tables 1.4.3 and 1.4.6 (Gaussian signal waveform). However, this method of improving bispectrum estimator performance works well for small 𝜎𝑛 values and RTT approaches to unity with 𝜎𝑛 increasing. (4) The RTT value for 𝑇 increasing grows more slowly for zero DC component signal bispectrum estimate in comparison to nonzero DC component signal bispectrum estimate (compare the corresponding results in Tables 1.4.1, 2 and 3, and the results in Tables 1.4.4–1.4.6). Moreover, RTT corresponding to the zero DC component signal is less in comparison to RTT for nonzero DC component signals.

1.4 Bispectrum-based filtering and signal shape reconstruction

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27

Table 1.4.7. The results obtained for rectangular signal waveform weighed by the Hann window (𝑀 = 100 Monte Carlo runs; nonzero DC component).

𝑇 TOSD

TRSD

𝜎𝑛 0.1

0.2 2.49 ⋅ 10

0.3 3

5.85 ⋅ 10

0.4 3

1.05 ⋅ 10

0.5 4

2.01 ⋅ 104

2

849

6 10

3

1.89 ⋅ 10 2.50 ⋅ 103

3

4.83 ⋅ 10 6.31 ⋅ 103

3

9.17 ⋅ 10 1.22 ⋅ 104

4

1.69 ⋅ 10 2.03 ⋅ 104

2.71 ⋅ 104 3.2 ⋅ 104

2

728

2.12 ⋅ 103

5.03 ⋅ 103

9.06 ⋅ 103

1.74 ⋅ 104

6 10

3

3

3

4

2.24 ⋅ 104 2.40 ⋅ 104

1.39 ⋅ 10 1.58 ⋅ 103

3.66 ⋅ 10 4.18 ⋅ 103

7.13 ⋅ 10 8.48 ⋅ 103

1.34 ⋅ 10 1.44 ⋅ 104

In order to verify expected decreasing of spectral leakage, the windowing has been performed in temporal domain. For this reason, typical Hann window of the form of 𝑤(𝑖) = 1/2[1 − cos(2𝜋(𝑖 − 𝐼/2 − 1)/𝐼)] has been examined in computer simulations. Let us consider the results of computer simulations for bispectrum estimate obtained for window smoothing by the Hann window. These results are represented in Table 1.4.7. As seen from comparison of the results in Tables 1.4.1 and 1.4.7, the Hann window is able to only slightly improve bispectrum estimate for the considered signal and noise parameters. This improvement can reach approximately 6% for small standard deviations 𝜎𝑛 and it is practically negligible for large 𝜎𝑛 values. Thus, the statement about improvement of bispectrum estimate by windowing [2] is absolutely correct, but it is not reasonably large for the considered range of signal and noise parameters. Thus, the results of studying the performance of the bispectrum estimator performed for three aforementioned methods and for real-life cases in signal processing are the following. (1) Although the subtraction of DC component from the original signal slightly improves the bispectrum estimate, the latter still remains severely distorted due to finite data block length, limited number of observed realizations, and spectral leakage. (2) Improving bispectral estimator performance by removing the bispectrum samples belonging to borders of the principal triangular region and by subtracting the DC signal component at the input have shown relatively good effectiveness only for low intensity AWGN. However, spectral leakage severely restricts efficiency of this approach. (3) Conventional windowing of input data leads to rather small improvements of bispectrum estimator performance for high intensity of AWGN.

28 | 1 General properties of bispectrum-based digital signal processing The quality of bispectral density estimate can be improved by using the Kravchenko weight functions (windows) [33, 121–124]. Let us demonstrate improving bispectral estimate of additive mixture of harmonic signals with a large dynamic range of amplitudes using Kravchenko weight functions. For this purpose we consider the following test signal with the dynamic amplitude range of 60 dB:

𝑥(𝑡) = exp(𝑗30𝜋𝑡) + 0.005(𝑗21𝜋𝑡) + 0.001(𝑗43𝜋𝑡) .

(1.4.17)

In order to study the windowing related to bispectral estimation, several different smoothed bispectral estimates have been computed with exploiting different windows for test signals (1.4.17). Bispectral samples belonging to the slice performed along the main bispectral diagonal (see (1.2.11) under condition 𝑝 = 𝑞) are demonstrated in Figure 1.4.9. Result of applying the window of uniform shape (see Figure 1.4.9 (a)) does not allow us to detect a weak signal component. This weak signal contribution contained in additive mixture (1.4.17) is completely contaminated and masked by bispectral side lobes. In order to extract the weak harmonics contained closely to the huge harmonic in the test signal (1.4.7), the signal was multiplied by the following weight functions: Kaiser function 𝐾(𝑡, 𝛽 = 6); Chebyshev function with the level of side lobes equal to −60 dBCh(𝑡, −60); Kravchenko–Kaiser function [fup0.05 2 (𝑡)(1 − 0.02) + 0.02]𝐾(𝑡, 𝛽 = 6). The bispectral slices corresponding to the mentioned weight functions are demonstrated in Figure 1.4.9 (b)–(d). The values of the frequency and magnitude estimates are denoted by (𝑋, Hz) and (𝑌, dB) in the graphs, respectively. The comparison of original value of the second (−46 dB) and third (−60 dB) harmonic components with the estimated values indicates that the minimum total estimation error corresponds to the weight function Kravchenko–Kaiser. Therefore, the usage of weight function Kravchenko–Kaiser for bispectral estimation of multifrequency signals with large dynamic range of the magnitudes is a prospective tool regarding different applications. It will allow to confidently extract the weak contributions contained in multicomponent signals and estimate their magnitudes. Below, in the next Chapter, it will be demonstrated that the use of smoothing the bispectrum estimates by nonlinear filters and robust approaches to processing a set of realizations seems to be a more efficient way for improving the performance of a bispectrum estimator especially for small input SNR. Least-square method and all bispectrum samples belonging to the hexagonal domain are used in bispectrum-based signal reconstruction techniques proposed in [34]. For solving the problem of unknown signal phase Fourier spectrum reconstruction by using phase bispectrum estimate, equation (1.4.3) in [34] is represented in the follow-

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1.4 Bispectrum-based filtering and signal shape reconstruction

0

0

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–10

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–60

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X: –10.65 Y: –44.63

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–40

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–30

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0

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–30

X: –11.03 Y: –46.04

–60

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–70

–70

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0

X: –10.78 Y: –44.35

–40

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(d)

(c)

Fig. 1.4.9. The bispectral slices performed along the main bispectrum diagonal with using the following weight functions: window of uniform shape (a); Kaiser window (b); Chebyshev window (c); Kravchenko–Kaiser window (d).

ing matrix form

𝐴𝜙 = 𝛾 ,

(1.4.18)

where 𝜙 is the vector of unknown Fourier phase values equal to

𝜙 = (𝜙1 , 𝜙2 , 𝜙3 , . . ., 𝜙𝑁 )𝑇 ,

(1.4.19)

and 𝑁 is the total number of unknown Fourier phases; 𝑇 denotes transpose procedure; 𝛾 is the vector of phase bispectrum values defined as

𝛾 = (𝛾1,1 , 𝛾1,2 , . . ., 𝛾1,𝑁−1 , 𝛾2,2 , 𝛾2,3 , . . ., 𝛾2,𝑁−2 , . . ., 𝛾𝑁/2,𝑁/2 )𝑇 .

(1.4.20)

30 | 1 General properties of bispectrum-based digital signal processing The matrix of the coefficients 𝐴 in (1.4.18) has the following form

2 −1 0 0 0 ⋅ ⋅ 0 ] [1 1 −1 0 0 ⋅ ⋅ 0 ] [ [1 0 1 −1 0 ⋅ ⋅ 0] ] [ [⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ] ] [ [⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ] ] [ ] [ 0 0 0 ⋅ ⋅ 1 −1] . 𝐴 = [1 0 ] [ [0 2 0 −1 0 ⋅ ⋅ 0] ] [ [0 1 1 0 −1 ⋅ ⋅ ⋅ ] ] [ [⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ] ] [ [⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 0] ⋅ ⋅ ⋅ ⋅ ⋅ 0 −1] [⋅ ⋅

(1.4.21)

Dimension of the matrix in (1.4.21) is (𝑁/2)2 × (𝑁 − 1) for even and [(𝑁 − 1)(𝑁 + 1)]/4 × (𝑁 − 1) for odd 𝑁 values, respectively. For signal phase Fourier spectrum reconstruction it is proposed in [34] to solve the following equation by the least-square method

𝜙 = (𝐴𝑇 𝐴)−1 𝐴𝑇 𝛾 .

(1.4.22)

It should be noted that two typical obstacles occur with solving the equation (1.4.22) by using the least-square method. The first one is performing of the complicated operations for large dimension matrix inversion. The second one is the ambiguity of the obtained solution and as a result, large phase errors arise due to abovenoted phase wrapping. According to phase unwrapping algorithm proposed in [35], phase bispectrum is defined in the following form 󸀠 𝛾𝑖,𝑗 = 𝛾𝑖,𝑗 + 2𝜋𝑘𝑖,𝑗 ,

(1.4.23)

󸀠 is the principal argument value in the complex-valued bispectrum given where 𝛾𝑖,𝑗 within the interval of [0, 2𝜋) and 𝑘𝑖,𝑗 is the integer value that determines the number of bispectrum phase wrapping in an arbitrary point on the bispectral plane (𝑖, 𝑗). According to this approach, matrix equation (1.4.18) can be transformed to the form of 𝐴𝜙 = 𝛾󸀠 + 2𝜋k , (1.4.24) 󸀠 󸀠 󸀠 󸀠 󸀠 󸀠 󸀠 , 𝛾1,2 , . . ., 𝛾1,𝑁−1 , 𝛾2,2 , 𝛾2,3 , . . ., 𝛾2,𝑁−2 , . . ., 𝛾𝑁/2,𝑁/2 )𝑇 is the vector of prinwhere 𝛾󸀠 = (𝛾1,1 cipal values in the phase bispectrum; the vector k = (𝑘1,1 , 𝑘1,2 , . . ., 𝑘1,𝑁−1 , 𝑘2,2 , 𝑘2,3 , . . ., 𝑘2,𝑁−2 , . . ., 𝑘𝑁/2,𝑁/2 )𝑇 . In addition to the equation (1.4.24), a new matrix C is formed to accomplish the following condition CA = 0 . (1.4.25)

1.5 Reduction of waveform distortions in bispectrum-based signal reconstruction systems

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By multiplying two parts in the equation (1.4.24) with the matrix C, one can obtain

𝐶𝐴𝜙 = 𝐶(𝛾󸀠 +2𝜋k) , 󸀠

𝐶k = (−1/2𝜋)𝐶𝛾 .

(1.4.26) (1.4.27)

Solving the latter equation with respect to k and by substitution of the solution to (1.4.24) allows performing phase bispectrum unwrapping [35]. The main drawback of the algorithm proposed in [35] is its high sensitivity to noise. Due to a noise influence, real-valued quantities arise in the right part in the equation (1.4.27) and it is necessary to approximate them to the nearest integers. Phase errors arise as a result. Thus, the problem of phase wrapping is of great importance and the next Subsection is dedicated to some approaches developed to avoid this problem and, hence, to decrease phase errors in bispectrum-based signal reconstruction techniques.

1.5 Reduction of waveform distortions in bispectrum-based signal reconstruction systems A novel approach [36, 37] based on usage of continuous sine and cosine functions instead of commonly-used discontinuous bispectrum phase arc tangent functions is considered for signal reconstruction systems in this Subsection. First, we discuss the problem in deterministic statement, and after that we will pay attention to noise influence. In addition to the bispectrum-based signal processing algorithms mentioned in the previous Subsections, most authors directly or indirectly exploit phase information in their approaches [10, 29, 38–41]. The authors of the cited papers assume that it is obligatory to compute the phase bispectrum for signal reconstruction directly [10, 29, 38]; or to restore a signal without direct consideration of phase bispectrum but using phase information inherent in signal log-bispectrum (bicepstrum) [39] or differential cepstrum [40]; or to recover the phase of a system transfer function from any pair of consecutive phase bispectrum slices [41]. Note that, on one hand, errorless recovery of phase information is possible only under certain conditions and constraints imposed on the processed signals. On the other hand, the errors that arise in phase recovery may provoke significant distortions in reconstructed signal waveform. For example, 2-D (see the references [10, 29, 38]) or 1-D (see nonparametric algorithm described in [39] and [41]) phase unwrapping is required for reconstruction of the true phase of a signal Fourier spectrum. Effectiveness of the parametric algorithm [39] largely depends on the location of signal zeros and poles towards the unit circle on z-plane. It does not allow having zeros or poles on the unit circle since the parametric algorithm [39] is based on the cepstral coefficients that are not defined in this case. The additional a priori information

32 | 1 General properties of bispectrum-based digital signal processing in the form of signal power cepstrum, as well as a priori knowledge of signal magnitude to transform the processed sequence to minimum or maximum phase signal are necessary for successful operation of the parametric algorithm [39]. The bispectrum iterative reconstruction algorithm [39] also operates under certain limitations, such as that a phase of bispectrum is supposed to be known and, hence, differences of cepstrum coefficients are also known. However, high convergence rate in iterative algorithm [39] is possible only when a priori information about the signal is available. Strict requirements must be imposed on differentiability of z-transform of a processed sequence that cannot have zeros on the unit circle in algorithm [40] using differential cepstrum. According to our strategy, although phase information can be interesting and important for a number of applications, its direct measurement is quite unnecessary for solving bispectrum-based signal shape reconstruction problems. In this Subsection we describe a new approach to solving this problem without addressing direct phase evaluation. Instead of traditional approaches and their abovementioned intrinsic limitations, we propose to use recursive signal reconstruction procedures for quadrature components of complex-valued normalized bispectrum and signal Fourier spectrum. According to the direct method 1.3.4–1.3.6, for a real-valued, deterministic, discrete-time and time-limited signal 𝑥𝑖 (𝑖 = 0, 1, . . ., 𝐼 − 1, and, traditionally, unitary and uniform temporal sampling is assumed) its bispectrum 𝐵𝑝,𝑞 can be defined as the following 2-D complex-valued function

󵄨󵄨 𝑗𝛽𝑝,𝑞 󵄨󵄨 Im 󵄨 󵄨 𝐵𝑝,𝑞 = 𝑋𝑝 𝑋𝑞 𝑋∗𝑝+𝑞 = 𝐵Re , 𝑝,𝑞 + 𝑗𝐵𝑝,𝑞 = 󵄨󵄨𝐵𝑝,𝑞 󵄨󵄨 𝑒

(1.5.1)

Im where 𝑋𝑝 = |𝑋𝑝 |𝑒𝑗𝜑𝑝 is the discrete Fourier transform of 𝑥𝑖 ; 𝐵Re 𝑝,𝑞 and 𝐵𝑝,𝑞 are the real and imaginary parts of bispectrum, respectively; Im Re 2 Im 2 |𝐵𝑝,𝑞 | = √(𝐵Re 𝑝,𝑞 ) + (𝐵𝑝,𝑞 ) is the magnitude bispectrum; 𝛽𝑝,𝑞 = arctan(𝐵𝑝,𝑞 /𝐵𝑝,𝑞 )

is the phase bispectrum determined within the principal interval of inverse tangent function, that is, 𝛽𝑝,𝑞 ∈ (−𝜋, 𝜋]; 𝜑𝑝 and |𝑋𝑝 | are the signal phase and magnitude Fourier spectra, respectively. In order to recover the Fourier spectrum magnitude |𝑋̂ 𝑝 | and phase 𝜑̂ 𝑝 from bispectrum (1.5.1), it is commonly accepted to address to the recursive algorithm represented by equations (1.4.4) and (1.4.5). Since the recovery of a magnitude Fourier spectrum by (1.4.5) does not provoke any distortions and errors, we are focusing on the phase spectrum recovery procedure (1.4.4). The recursive strategy proposed in [10] and repeated later by many authors for signal Fourier phase spectrum recovery by the procedure (1.4.4) is implemented practically for bispectrum samples that belong to the principal bispectrum triangular domain. Note that two different possible sequences of recursive steps are possible. The first approach exploits the parallel direction to the frequency axis 𝑝 = 0 and the second one is parallel to the axis 𝑞 = 0. These two different approaches are represented

1.5 Reduction of waveform distortions in bispectrum-based signal reconstruction systems

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33

by the following corresponding two sets of equations

𝜑̂ 2 = −𝛽1,1 𝜑̂ 3 = 𝜑̂ 2 − 𝛽1,2 𝜑̂ 4 = 𝜑̂ 3 − 𝛽1,3 𝜑̂ 2 = −𝛽1,1

𝜑̂ 5 = 𝜑̂ 4 − 𝛽1,4

𝜑̂ 3 = 𝜑̂ 2 − 𝛽1,2

... 𝜑̂ 𝑁 = 𝜑̂ 𝑁−1 − 𝛽1,𝑁−1

𝜑̂ 4 = 2𝜑̂ 2 − 𝛽2,2

𝜑̂ 4 = 2𝜑̂ 2 − 𝛽2,2

𝜑̂ 4 = 𝜑̂ 3 − 𝛽1,3 ...

𝜑̂ 5 = 𝜑̂ 2 + 𝜑̂ 3 − 𝛽2,3

𝜑̂ 𝑁 = 𝜑̂ 𝑁−1 + 𝜑̂ 𝑁+1 − 𝛽 𝑁−1 , 𝑁+1

...

2

2

2

2

𝜑̂ 5 = 𝜑̂ 4 − 𝛽1,4

𝜑̂ 𝑁 = 𝜑̂ 2 + 𝜑̂ 𝑁−2 − 𝛽2,𝑁−2 ...

...

𝜑̂ 𝑁 = 𝜑̂ 𝑁−1 + 𝜑̂ 𝑁+1 − 𝛽 𝑁−1 , 𝑁+1 , (1.5.2a) 2

2

2

2

𝜑̂ 𝑁 = 𝜑̂ 𝑁−1 − 𝛽1,𝑁−1 .

(1.5.2b)

It was shown in [10] that the procedures (1.5.2a) and (1.5.2b) may produce different results at the reconstruction system output in case of noise presence at the bispectrum estimator input. Note that the behavior of the function 𝛽𝑝,𝑞 in (1.5.2a) and (1.5.2b) is characterized by discontinuities (wrappings) contained within the interval of [−𝜋, 𝜋] radians that may cause the abruptions in the recovered signal phase Fourier spectrum 𝜑̂ 𝑝+𝑞 . The

Im behavior (steepness and oscillation frequency) of the functions 𝐵Re 𝑝,𝑞 and 𝐵𝑝,𝑞 along the frequency axes 𝑝 = 0 and 𝑞 = 0 may vary considerably depending on a processed signal waveform. For fixed uniform sampling interval Δ𝑝 = Δ𝑞 = 1/𝐼 given in the bispectrum plane (𝑝, 𝑞), sampling errors of the function 𝛽𝑝,𝑞 are dependent on its behavior that is usually unknown in practice. Thus, for fixed sampling interval that is usually limited by computer processing rate and memory capacity and under a priori unknown Nyquist frequency, one should expect the distortions of Fourier phase spectrum recovered by the procedures (1.5.2a) or (1.5.2b). Let us study the accuracy of reconstructed signal achieved by the conventional algorithms exploiting direct phase measurements. As a typical example, we consider the algorithm developed in [10]. For this purpose, we introduce the following quantitative measure of distortion for the reconstructed waveform

󵄨󵄨 󵄨󵄨 ∑𝐼−1 𝑖=0 󵄨󵄨𝑥𝑖−𝑡 − 𝑥̂ 𝑖 󵄨󵄨 (1.5.3) 󵄨󵄨 󵄨󵄨 } , 𝑡 ∑𝐼−1 𝑖=0 󵄨󵄨𝑥𝑖 󵄨󵄨 where minimization over 𝑡 = 0, 1, . . ., 𝐼 − 1 is used to take into account possible temporal shift in the reconstructed signal 𝑥̂ 𝑖 . 𝛿 = min {

We now pay attention to the methodical errors that are typical for conventional algorithm [10] and its implementation in a number of papers.

34 | 1 General properties of bispectrum-based digital signal processing First, we consider the problem of reconstruction of the single pulse signals of rectangular shape with different pulse lengths Δ𝑡 observed in the fixed limited sample grid, for example, of 𝐼 = 256 samples. Let us study the test signal centered with respect to the point of origin. In this case, distortion values calculated according to (1.5.3) for the procedures (1.5.2a) and (1.5.2b) are represented in Table 1.5.1. Table 1.5.1. Waveform distortions 𝛿 referred to the procedures (1.5.2a) and (1.5.2b).

Δ𝑡 9 13 17 21 25 29 33 37 41 45 49 𝛿 0.0002 0.003 0.017 0.059 0.105 0.145 0.097 0.173 0.156 0.204 0.192

It should be stressed that the procedures (1.5.2a) and (1.5.2b) cause the same distortions and analysis of the distortion values given in Table 1.5.1 demonstrates that 𝛿 value tends to increase with Δ𝑡 increasing. As a typical example, the test signal of the length of Δ𝑡 = 37 samples and the amplitude of 𝐴 = 1 reconstructed by the procedure (1.5.2a) or (1.5.2b) is shown in Figure 1.5.1. It is clearly seen from Figure 1.5.1 that the original rectangular signal waveform is considerably distorted. Second type of the test signal under study is given in the form of two short-time rectangular pulses. Suppose that the pulses have the same length of Δ𝑡1 = Δ𝑡2 = 3 samples, different amplitudes equal to 𝐴 1 = 1 and 𝐴 2 = 4, and mutual pulse center shift equal to 7 samples. The two-pulse signal reconstructed by the procedure (1.5.2a) or (1.5.2b) is shown in Figure 1.5.2. As it is seen from Figure 1.5.2, the reconstructed signal waveform is con1.4 1.2

0.8 0.6 0.4

-0.2 -0.4

Sample number Fig. 1.5.1. Signal reconstructed by the procedure (1.5.2a) or (1.5.2b), 𝛿 = 0.173.

125

116

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1.5 Reduction of waveform distortions in bispectrum-based signal reconstruction systems

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Signal amplitude

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Sample number Fig. 1.5.2. Signal reconstructed by the procedure (1.5.2a) or (1.5.2b), 𝛿 ≈ 0.6515.

siderably distorted. Note that the distortion values (1.5.3) computed for this two-pulse test signal reconstructed by the procedure (1.5.2a) and (1.5.2b) are equal to approximately the same value 𝛿 ≈ 0.6515. Thus, in the cases when bispectral data are available on a sparse set of points and a signal waveform is a priori unknown, the conventional algorithm [10] may provoke large distortions in the reconstructed signal shape. A novel approach [36] that provides considerably decreasing the distortions in the reconstructed signal waveform is described below. The main idea of this approach is to avoid the discontinuous bispectrum phase function 𝛽𝑝,𝑞 from the signal processing. We propose: first, to replace the function 𝛽𝑝,𝑞 with smooth and continuous functions as cos 𝛽𝑝,𝑞 and sin 𝛽𝑝,𝑞 , and, second, to recover cos 𝜑̂ 𝑝+𝑞 and sin 𝜑̂ 𝑝+𝑞 instead of 𝜑̂ 𝑝+𝑞 in the conventional reconstruction recursive procedure (1.5.2a) or (1.5.2b). The proposed algorithm is based on rewriting the equation (1.5.1) using normalized bispectrum as

𝐵𝑝,𝑞 󵄨󵄨 󵄨 = cos 𝛽𝑝,𝑞 +𝑗 sin 𝛽𝑝,𝑞 = (cos 𝜑̂ 𝑝 +𝑗 sin 𝜑̂ 𝑝 )(cos 𝜑̂ 𝑞 +𝑗 sin 𝜑̂ 𝑞 )(cos 𝜑̂ 𝑝+𝑞 −𝑗 sin 𝜑̂ 𝑝+𝑞 ). 󵄨󵄨𝐵𝑝,𝑞 󵄨󵄨󵄨 󵄨 󵄨

(1.5.4) From equation (1.5.4), the recovery procedures for cosine and sine of signal Fourier phase from cosine and sine of phase bispectrum can be written as

cos 𝜑̂ 𝑝+𝑞 = (cos 𝜑̂ 𝑝 cos 𝜑̂ 𝑞 − sin 𝜑̂ 𝑝 sin 𝜑̂ 𝑞 ) cos 𝛽𝑝,𝑞 + (cos 𝜑̂ 𝑝 sin 𝜑̂ 𝑞 + sin 𝜑̂ 𝑝 cos 𝜑̂ 𝑞 ) sin 𝛽𝑝,𝑞 ,

(1.5.5a)

sin 𝜑̂ 𝑝+𝑞 = (cos 𝜑̂ 𝑝 sin 𝜑̂ 𝑞 + sin 𝜑̂ 𝑝 cos 𝜑̂ 𝑞 ) cos 𝛽𝑝,𝑞 − (cos 𝜑̂ 𝑝 cos 𝜑̂ 𝑞 − sin 𝜑̂ 𝑝 sin 𝜑̂ 𝑞 ) sin 𝛽𝑝,𝑞 .

(1.5.5b)

36 | 1 General properties of bispectrum-based digital signal processing Just as in the conventional approach (see equations (1.5.2a) or (1.5.2b)), we propose below two different sequences of recursive steps. The first one, in accordance with (1.5.5a) and (1.5.5b) (parallel to the frequency axis 𝑝 = 0), can be defined as

{

cos 𝜑̂ 2 = cos 𝛽1,1 sin 𝜑̂ 2 = − sin 𝛽1,1

,

... , {

cos 𝜑̂ 𝑁 = cos 𝜑̂ 𝑁−1 cos 𝛽1,𝑁−1 + sin 𝜑̂ 𝑁−1 sin 𝛽1,𝑁−1 , sin 𝜑̂ 𝑁 = sin 𝜑̂ 𝑁−1 cos 𝛽1,𝑁−1 − cos 𝜑̂ 𝑁−1 sin 𝛽1,𝑁−1

.

... , cos 𝜑̂ 𝑁 = (cos 𝜑̂ 𝑁−1 cos 𝜑̂ 𝑁+1 − sin 𝜑̂ 𝑁−1 sin 𝜑̂ 𝑁+1 ) cos 𝛽 𝑁−1 , 𝑁+1 { { 2 2 2 2 2 2 { { { { +(sin 𝜑̂ 𝑁−1 cos 𝜑̂ 𝑁+1 + cos 𝜑̂ 𝑁−1 sin 𝜑̂ 𝑁+1 ) sin 𝛽 𝑁−1 , 𝑁+1 2 2 2 2 2 2 { { 𝑁−1 cos 𝜑̂ 𝑁+1 + cos 𝜑̂ 𝑁−1 sin 𝜑̂ 𝑁+1 ) cos 𝛽 𝑁−1 𝑁+1 ̂ ̂ sin 𝜑 = (sin 𝜑 { 𝑁 , 2 { 2 2 2 2 2 { { +(sin 𝜑̂ 𝑁−1 sin 𝜑̂ 𝑁+1 − cos 𝜑̂ 𝑁−1 cos 𝜑̂ 𝑁+1 ) sin 𝛽 𝑁−1 , 𝑁+1 2 2 2 2 2 2 {

.

(1.5.6)

Since the sequence of recursive steps in the direction parallel to the axis 𝑞 = 0 differs from (1.5.6) only by succession of the equations, we omit the equations corresponding to the sequence of recursive steps parallel to the frequency axis 𝑞 = 0. See the reference on how the samples are actually collected from the main bispectrum triangular domain in [10] where detailed illustrations of recursive strategy are given. To compare the performance of the conventional algorithm employing the procedures (1.5.2a) or (1.5.2b) and the proposed algorithm using (1.5.6), we begin with example of single pulse test signals with different pulse lengths Δ𝑡 and centered at the origin. As a comparative example, the signal of the length of Δ𝑡 = 37 samples and amplitude of 𝐴 = 1 reconstructed by algorithm (1.5.6) is shown in Figure 1.5.3. It is clearly seen from Figure 1.5.3 that the original rectangular signal waveform is not distorted. Note that both proposed procedures (parallel to the frequency axis 𝑝 = 0 or 𝑞 = 0) give errorless results for other considered pulse lengths Δ𝑡. In Figure 1.5.4, the two-pulse test signal (pulse lengths Δ𝑡1 = Δ𝑡2 = 3 samples, pulse amplitudes 𝐴 1 = 1 and 𝐴 2 = 4 and the mutual center shift are the same as in Figure 1.5.2) recovered by the proposed algorithm (1.5.6) is shown. Comparison of the two-pulse signal waveform reconstructed by conventional procedure (1.5.2a) or (1.5.2b) (see Figure 1.5.2) and by the proposed procedures (see Figure 1.5.4) permits to notice considerable benefit referred to the suggested algorithm (1.5.6). The main distinction of our approach from the known approaches is in abandonment of recovery of phase information, because the phase is not interesting itself for signal reconstruction from bispectrum. In our opinion, traditional computation of bispectrum phase is unnecessary since we need only the normalized bispectrum for solv-

1.5 Reduction of waveform distortions in bispectrum-based signal reconstruction systems

|

37

1.2

Signal amplitude

1 0.8 0.6 0.4 0.2

125

111

97

83

69

55

41

27

13

255

241

227

213

199

185

171

157

143

129

0

Sample number Fig. 1.5.3. Signal reconstructed by the suggested algorithm (1.5.6), 𝛿 = 0. 4.5 4

Signal amplitude

3.5 3 2.5 2 1.5 1 0.5

125

116

98

107

89

71

80

62

53

44

35

17

26

8

255

246

237

228

219

210

192

201

174

183

165

156

147

129 –0.5

138

0

Sample number Fig. 1.5.4. Signal reconstructed by the proposed procedure (1.5.6), 𝛿 = 0.0335.

ing a reconstruction problem. The use of the normalized bispectrum is free from typical constraints imposed on the processed signals and gives considerable reduction of reconstructed signal errors. The proposed approach gives us a possibility to decrease algorithm sensitivity to sampling rate in bispectrum domain that is an important peculiarity for reconstruction of an unknown signal waveform. The sampling frequency in our approach can be chosen approximately by Nyquist criterion and no extension of data file by zero padding is required for sampling errors decreasing. Hence, the proposed algorithm is computationally effective and one of its main advantages is the simplicity.

38 | 1 General properties of bispectrum-based digital signal processing Now we pass to a study of noise immunity of the algorithm (1.5.6) in a bispectrumbased signal reconstruction system operating under influence of both AWGN and mixture of AWGN and impulsive noise [37]. The following set of values has been computed for comparative performance analysis of the proposed algorithm (1.5.6) and conventional algorithm [10]. 2 (1) The sampling variance 𝜎̄ inp , that is, variance calculated by practically limited number of realizations 𝑀 and SNRinp assessed at the input of the signal reconstruction system and, respectively, computed as 2 𝜎̄inp =⟨

2 1 𝐼−1 (𝑚) ∑ [𝑥 (𝑖) − 𝑠(𝑖)] ⟩ , 𝐼 − 1 𝑖=0 𝑀 𝑃 SNRinp = 𝑠 , 𝜎2 ̄

(1.5.7) (1.5.8)

inp

where ⟨. . . ⟩𝑀 denotes expectation computed over 𝑀 observed realizations; 𝑃𝑠 = 𝐼−1 2 (1/𝐼) ∑𝐼−1 𝑖=0 [𝑠(𝑖) − 𝑚𝑠 ] is the power of original signal 𝑠(𝑖); 𝑚𝑠 = (1/𝐼) ∑𝑖=0 𝑠(𝑖). (2) The sampling variance computed at the output of the signal reconstruction system as

𝜎̄ 2out =

1 𝐾 2 ∑𝜎 𝑘, 𝐾 𝑘=1 out

𝐼−1

(1.5.9) 𝐼−1

2 = min𝑡 (1/𝐼) ∑𝑖=0 [(𝑠𝑘̂ (𝑖) − 𝑠(𝑖 − 𝑡)) − (1/𝐼) ∑𝑖=0 (𝑠𝑘̂ (𝑖) − 𝑠(𝑖 − 𝑡))]2 ; where 𝜎out𝑘 𝑠𝑘̂ (𝑖) is the reconstructed signal waveform estimated for 𝑘th experiment (𝑘 = 1, 2, . . ., 𝐾); 𝐾 is the number of the repetitions (number of Monte Carlo runs) executed in computer statistical simulations for obtaining reliable estimates; 𝑡 is the signal shift index (𝑡 = 0, 1, . . ., 𝐼 − 1) introduced taking into consideration the well-known bispectrum invariance property for the signal translation. (3) The SNRout at the output ofthe signal reconstruction system computed as

SNRout =

𝑃𝑠 . 2 𝜎̄out

(1.5.10)

(4) The value 𝜀 which demonstrates the improvement of SNR value evaluated at the signal reconstruction system output compared to its input

𝜀=

SNRout SNRinp

(1.5.11)

Again, we consider a test signal 𝑠(𝑖) (𝑖 = 1, 2, . . ., 256) given in the form of two pulses of rectangular shape and different amplitudes. The two-pulse signal location in the temporal axis has been changed randomly from one 𝑚th observed realization to another. The total number of 𝑀 = 200 realizations was exploited in computer simulations. The signal random shift deviation was equal to 𝜏(𝑚) = 40 samples.

1.5 Reduction of waveform distortions in bispectrum-based signal reconstruction systems

s(i)

s(i)

1

0.8

0.6

0.6

0.4

0.4

0.2

0.2 i 0

(a)

64

128

192

i

0

256

39

1

0.8

0

|

0

64

128

192

256

(b)

Fig. 1.5.5. Reconstructed signal (the length of each signal pulse is equal to 15 samples): (a) the technique [10], 𝜀 = 2.63; (b) the technique proposed in [37], 𝜀 = 4.99. 2 Two kinds of interferences have been studied. The first, AWGN of variance of 𝜎̄inp has been added to each observed realization. The second, original signal has been corrupted by additive mixture of AWGN and impulsive noise. Impulsive noise component has been generated by the given pulse amplitudes and probabilities of appearance the pulses. The so-called “salt and pepper” model of impulsive noise has been utilized. The noise pulse amplitudes were equal to 𝐴 pos = 2 and varying probability of appearance of the pulses was given by the value 𝑃. The test pulse signal length has been varied over the wide limits from 3 to 35 samples. It makes it possible to investigate bispectrum-based signal reconstruction performance depending on the pulse signal length in noise environments. Two examples 2 of the test signal reconstructed in AWGN environment (𝜎̄inp = 0.3) by the technique proposed in [37] and the conventional technique [10] are demonstrated in Figures 1.5.5 and 1.5.6. It is clearly seen from Figures 1.5.5 (a) and 1.5.6 (a) that the original signal rectangular waveform is distorted considerably for the signals reconstructed by the technique [10] due to the abovementioned phase wrapping. Note that the level of distortions increases with increasing of signal pulse length since the number of phase “𝜋jumps” in biphase estimate increases with increasing of signal pulse length. Because of this, the reconstructed signal waveform distortions increase. At the same time, the proposed technique using the continuous sine and cosine functions preserves the original rectangular signal waveform (see Figures 1.5.5 (b) and 1.5.6 (b)). Comparison of the values 𝜀 (1.5.11) that characterize the signal reconstruction performance including both improvement of signal-to-noise ratio at the signal reconstruction system output in comparison to its input and preservation of the original

40 | 1 General properties of bispectrum-based digital signal processing s(i)

s(i)

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2 i

0 0

64

128

192

(a)

i

0

256

0

64

128

192

256

(b)

Fig. 1.5.6. Reconstructed signal (the length of each signal pulse is equal to 35 samples): (a) the technique [10], 𝜀 = 1.26; (b) the technique proposed in [37], 𝜀 = 3.91.

6

Improvement

5 4 3 2 1 0 0

5

10

15

20 25 Pulse signal length

BLW

30

35

40

Technique proposed

2 Fig. 1.5.7. Improvement 𝜀 as a function of pulse signal length, 𝜎̄ inp = 0.3.

signal waveform permits to note that the technique proposed provides better performance compared to the conventional technique. The plots of improvement parameter 𝜀 as a function of signal pulse length computed for technique [10] (Bartelt, Lohmann and Wirnitzer – BLW technique in the Figures) and for the technique suggested in [37] are shown in Figures 1.5.7 and 1.5.8, respectively.

1.5 Reduction of waveform distortions in bispectrum-based signal reconstruction systems

|

41

12

Improvement

10 8 6 4 2 0 0

5

10

15

20 25 Pulse signal length

BLW

30

35

40

Technique proposed

2 Fig. 1.5.8. Improvement 𝜀 as a function of pulse signal length, 𝜎̄ inp = 1.

6

Improvement

5 4 3 2 1 0 0

5

10

15

20 25 Pulse signal length

BLW

30

35

40

Technique proposed

2 Fig. 1.5.9. Improvement 𝜀 as a function of pulse signal length, 𝜎̄ inp = 0.3; 𝑃 = 5%.

The plots in Figures 1.5.7 and 1.5.8 demonstrate the benefit of the suggested technique compared to the BLW technique almost within the total range of the pulse signal lengths studied. The graphs illustrating the behavior of the improvement value 𝜀 as a function of pulse signal length for additive mixture of AWGN and impulsive noise are shown in the Figures 1.5.9 and 1.5.10.

42 | 1 General properties of bispectrum-based digital signal processing 7 6 Improvement

5 4 3 2 1 0 0

5

10

15

20 25 Pulse signal length

BLW

30

35

40

Technique proposed

2 Fig. 1.5.10. Improvement 𝜀 as a function of pulse signal length, 𝜎̄ inp = 0.3; 𝑃 = 30%.

Thus, it can be concluded that the technique proposed provides better performance compared to the BLW technique apart from the very short pulse signal length.

1.6 Performance of the bispectral density estimator Accuracy of signal waveform restoration by using bispectral density estimates largely depends upon the accuracy of the latter function. In order to assess the accuracy, we consider bispectral density estimates in the form of (1.4.8). Bispectral density estimate tends to a consistent value if after 𝐾 statistical examinations and averaging of the sampling estimates (1.4.8) over 𝐾 → ∞ bispectral density estimate will tend to bispectrum. Bispectral density estimate computed for an arbitrary 𝑘th (𝑘 = 1, 2, . . ., 𝐾) examination or 𝑘th arbitrary experiment can be written as

1 𝑀 ̂̇ ∑ 𝐵 (𝑝, 𝑞) , 𝐵̂̇ 𝑥𝑘 (𝑝, 𝑞) = ⟨𝐵̂̇ 𝑥𝑚𝑘 (𝑝, 𝑞)⟩ = 𝑀 𝑀 𝑚=1 𝑥𝑚𝑘

(1.6.1)

where 𝐵̂̇ 𝑥𝑚𝑘 (𝑝, 𝑞) is the 𝑚th sampling bispectral estimate (1.4.8) obtained by 𝑘th examination.

1.6 Performance of the bispectral density estimator

| 43

Taking into account (1.4.1), we can represent (1.6.1) in the form of ∗ (𝑝 + 𝑞) 𝑒−𝑗2𝜋𝜏𝑚𝑘 (𝑝+𝑞) ⟩ 𝐵̂̇ 𝑥𝑘 (𝑝, 𝑞) = 𝑆̇ (𝑝) 𝑆̇ (𝑞) 𝑆̇∗ (𝑝 + 𝑞) + 𝑆̇ (𝑝) 𝑆̇ (𝑞) ⟨𝑁̇ 𝑚𝑘

+ 𝑆̇ (𝑝) 𝑆̇ ∗ (𝑝 + 𝑞) ⟨𝑁̇ 𝑚𝑘 (𝑞) 𝑒𝑗2𝜋𝜏𝑚𝑘 𝑞 ⟩𝑀

𝑀

+ 𝑆̇ (𝑞) 𝑆̇ ∗ (𝑝 + 𝑞) ⟨𝑁̇ 𝑚𝑘 (𝑝) 𝑒𝑗2𝜋𝜏𝑚𝑘 𝑞 ⟩𝑀

∗ + 𝑆̇ (𝑝) ⟨𝑁̇ 𝑚𝑘 (𝑝 + 𝑞) 𝑁̇ 𝑚𝑘 (𝑞) 𝑒𝑗2𝜋𝜏𝑚𝑘𝑝 ⟩𝑀

∗ + 𝑆̇ (𝑞) ⟨𝑁̇ 𝑚𝑘 (𝑝 + 𝑞) 𝑁̇ 𝑚𝑘 (𝑝 + 𝑞) 𝑒−𝑗2𝜋𝜏𝑚𝑘𝑞 ⟩𝑀

+ 𝑆̇ ∗ (𝑝 + 𝑞) ⟨𝑁̇ 𝑚𝑘 (𝑝) 𝑁̇ 𝑚𝑘 (𝑞) 𝑒𝑗2𝜋𝜏𝑚𝑘 (𝑝+𝑞) ⟩

𝑀

∗ + ⟨𝑁̇ 𝑚𝑘 (𝑝) 𝑁̇ 𝑚𝑘 (𝑞) 𝑁̇ 𝑚𝑘 (𝑝 + 𝑞)⟩𝑀 .

.

= 𝐵̇ 𝑠 (𝑝, 𝑞) + 𝐵̇ 𝑛𝑘 (𝑝, 𝑞) ,

(1.6.2)

.

𝐼−1

where 𝑆(𝑝) = ∑𝑖=0 𝑠(𝑖) exp(−𝑗(2𝜋/𝐼)𝑖𝑝) is the signal Fourier spectrum; .

𝐼−1 𝑁̇ 𝑚𝑘 (𝑝) = ∑𝑖=0 𝑛𝑚𝑘 (𝑖) exp(−𝑗(2𝜋/𝐼)𝑖𝑝) is the Fourier transform referred to an arbi. trary 𝑚th realization of Gaussian noise and 𝑘th examination; 𝐵̇ 𝑠 (𝑝, 𝑞) is the true signal .

bispectrum; 𝐵̇ 𝑛𝑘 (𝑝, 𝑞) denotes the noise contribution. Since Gaussian noise 𝑛𝑚𝑘 (i) is assumed to be of zero-mean, the terms contained the statistical mean value ⟨𝑁𝑚𝑘 (. . .)⟩𝑀 in (1.6.2), that is, the terms ∗ (𝑝 + 𝑞) 𝑒−𝑗2𝜋𝜏𝑚𝑘(𝑝+𝑞) ⟩ ≅ 0 , 𝑆̇ (𝑝) 𝑆̇ (𝑞) ⟨𝑁̇ 𝑚𝑘 . ∗

𝑆̇ (𝑝) 𝑆 (𝑝 + 𝑞) ⟨𝑁̇ 𝑚𝑘 (𝑞) 𝑒

𝑀

𝑗2𝜋𝜏𝑚𝑘 𝑞

⟩𝑀 ≅ 0 ,

𝑆̇ (𝑞) 𝑆̇∗ (𝑝 + 𝑞) ⟨𝑁̇ 𝑚𝑘 (𝑝) 𝑒𝑗2𝜋𝜏𝑚𝑘𝑞 ⟩𝑀 ≅ 0 ,

(1.6.3a)

tend to zero for quite large sample number 𝑀, and the last term in (1.6.2) also tends asymptotically to zero according to the bispectrum property (1.2.8): ∗ ⟨𝑁̇ 𝑚𝑘 (𝑝) 𝑁̇ 𝑚𝑘 (𝑞) 𝑁̇ 𝑚𝑘 (𝑝 + 𝑞)⟩𝑀 ≅ 0 .

(1.6.3b)

.

Taking into account (1.6.3), the noise contribution 𝐵𝑛𝑘 (𝑝, 𝑞) in (1.6.2) can be represented as: ∗ (𝑝 + 𝑞) 𝑁̇ 𝑚𝑘 (𝑞) 𝑒𝑗2𝜋𝜏𝑚𝑘 𝑝 ⟩𝑀 𝐵̂̇ 𝑛𝑘 (𝑝, 𝑞) ≅ 𝑆̇ (𝑝) ⟨𝑁̇ 𝑚𝑘

+ 𝑆̇ (𝑞) ⟨𝑁̇ 𝑚𝑘 (𝑝 + 𝑞) 𝑁̇ ∗ 𝑚𝑘 (𝑝 + 𝑞) 𝑒−𝑗2𝜋𝜏𝑚𝑘 𝑞 ⟩𝑀 + 𝑆̇∗ (𝑝 + 𝑞) ⟨𝑁̇ 𝑚𝑘 (𝑝) 𝑁̇ 𝑚𝑘 (𝑞) 𝑒𝑗2𝜋𝜏𝑚𝑘(𝑝+𝑞) ⟩

𝑀

.

(1.6.4)

44 | 1 General properties of bispectrum-based digital signal processing Therefore, bispectral density estimate (1.6.2) can be written in the form of

𝐵̂̇ 𝑥𝑘 (𝑝, 𝑞) ≅ {Re [𝐵̇ 𝑠 (𝑝, 𝑞)] + Re [𝐵̇̂ 𝑛𝑘 (𝑝, 𝑞)]}+𝑗 {Im [𝐵̇̂ 𝑠 (𝑝, 𝑞)] + Im [𝐵̇̂ 𝑛𝑘 (𝑝, 𝑞)]} . (1.6.5) By using the results obtained in [42], we suppose that for quite large number 𝐾 of examinations, distribution law of the estimates Re[𝐵̂̇ 𝑥𝑘 (𝑝, 𝑞)] and Im[𝐵̂̇ 𝑥𝑘 (𝑝, 𝑞)] will tend to Gaussian law. According to the Gaussianity assumption, a likelihood function that is multidimensional conditional probability density of the vector estimate 𝐵⃗ 𝑥 (𝑝, 𝑞) under as.

sumption of unknown bispectrum 𝐵̇ 𝑠 (𝑝, 𝑞) can be written as

󵄨󵄨 󵄨󵄨 Re 󵄨󵄨 Im ⃗̂ Im 󵄨󵄨 󵄨󵄨 𝑃 [𝐵⃗ 𝑥 (𝑝, 𝑞)󵄨󵄨󵄨󵄨𝐵̇ 𝑠 (𝑝, 𝑞)] = 𝑃 [𝐵⃗̂ Re 𝑥 (𝑝, 𝑞)󵄨󵄨 𝐵𝑠 (𝑝, 𝑞)] 𝑃 [𝐵𝑥 (𝑝, 𝑞)󵄨󵄨 𝐵𝑠 (𝑝, 𝑞)] , 󵄨 󵄨 󵄨

(1.6.6)

⃗ (𝑝, 𝑞)| 𝐵Re (𝑝, 𝑞)] and [𝐵⃗̂ Im (𝑝, 𝑞)| 𝐵Im (𝑝, 𝑞)] are the conditional probability where [𝐵̂ Re 𝑥 𝑠 𝑥 𝑠 densities referred to both as real (Re) and imaginary (Im) parts under assumption of Im unknown signal contributions 𝐵Re 𝑠 (𝑝, 𝑞) and 𝐵𝑠 (𝑝, 𝑞), respectively;

̇̂ ̇̂ ̇̂ 𝐵⃗̂ Re 𝑥 (𝑝, 𝑞) = {Re[𝐵𝑥1 (𝑝, 𝑞)], Re[𝐵𝑥2 (𝑝, 𝑞)], . . . , Re[𝐵𝑥𝐾 (𝑝, 𝑞)]} ; ̇̂ ̇̂ ̇̂ 𝐵⃗̂ Im 𝑥 (𝑝, 𝑞) = {Im[𝐵𝑥1 (𝑝, 𝑞)], Im[𝐵𝑥2 (𝑝, 𝑞)], . . . , Im[𝐵𝑥𝐾 (𝑝, 𝑞)]} . Since the mathematical manipulations performed below are the same for the Re and Im parts of bispectral estimate, it seems to be reasonable to restrict further derivation of the formulas only for one part, for example, only for the Re part. Conditional probability density referred to the Re part of bispectral estimate for 𝑘th examination is equal to

󵄨󵄨 Re 󵄨󵄨 𝑃 [ 𝐵⃗̂ Re 𝑥𝑘 (𝑝, 𝑞)󵄨󵄨 𝐵𝑠 (𝑝, 𝑞)] 󵄨 2 1 1 ̂̇ (𝑝, 𝑞)] − ⟨Re [𝐵̂̇ (𝑝, 𝑞)]⟩ } } , exp {− {Re [ 𝐵 = 𝑥𝑘 𝑥𝑘 2 2 √2𝜋𝜎̂Re𝑘 2𝜎̂Re𝑘 (𝑝, 𝑞) (𝑝, 𝑞) 𝐾 (1.6.7) 𝑀

𝑀

2 (𝑝, 𝑞) = (1/𝑀 − 1) ∑𝑚=1 {Re[𝐵̇ 𝑛𝑚𝑘 (𝑝, 𝑞)] − Re[(1/𝑀) ∑𝑚=1 𝐵̇ 𝑛𝑚𝑘 (𝑝, 𝑞)]}2 where 𝜎̂Re𝑘 is the sampling variance, that is, the value computed with limited number of 𝑀 realizations for 𝑘th examination. Note, that the estimate Re⟨𝐵̂̇ 𝑥𝑘 (𝑝, 𝑞)⟩𝐾→∞ → 𝐵Re 𝑠 (𝑝, 𝑞), that is, this estimate is unbiased except the samples belonging to the axes 𝑝 = 0, 𝑞 = 0 and 𝑝 = 𝑞.

1.6 Performance of the bispectral density estimator

| 45

The following logarithmic function is commonly more suitable for further analysis

󵄨󵄨 Re 󵄨󵄨 𝐵 (𝑝, 𝑞)]} ln {𝑃 [ 𝐵⃗̂ Re (𝑝, 𝑞) 󵄨󵄨 𝑠 𝑥𝑘 󵄨 = ln {

1 2 √2𝜋𝜎̂ Re𝑘 (𝑝, 𝑞)

}−

2 {Re [𝐵̂̇ 𝑥𝑘 (𝑝, 𝑞)] − Re ⟨𝐵̂̇ 𝑥𝑘 (𝑝, 𝑞)⟩ } 𝐾

2 2𝜎̂ Re𝑘 (𝑝, 𝑞)

, (1.6.8)

𝐾

where Re⟨𝐵̂̇ 𝑥𝑘 (𝑝, 𝑞)⟩𝐾 = Re[(1/𝐾) ∑𝑘=1 𝐵̂̇ 𝑥𝑘 (𝑝, 𝑞)]. Re As conditional estimate 𝐵̂ Re 𝑠ML (𝑝, 𝑞) referred to the real part of bispectrum 𝐵𝑠 (𝑝, 𝑞) Re we select such value 𝐵𝑠 (𝑝, 𝑞) for which logarithmic likelihood function will tend to maximum value after ensemble averaging by 𝐾 observations of the function (1.6.8). Conditional estimate 𝐵̂ Re 𝑠ML (𝑝, 𝑞) that satisfies with the maximum likelihood function criterion will be assumed as a maximum likelihood function. The likelihood equation for real part of bispectral estimate can be written as:

⟨

󵄨󵄨 Re 𝜕{ln 𝑃[𝐵⃗̂ Re 𝑥𝑘 (𝑝, 𝑞)󵄨󵄨 𝐵𝑠 (𝑝, 𝑞)]} 𝜕 𝐵Re 𝑠 (𝑝, 𝑞)

1 𝐾 ⟩ = ∑ 𝐾 𝑘=1 𝐾

󵄨󵄨 Re 𝜕 {ln 𝑃[𝐵⃗̂ Re 𝑥𝑘 (𝑝, 𝑞)󵄨󵄨 𝐵𝑠 (𝑝, 𝑞)]} 𝜕 𝐵Re 𝑠 (𝑝, 𝑞)

= 0.

(1.6.9)

Each solution 𝐵̂ Re 𝑠ML (𝑝, 𝑞) in the likelihood equation (1.6.9) is a function of the set of 𝐾 bispectral estimates Re[𝐵̂̇ 𝑥𝑘 (𝑝, 𝑞)], 𝑘 = 1, 2, . . ., 𝐾, and each solution corresponds to

the maximum of the likelihood function. The problem is to find such a solution that corresponds to the maximum of the likelihood function. If an unbiased effective estimate 𝐵̂ Re 𝑠eff exists and it provides minimum root-meansquare error with respect to the true bispectrum 𝐵Re 𝑠 (𝑝, 𝑞) between all possible esti(𝑝, 𝑞) , that is, the following condition is valid mates 𝐵̂ Re 𝑠ML 2

2

̂ Re ̂ Re ̂ Re ⟨[𝐵̂ Re 𝑠eff (𝑝, 𝑞) − 𝐵𝑠 (𝑝, 𝑞)] ⟩ ≤ ⟨[𝐵𝑠ML (𝑝, 𝑞) − 𝐵𝑠 (𝑝, 𝑞)] ⟩ ,

(1.6.10)

the likelihood equation (1.6.9) will have singular solution equal to 𝐵̂ Re 𝑠eff (𝑝, 𝑞). Therefore, the likelihood equation (1.6.9) provides necessary but not sufficient condition of the maximum. Taking into account (1.6.8), the likelihood equation (1.6.9) can be written as Re Re Re 1 𝐾 𝐵𝑥𝑘 (𝑝, 𝑞) − 𝐵𝑠 (𝑝, 𝑞) − ⟨𝐵𝑛𝑘 (𝑝, 𝑞)⟩𝐾 ∑ =0. 2 𝐾 𝑘=1 (𝑝, 𝑞) 𝜎̂ Re𝑘

(1.6.11)

Accuracy of bispectral estimate 𝐵Re 𝑠 (𝑝, 𝑞) depends upon the “sharpness” of the Gaussian likelihood function. Therefore, in order to estimate the accuracy, one can utilize the second derivative of logarithmic likelihood function referred to as its maximum and taken with an opposite sign.

46 | 1 General properties of bispectrum-based digital signal processing Then, the error of bispectral estimate can be defined by a derivative taken with

𝐵Re 𝑠 (𝑝, 𝑞) in the equation (1.6.11) and with an opposite sign. As a result, we obtain: 󵄨󵄨 Re ⃗ Re 2 󵄨 𝐾 𝜕 {ln 𝑃 [ 𝐵̂ 𝑥𝑘 (𝑝, 𝑞)󵄨󵄨 𝐵𝑠 (𝑝, 𝑞)]} 1 1 1 𝐾 󵄨󵄨 − ∑ ∑ 2 (1.6.12) = . 𝐾 𝑘=1 𝐾 ̂ 𝜎 𝜕2 𝐵Re (𝑝, 𝑞) 𝑠 𝑘=1 Re𝑘 (𝑝, 𝑞) Variance of bispectral estimate obtained by the maximum likelihood function and defined according to the Cramér–Rao criterion can be written by the following inequality

VAR [𝐵̂ Re 𝑠ML (𝑝, 𝑞)] ≥

1 − 𝐾1

𝐾 ∑𝑘=1

𝜕2 {ln 𝑃[ 𝐵⃗̂ Re 𝑥𝑘

󵄨󵄨

(𝑝,𝑞)󵄨󵄨󵄨󵄨󵄨 𝐵Re 𝑠 (𝑝,𝑞)]} 𝜕2 𝐵Re 𝑝,𝑞 ) 𝑠 (

.

(1.6.13)

Taking into account (1.6.12) and (1.6.13), the extreme accuracy of bispectral estimate can be evaluated according to the Cramér–Rao lower bound as

VAR [𝐵̂ Re 𝑠ML (𝑝, 𝑞)] ≥

1 1 𝐾

𝐾 ∑𝑘=1 𝜎̂ 2 1𝑝,𝑞 ) Re𝑘 (

.

(1.6.14)

The same result can be obtained for the imaginary part of bispectral estimate

VAR [𝐵̂ Im 𝑠ML (𝑝, 𝑞)] ≥ 𝑀

1 1 𝐾

1 ∑𝐾 𝑘=1 𝜎̂ 2 (𝑝,𝑞) Im𝑘

,

(1.6.15) 𝑀

2 where 𝜎̂ Im𝑘 (𝑝, 𝑞) = (1/𝑀 − 1) ∑𝑚=1 {Im[𝐵̇ 𝑛𝑚𝑘 (𝑝, 𝑞)] − Im[(1/𝑀) ∑𝑚=1 𝐵̇ 𝑛𝑚𝑘 (𝑝, 𝑞)]}2 is the sampling variance obtained with limited sample contained 𝑀 realizations referred to the imaginary part of interference in (1.6.4) for 𝑘th examination. Taking into consideration (1.6.14) and (1.6.15), the extreme root-mean-square error referred to the complex-valued bispectral estimate can be defined as

̂ Im VAR [𝐵̂ 𝑠ML (𝑝, 𝑞)] = VAR [𝐵̂ Re 𝑠ML (𝑝, 𝑞)] + VAR [𝐵𝑠ML (𝑝, 𝑞)] 1 1 ≥ 1 𝐾 + 1 𝐾 . 1 ∑𝑘=1 𝜎̂ 2 𝑝,𝑞 ∑𝑘=1 𝜎̂ 2 1𝑝,𝑞 𝐾 𝐾 ( ) ( ) Re𝑘 Im𝑘

(1.6.16)

Thus, formula (1.6.16) can serve for evaluation of the extreme accuracy of bispectral estimate necessary for bispectrum-based signal processing in different applications.

1.7 Conclusions The performed analysis demonstrates that the bispectrum and bispectrum-based data processing possess several important features. They are highly immune to noise which is zero mean and has symmetric PDF, insensitive to random delays of a signal component in observed realizations, able to preserve phase information and phase coupling

1.7 Conclusions

| 47

between the frequencies contained in processed signal, and so on. These useful properties have already been exploited in abovementioned applications and they open new perspectives. However, it is not always clear how positive features of bispectrum-based signal processing can be exploited in a particular application. It should be noted that bispectrum properties’ study is far away from completeness. First, there are several strategies of bispectrum estimation. In the next Chapters we prefer to rely on direct technique of bispectrum estimation defined by expressions (1.3.4)–(1.3.6). This way is rather simple and fast since it allows exploiting fast Fourier transform algorithms. Second, noise contained in observed input realizations leaks to bispectrum domain and makes bispectrum estimate noisy. Moreover, even if noise in observed input realizations is purely additive, zero mean, i.i.d. and Gaussian, noise contained in real and imaginary parts of bispectrum estimate occurs to be of quite complex statistical properties (see expression (1.4.8) and investigation results presented in Subsection 1.4). If input SNR is low and a number of observed realizations 𝑀 is small, the obtained estimate of bispectrum can be rather noisy. This noise leaks to the reconstructed signal shape estimate via nonlinear procedures of signal spectrum recovery from bispectrum estimate and further signal waveform reconstruction. Thus, improvement of bispectrum estimate is an important task. Third, although there exists the conventional procedure of signal spectrum recovery from bispectrum estimate [10] and it performs well enough, there are also some alternatives and they can be used as well. Fourth, bispectrum-based processing does not produce perfect reconstruction of ̂ (see Subseca signal waveform due to uncertainties in setting the phase sample 𝜑(1) tion 1.4). Besides, reconstructed signal estimates are “centered” with respect to the origin (see examples in Figures 1.5.3 and 1.5.6). This should be taken into account in practical applications and in selection of quantitative criteria to characterize accuracy of signal waveform reconstruction. Fifth, although bispectrum-based signal processing is usually applied for zero mean and symmetric PDF noise that affects an input signal, recent results [32] show that bispectrum-based reconstruction can be carried out also for non-Gaussian and mixed noise environments. However, this problem is not well studied yet. Sixth, bispectral signal processing operates with 2-D functions (2-D bispectrum or 2-D TAF estimates) even if an input signal is a 1-D function, that is, dimensionality of processed data increases. Therefore, special attention should be paid to computational efficiency of bispectrum-based techniques. Finally, bispectrum-based technique performance should be compared to that of other techniques that could be applied for particular practical tasks in order to draw final conclusions.

2 Unknown noisy signal shape estimation by bispectrum-filtering techniques Performance of a bispectrum-based signal processing system largely depends on performance of the bispectrum estimator commonly operating in adverse noise environment, as well as under a priori uncertainty regarding signal and noise parameters at the system input. Recently, several novel techniques have been developed [42–49] for noise suppression in bispectrum estimates. The objective of this Chapter is a discussion of the results obtained by these novel techniques. In the previous Chapter, it was mentioned that a useful property of bispectrumbased signal processing is its ability of quasi-coherent accumulation of mutually and randomly shifted separate contributions of noisy signals where a signal shape is a priori unknown. A general assumption related to the signal is that its shape is the same or almost the same for all observed realizations. This assumption, for example, holds for high-resolution radar target range profile estimates obtained in a set of sequential scans following each other rather frequently under condition that a target aspect angle does not change from one scan to another. An example of such high-resolution radar range profile estimates for a missile model developed in [50] is presented in Figure 2.1. As seen, these radar signatures are practically of the same shape. A simple model of such radar range profiles could be represented in the form of two rather short pulses with different pulse magnitudes. Analysis of the presented range profiles also demonstrates another peculiarity of the observed signal. As seen, there are such sample minimal and maximal indices 𝑖min and 𝑖max that 𝑠(𝑖) = 0 for 𝑖 = 0, . . ., 𝑖min and 𝑖 = 𝑖max , . . ., 𝐼 − 1. Moreover, 𝑖max − 𝑖min is considerably less than 𝐼. These properties will later be used in computer simulations performed in this Chapter. Note that radar range profile estimates can be used for aerial target classification. As shown in [50], classification accuracy depends on many factors but one of the basic ones is noise intensity in an obtained estimate of radar range profile. Moreover, it is also mentioned that for radar target classification it is desirable to center an estimate with respect to origin. Then a classifier, for example, that one based on neural network is able to perform better. Thus, it is possible to assume that the better, that is, less noisy a range profile estimate is, the better probability of correct classification of aerial target type can be provided. Within the bispectrum-based data processing intended on unknown signal shape reconstruction this means the following. First, better estimate of a signal shape can be obtained if a better estimate of a signal Fourier spectrum recovered from bispectrum is provided. Second, better estimate of a signal Fourier spectrum recovered from bispectrum can be provided if a better estimate of bispectrum is produced. Thus, it is reasonable to apply any methods that provide better estimates of bispectrum and/or signal Fourier spectrum recovered from bispectrum estimates.

2 Unknown noisy signal shape estimation by bispectrum-filtering techniques |

49

Filtering is one typical approach to obtaining better estimates of 1-D or 2-D processes under study. Recently, a great number of various filters have been proposed and used in numerous applications. Thus, it is a problem to find and select or to design a proper filter for a given bispectrum-based signal processing. Very often, real-life situation is such that more a priori information about signal parameters and properties of noise is available, more effective and efficient filters can be found or designed. Because of this, below we consider an opportunity to combine bispectrum-based signal processing with linear or nonlinear filtering and other approaches devoted to improving the performances of bispectrum and spectrum estimators. One more problem is that both bispectrum and noisy signal Fourier spectrum recovered from bispectrum are complex-valued 2-D and 1-D processes, respectively. This opens a question of what could be a proper way of filtering. For example, filtering can be applied to magnitude and phase functions or to the real and imaginary parts. The results discussed in this Chapter partly address all these issues. S 0.15

S 0.15

0.1 0.1 0.05

0.05 i

0

0

50

100

150

200

0

250

(a)

(b)

S 0.15

S 0.15

0.1

0.1

0.05

0.05

0 (c)

i 0

50

100

150

200

0

250

i 0

50

100

150

200

250

0

50

100

150

200

250

i

(d)

Fig. 2.1. Noise-free case: Centered radar range profiles of a missile model evaluated for aspect angles 180∘ (a), 175∘ (b), 174.95∘ (c), and 174.90∘ (d).

50 | 2 Unknown noisy signal shape estimation by bispectrum-filtering techniques

2.1 Smoothing the noisy bimagnitude and biphase or the real and imaginary parts of bispectrum estimates by using nonadaptive 2-D linear and nonlinear filtering An approach suggested for improving the performance of bispectrum-based signal shape reconstruction system is developed in [8, 32]. The main idea of this approach is in smoothing the noisy bispectrum estimates by using 2-D linear and nonlinear filtering. First, let us study by visual analysis the distortions of the bispectrum estimate for the test signal shown in Figure 2.1.1. The bimagnitude and biphase of such noisefree signals are demonstrated in Figure 2.1.2 (a) and 2.1.2 (b), respectively. Obviously, the bimagnitude is a rather smooth function of two frequencies. It is seen from Figure 2.1.2 (a) that bimagnitude does not contain discontinuities. At the same time, biphase contains the number of discontinuities observed in the form of phase wrapping (see Figure 2.1.2 (b)). Examples of bimagnitude and biphase estimates contaminated by noise are demonstrated in Figure 2.1.3 and Figure 2.1.4, respectively. They have been computed by ensemble averaging with 𝑀 = 256 realizations and for input signal-to-noise ratio SNRinp = 0.5 It is clearly seen that despite the relatively large realization number 𝑀 = 256 participating in ensemble averaging, bimagnitude and biphase functions are severely contaminated by AWGN contained in the input process that leaked into bispectral domain. Our strategy suggested in [32] is based on the reduction of the reconstructed signal estimation errors by smoothing the noisy bispectrum estimates with 2-D nonadaptive linear or nonlinear filters before signal Fourier spectrum recovery. It was proposed to improve the bispectrum estimate by either smoothing the phase and magnitude bispectra or filtering the real and imaginary components of the bispectrum estimate. Our preliminary investigations [42] have demonstrated that the smoothing of the TAF estimates is inefficient for short pulse signals. The reason is that the main lobe width of the TAF estimate in the considered case is small and, thus, the nonlinear/linear filters not only reduce noise but also deteriorate useful information features of these triple correlation estimates. On the contrary, for short pulse signals, the corresponding bimagnitude

̂ (𝑝, 𝑞) are the relatively slowly varying functions in bifre|𝐵̂̇ 𝑠ind (𝑝, 𝑞)| and biphase 𝛾𝑠ind quency domain (subscript ind denotes here the indirect technique: (1.3.1)–(1.3.3)). Therefore, it can be expected that effective noise suppression can be provided by linear or nonlinear filters if they are applied both to the functions |𝐵̂̇ 𝑠ind (𝑝, 𝑞)| and

̂ (𝑝, 𝑞) or to filtering the real Re{𝐵̂̇ 𝑠ind(𝑝, 𝑞)} and imaginary Im{𝐵̂̇ 𝑠ind(𝑝, 𝑞)} parts 𝛾𝑠ind of the bispectrum estimate.

2.1 Smoothing bispectrum estimates by using nonadaptive linear and nonlinear filtering | 51

6

s(i)

4

2

i

0 0

50

100

150

200

250

Fig. 2.1.1. Noise-free original test signal.

4000 3000 2000 1000 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250

0

(a)

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250

3 2 1 0 –1 –2 –3

(b) Fig. 2.1.2. (a) Noise-free bimagnitude as a function of two frequencies. (b) Noise-free biphase as a function of two frequencies.

52 | 2 Unknown noisy signal shape estimation by bispectrum-filtering techniques

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250

6000 5000 4000 3000 2000 1000 0

Fig. 2.1.3. Noisy bimagnitude as a function of two frequencies.

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250

3 2 1 0 –1 –2 –3

Fig. 2.1.4. Noisy biphase as a function of two frequencies.

̃

̃̂ (𝑝, 𝑞) estimates or the The smoothed bimagnitude |𝐵̂̇ 𝑠ind (𝑝, 𝑞)| and biphase 𝛾𝑠ind

̃ 𝐵̂̇ 𝑠ind (𝑝, 𝑞)} parts obtained at the output filtered real ̃ Re{𝐵̂̇ 𝑠ind(𝑝, 𝑞)} and imaginary Im{ of linear, for example, mean or nonlinear, for instance, median filter can be, respectively, written as follows 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 ̃ 󵄨 ̃̂ 󵄨󵄨𝐵̂̇ ̂ (𝑝, 𝑞)} , (𝑝, 𝑞) = MEAN/MED{󵄨󵄨󵄨󵄨𝐵̂̇ 𝑠ind (𝑝, 𝑞)󵄨󵄨󵄨󵄨}, {𝛾𝑠ind 󵄨󵄨 Mean/Median (𝑝, 𝑞)󵄨󵄨󵄨, 𝛾Mean/Median 󵄨 󵄨 󵄨 󵄨

(2.1.1)

̃ Mean/Median {𝐵̂̇ 𝑠ind(𝑝, 𝑞)} ̃Mean/Median {𝐵̂̇ 𝑠ind (𝑝, 𝑞)}, Im Re = MEAN/MED{Re{𝐵̇̂ 𝑠ind (𝑝, 𝑞)}}, {Im{𝐵̇̂ 𝑠ind (𝑝, 𝑞)}} , (2.1.2)

2.1 Smoothing bispectrum estimates by using nonadaptive linear and nonlinear filtering | 53

where MEAN {. . .} denotes the 2-D mean operation, that is, Ĩ mMean {𝐵̂̇ 𝑠ind (𝑝, 𝑞)} =

𝑁 𝑁 (1/(2𝑁 + 1)2 ) ∑𝑛1 =−𝑁 ∑𝑛2 =−𝑁 Im{𝐵̂̇ 𝑠ind (𝑝 + 𝑛1 , 𝑞 + 𝑛2 )}; MED {. . .} denotes the 2-D median operation, that is, ̃ ReMedian {𝐵̂̇ 𝑠ind(𝑝, 𝑞)} = MED{Re{𝐵̂̇ 𝑠ind (𝑝1 , 𝑞1 )}}, 𝑝1 = 𝑝 − 𝑁, . . ., 𝑝 + 𝑁; 𝑞1 = 𝑞 − 𝑁, . . ., 𝑞 + 𝑁; (2𝑁 + 1)(2𝑁 + 1) is the scanning 2-D window

size (WS). Note that the selection of the standard median or mean filters [53] is a rather arbitrary step because we study the possibility of the bispectral estimator improving by typical linear and nonlinear filters. In order to obtain the best possible performance one should optimize the filter for this particular application. In addition to the set of parameters (1.5.7–1.5.11) serving for performance evalu2 ation, we also introduce the MS bias 𝛿out T# 1. . . T# 10 that relates to the reconstructed signal and determines the value of integrated dynamic error 2 𝛿out T# 1. . . T# 10 =

1 𝐼−1 2 ̄̂ ∑ [𝑠(𝑖) − 𝑠T# 1. . . # 10 (𝑖)] . 𝐼 𝑖=0

(2.1.3)

Note that consideration of only mean and median smoothing filters in our computer simulations performed, for example, in [32] is not exhaustive and serves only the purpose of demonstrating that the proposed approach is promising and challenging. It has been shown using different sets of simulation examples that the techniques proposed in [32] may yield a considerably better waveform reconstruction than the conventional bispectrum-based techniques. It should especially be noted that the proposed combined bispectrum-filtering techniques are robust not only to random signal shifts and additive Gaussian noise but to impulsive noise as well. The statistical experiments obtained for a variety of test cases show that the filtering of the real and imaginary parts of noisy bispectrum estimates is more efficient than the processing of bimagnitude and biphase estimates. One reason is that the real and imaginary parts of noisy bispectrum estimates are more “suitable” 2-D processes for filtering than bimagnitude and biphase estimates since the latter functions contain discontinuities that are difficult to preserve after filtering. The selection of the appropriate technique and its parameters among the proposed ones in [32] depends on a particular application, noise environment properties, criterions used and priority of requirements to signal reconstruction system accuracy. For solving the problems of target recognition and classification, it is more important to take into account the output variance rather than MS bias. For the case of signal reconstruction with the aim of further estimation of its parameters, both MS bias and output variance should be considered. In our opinion, the proposed techniques [32] can be useful for solving aforementioned tasks in practical applications in high resolution radar range profile estimations in the cases of a priori unknown signals and properties of noise. Note that the performance of the combined bispectral-filtering methods depends on the following conditions and circumstances:

54 | 2 Unknown noisy signal shape estimation by bispectrum-filtering techniques (a) the statistical properties of noise leaked into a 2-D bispectral domain from 1-D input signal; (b) the peculiarities of behavior of information contributions contained in bispectral estimates that are dependent on the original signal and noise mixture properties; these properties are often a priori unknown; (c) the type and parameters of a 2-D filter applied for smoothing of the bispectral estimate. In such situations, it is intuitively clear that the best performance could be provided in a case of application of adaptive 2-D filters or, at least, adaptive selection of the best nonadaptive filter from a filter bank – taking into account the available information about signal and noise properties that is commonly limited. However, to ensure an opportunity for such adaptation, quite intensive study and analysis of different filter performances for a wide variety of possible signals and noise environments should be carried out. So, our attempts below can be focused on the consideration of initial steps intended for solving this complicated task. First, we started with studying the performance of two other types of nonlinear and nonadaptive filters [43]. Among a large number of modern nonlinear filters the K-nearest neighbor (KNN) filter [54] and the FIR-median hybrid (FMH) filter [53] have been selected for solving a problem of bispectrum-based signal reconstruction. The output of the KNN filter is given as the mean value of the 𝐾𝑁𝑁 (1 ≤ 𝐾𝑁𝑁 ≤ 𝑁 × 𝑁) samples whose values are the closest to the value of the central sample of the filter window with the size of 𝑁×𝑁 samples. The output of the FMH filter is defined as the median value of the set of the FIR subfilters. The scanning window size of the KNN filter can be different while the FMH filter only has the version with the fixed window size of 5 × 5 samples. Two types of scanning window filters listed above do not presume the availability of a priori information about noise type: additive, multiplicative, and so on, and characteristics, for example, variance, like some other types of filters do, for instance, the standard sigma or the local statistic Lee filters. This is, at the same time, good and bad, since, on one hand, the selected two filters can be easily applied to processing the noisy bispectrum estimates without making up preliminary analysis of their statistical properties. On the other hand, one can expect a better performance for the case of more sophisticated filter applications that takes into account the characteristics of noise in bispectrum estimates to be processed. But this is a direction of further investigations in the next paragraphs of this book.

̃ The smoothed bimagnitude |𝐵̂̇ 𝑠 (𝑝, 𝑞)| and biphase 𝛾𝑠̃̂ (𝑝, 𝑞) or the filtered real

̃ ̃ 𝐵̂̇ 𝑠 (𝑝, 𝑞)} parts of bispectrum estimates computed at Re{𝐵̂̇ 𝑠 (𝑝, 𝑞)} and imaginary Im{

2.1 Smoothing bispectrum estimates by using nonadaptive linear and nonlinear filtering | 55

the output of the KNN or FMH filters are, respectively, represented in [43] as

̃ |𝐵̂̇ KNN (𝑝, 𝑞)| = KNN{𝐵̂̇ 𝑠 (𝑝, 𝑞)} , 𝛾̃̂ (𝑝, 𝑞) = KNN{𝛾̂ (𝑝, 𝑞)} , KNN

𝑠

̃ |𝐵̂̇ FMH (𝑝, 𝑞)| = FMH{𝐵̂̇ 𝑠 (𝑝, 𝑞)} , 𝛾̃̂ (𝑝, 𝑞) = FMH{𝛾̂ (𝑝, 𝑞)} , FMH

𝑠

̃ ReKNN {𝐵̂̇ 𝑠 (𝑝, 𝑞)} = KNN{Re[𝐵̂̇ 𝑠 (𝑝, 𝑞)]} , ̃ {𝐵̇̂ (𝑝, 𝑞)} = KNN{Im[𝐵̇̂ (𝑝, 𝑞)]} , Im KNN

𝑠

𝑠

̃ ReFMH {𝐵̂̇ 𝑠 (𝑝, 𝑞)} = FMH{Re[𝐵̂̇ 𝑠 (𝑝, 𝑞)]} , Ĩ m {𝐵̂̇ (𝑝, 𝑞)} = FMH{Im[𝐵̂̇ (𝑝, 𝑞)]} , FMH

𝑠

𝑠

(2.1.4) (2.1.5) (2.1.6) (2.1.7) (2.1.8) (2.1.9) (2.1.10) (2.1.11)

where KNN {. . .} denotes the filtering procedure performed by the 2-D KNN filter; FMH {. . .} denotes the filtering procedure for the 2-D FMH filter. The performances are evaluated and studied for five following different techniques: (a) the conventional BLW technique (Technique #1); (b) the suggested combined technique based on the BLW technique and the bimagnitude and biphase smoothing by the 2-D KNN filter (2.1.4–2.1.5) (Technique #2); (c) the combined technique that uses the BLW technique and the bimagnitude and biphase smoothing by the 2-D FMH filter (2.1.6–2.1.7) (Technique #3); (d) the combined technique based on the BLW technique and smoothing of the real and imaginary bispectral estimate parts by the 2-D KNN filter (2.1.7–2.1.9) (Technique #4); (e) the combined technique based on the BLW technique and the smoothing of the real and imaginary bispectral estimate parts by the 2-D MFH filter (2.1.10–2.1.11) (Technique #5). In order to perform comparative analysis for signal reconstruction performances, we propose to follow two approaches. The first one assumes visual express analysis of the reconstructed signal plots. The second implies more detailed quantitative comparative analysis of the graphs demonstrated the fluctuation state and bias errors at the output of the reconstruction system, as well as the analysis of estimation consistency. The signal reconstructed by conventional Technique #1 for the case of AWGN is shown in Figure 2.1.5. The discrimination of the small intensity object given by small pulse signal magnitude from Figure 2.1.5 by visual analysis is impossible.

56 | 2 Unknown noisy signal shape estimation by bispectrum-filtering techniques

Fig. 2.1.5. Signal restored by Technique #1: AWGN, SNRinp = 0.6.

As clearly seen from Figure 2.1.6, the reliable discrimination of the small intensity object is provided by using the proposed Technique #5. The signal reconstructed in the case of the mixed noise by conventional Technique #1 is shown in Figure 2.1.7. It should be stressed, that the spikes are “smeared-out” in Figure 2.1.7 and reasonably good performance of the conventional Technique #1 is achieved. At the same time, the smoothing of the real and imaginary bispectrum parts by the 2-D FMH nonlinear filter provides a better performance for the proposed combined bispectrum-filtering Technique #5 (see Figure 2.1.8) than for the conventional bispectrum signal reconstruction Technique #1 (see Figure 2.1.7) in the sense of better mixed noise suppression in the reconstructed signal. Better efficiency of the proposed Technique #4 with respect to the proposed Technique #2 using bimagnitude and biphase filtering in the case of the 2-D KNN filter application is demonstrated for the case of mixed noise (compare Figures 2.1.9 and 2.1.10). In order to evaluate the reconstructed signal estimation consistency, the output 2 system ensemble averaged fluctuation variances 𝜎̄out T# 1,4,5 (1.5.9) as well as the MS

2 bias 𝛿out T# 1,4,5 (2.1.3) for Techniques ##1, 4 and 5 were calculated depending on the number of Monte Carlo runs (𝑀 = 1, 10, 50, 100, 256 and 512 realizations). While 2 2 calculating the statistical parameters 𝜎̄ out T# 1,4,5 and 𝛿out T# 1,4,5 , each set of 𝑀 realizations was repeated 𝐾 times, that is, the number of numerical statistical simulation experiments was equal to 𝐾 = 30.

2.1 Smoothing bispectrum estimates by using nonadaptive linear and nonlinear filtering | 57

Fig. 2.1.6. Signal restored by Technique:#5: AWGN, SNRinp = 0.6.

Fig. 2.1.7. Signal restored by Technique #1: 2 mixed noise, 𝜎̄ inp = 0.4; 𝑃pos = 𝑃neg = 12%.

The plots of output variances and MS bias depending on observed realization number 𝑀 for the case of AWGN are represented in Figures 2.1.11 and 2.1.12, respec2 tively (𝜎̄inp = 0.8 = const and the maximal random signal shift deviation of 24 samples were considered). The plots of output variances and MS bias depending upon 𝑀 2 for the case of mixed white AWGN (𝜎̄inp = 0.4 = const and the maximal random signal

58 | 2 Unknown noisy signal shape estimation by bispectrum-filtering techniques

Fig. 2.1.8. Signal restored by Technique #5: 2 mixed noise, 𝜎̄ inp = 0.4, 𝑃pos = 𝑃neg = 12%.

Fig. 2.1.9. Signal restored by Technique #2 for 𝑁 × 𝑁 = 7 × 7; 𝐾𝑁𝑁 = 24 samples: mixed 2 noise, 𝑃pos = 𝑃neg = 12%; 𝜎̄ inp = 0.4.

shift deviation of 24 samples were exploited) and impulsive noise (𝑃pos = 𝑃neg = 6%) are given in Figures 2.1.13 and 2.1.14, respectively. Analysis of these plots shows that the output variance values (see Figures 2.1.11 and 2.1.13) gradually approach to zero values, that is, they tend to be strictly consistent in mathematical sense. This means that for both conventional Technique #1 and the proposed Techniques ##4 and 5 (as well as for Techniques ##2 and 3 for which the plots have similar behavior but are not presented in Figures 2.1.11 and 2.1.13) there 2 exists some 𝑀0 that for 𝑀 ≥ 𝑀0 the random fluctuation errors 𝜎̄out T# 1,4,5 can be considered negligibly small. Therefore, for practical applications, it is recommended to find such 𝑀0 that satisfies some reasonable preset requirements to the reconstruction system accuracy.

2.1 Smoothing bispectrum estimates by using nonadaptive linear and nonlinear filtering |

59

Fig. 2.1.10. Signal restored by Technique #4 for 𝑁 × 𝑁 = 7 × 7; 𝐾𝑁𝑁 = 24 samples: mixed 2 noise, 𝑃pos = 𝑃neg = 12%; 𝜎̄ inp = 0.4.

0.6 T#1 T#4

0.5

Output variance

T#5 0.4 0.3 0.2 0.1 0 1

10 50 100 256 512 Number of realizations

Fig. 2.1.11. Output variance as a function of realization number 𝑀 (AWGN).

MS bias errors for the proposed bispectrum-filtering Techniques ##4 and 5 are practically the same as for the conventional Technique #1 (see Figures 2.1.12 and 2.1.14 for comparison) for realizations number 𝑀 ≥50 evaluated for AWGN (see Figure 2.1.12) and they slightly differ for the case of mixed noise (see Figure 2.1.14). 2 2 In Tables 2.1.1 and 2.1.2 the values of 𝜎̄out T# 1. . . 5 and 𝛿out T# 1. . . 5 depending on the 2 2 𝜎̄inp /SNRinp for the case of Gaussian and on the probability 𝑃pos = 𝑃neg (𝜎̄inp = 0.4 was fixed value in case of mixed noise) for the case of mixed noise are represented, respectively. The scanning window size for different filters was fixed to 𝑁 × 𝑁 = 5 × 5,

60 | 2 Unknown noisy signal shape estimation by bispectrum-filtering techniques 0.14 T#1 T#4

0.12

T#5 0.1

Bias

0.08 0.06 0.04 0.02 0 1

10 50 100 256 512 Number of realizations

Fig. 2.1.12. MS bias as a function of realization number 𝑀 (AWGN).

0.3 T#1 T#4 0.25

T#5

Output variance

0.2

0.15

0.1

0.05

0 1

10 50 100 256 512 Number of realizations

Fig. 2.1.13. Output variance as a function of realization number 𝑀 (mixed noise).

for KNN filter 𝐾𝑁𝑁 = 12. These data have been obtained for fixed number of 𝑀 = 256 realizations and 𝐾 = 30 experiments. 2 As seen from Table 2.1.1 the fluctuation error 𝜎̄out T# 1 decreases by approximately three times with the increase of SNRinp by eight times. At the same time, the MS bias 2 𝛿out T# 1 value decreases proportionally to increasing of SNRinp .

0.0363 0.0294 0.0134 0.0108

3.2/0.3 1.6/0.6 0.8/1.2 0.4/2.3

0.0322 0.0164 0.00846 0.00510

2 𝛿out T#1

0.0435 0.0186 0.00609 0.00335

0.0264 0.0154 0.00855 0.00540

2 𝛿out T#2

Technique #2 2 𝜎̄out T#2

0.0285 0.0110 0.00516 0.00226

0.0306 0.0197 0.0124 0.00733

2 𝛿out T#3

Technique #3 2 𝜎̄out T#3

0.0248 0.0100 0.00395 0.00251

0.00496 0.00451 0.00375 0.00308

24 12 6 3

0.0123 0.0103 0.0087 0.00746

2 𝛿out T#1

Technique #1

2 𝜎̄out T#1

𝑃pos = 𝑃neg , % 0.00358 0.00341 0.00326 0.00432

0.0194 0.0143 0.00937 0.00750

2 𝛿out T#2

Technique #2

2 𝜎̄out T#2

0.0025 0.00232 0.00217 0.00226

0.0328 0.0253 0.0185 0.0140

2 𝛿out T#3

Technique #3

2 𝜎̄out T#3

2 𝛿out T#4

0.00272 0.00227 0.00195 0.00229

0.019 0.0134 0.00927 0.00683

Technique #4 2 𝜎̄out T#4

0.0259 0.0136 0.00712 0.00520

2 𝛿out T#4

Technique #4 2 𝜎̄out T#4

2 Table 2.1.2. Numerical simulation results for the case of mixed Gaussian (𝜎̄ inp = 0.4 = const) and impulsive noise.

2 𝜎̄out T#1

2 𝜎̄inp /SNRinp

Technique #1

Table 2.1.1. Numerical simulation results for the case of AWGN.

0.00238 0.00201 0.00172 0.00184

0.0329 0.0250 0.0178 0.0134

2 𝛿out T#5

0.0344 0.0211 0.0128 0.00763

Technique #5 2 𝜎̄out T#5

0.0189 0.00885 0.00352 0.00193

2 𝛿out T#5

Technique #5 2 𝜎̄out T#5

2.1 Smoothing bispectrum estimates by using nonadaptive linear and nonlinear filtering |

61

62 | 2 Unknown noisy signal shape estimation by bispectrum-filtering techniques 0.1 T#1 0.09

T#4 T#5

0.08 0.07

Bias

0.06 0.05 0.04 0.03 0.02 0.01 0 1

10

50

100 256 512

Number of realizations

Fig. 2.1.14. MS bias as a function of realization number 𝑀 (mixed noise).

2 The fluctuation error 𝜎̄out T# 2 decreases by approximately thirteen times with the 2 increase of SNRinp by eight times. The MS bias 𝛿out T# 2 is about the same level as 2 𝛿out T# 1 . The KNN filter with window size of 𝑁 × 𝑁 = 5 × 5 and 𝐾𝑁𝑁 = 12 produces small dynamic errors in comparison to the FMH filter. 2 The fluctuation errors 𝜎̄out T# 3 are less than for Technique #1, and the MS bias val2 ues 𝛿out T# 3 are only slightly worse than for Technique #1 because any smoothing filter, clearly, always introduces some additional dynamic errors. It should be stressed, that the proposed strategy applied to smoothing of the real and imaginary bispectral estimation parts by 2-D KNN or FMH filters (compare the data for Techniques ##4 and 5 to the data for all other Techniques) provide the best results in the sense of producing smaller fluctuation errors. At the same time, the MS bias values for Technique #5 are the worst among the considered Techniques. For low SNRinp , for example, for SNRinp = 0.3 and 0.6, the MS bias values are minimal in the case of the KNN filter application (see the data for Technique#2). For the case of mixed noise, the minimum MS bias values are observed for the conventional Technique #1 (see Table 2.1.2). The smoothing of the real and imaginary bispectrum parts by a 2-D FMH nonlinear filter (see the data in Table 2.1.2 evaluated for the proposed Technique #5) still provides the minimum fluctuation errors in com2 parison to Techniques ##1–4. As one can see, the fluctuation errors 𝜎̄out T# 2–5 tend to decrease when the values 𝑃pos = 𝑃neg reduce. But for minimal considered value 𝑃pos = 𝑃neg = 3%, these errors slightly increase. This peculiarity, in our opinion, can be caused, on one hand, due to complicated contribution (“re-distribution”) of Gaus-

2.2 Statistical properties of bispectrum estimate contaminated by noise

| 63

sian and non-Gaussian noise contributions contained in the bispectrum estimate and, on the other hand, by individual peculiarities of the investigated types of nonlinear filters that smooth Gaussian and impulsive noise components in different manner. This problem is one more distinctive subject of our investigation and it will be discussed below.

2.2 Statistical properties of bispectrum estimate contaminated by noise It should be noted that the performance of digital filtering applied in bispectral domains largely depends on the statistical property of complex-valued bispectrum estimates and, first of all, on its PDF shape and stationary state. This property has not been studied yet and not considered in the literature. Because of this, we have paid attention to the problem of statistical assessment of the behavior of complex-valued noisy bispectrum estimates in [8, 45]. First, let us consider the error contribution 𝐵̂̇ err (𝑝, 𝑞) caused by noise at the input of the signal processing system and leaked in the bispectrum estimate (1.4.8). This component 𝐵̂̇ err (𝑝, 𝑞) contains six signal-dependent and one nonsignal-dependent terms. Analysis of these error contributions caused by noise leakage from the signal reconstruction system input performed in [8] permits noting the following important peculiarities. (1) With increasing realization number 𝑀 participating in ensemble averaging, the ∗ last nonsignal-dependent term 𝐸[𝑁̇ 𝑚 (𝑝)𝑁̇ 𝑚 (𝑞)𝑁̇ 𝑚 (𝑝 + 𝑞)] in formula (1.4.8) (𝑚) asymptotically tends to zero if AWGN {𝑛 (𝑖), 𝑖 = 0, 1, . . ., 𝐼 − 1; 𝑚 = 1, 2, . . ., 𝑀} in (1.4.1) is supposed to be of zero mean. It has been demonstrated in [8] that the real and imaginary parts of this complex-valued random process are not Gaussian (see the histogram in Figure 2.2.1) although for large 𝑀 it tends to Guassian PDF according to the central limit theorem (see the histogram in Figure 2.2.2). These histograms have been computed for SNRinp = 0, that is, for the signal-absence case when only the last term in (1.4.8) is of nonzero value. (2) Since AWGN in the considered problem is supposed to be of zero mean value, ∗ hence, for large 𝑀 the terms 𝐸[𝑁̇ 𝑚 (𝑝 + 𝑞)𝑒−𝑗2𝜋𝜏𝑚 (𝑝+𝑞) ] = 𝐸[𝑁̇ 𝑚 (𝑞)𝑒𝑗2𝜋𝜏𝑚 𝑞 ] = 𝑗2𝜋𝜏 𝑝 𝐸[𝑁̇ 𝑚 (𝑝)𝑒 𝑚 ] → 0. (3) The rest of the noise terms contained in (1.4.8) are the “complex-valued signal Fourier spectrum depending” and “random shift 𝜏(𝑚) depending”. In other words, they have signal-dependent properties and result in the presence of multiplicative behavior components in the bispectral estimate. More detailed analysis of bispectrum statistical characteristics has been carried out th in [45]. The following noisy component 𝐵(𝑚) 𝑛 (𝑝, 𝑞) contained in 𝑚 arbitrary realiza-

64 | 2 Unknown noisy signal shape estimation by bispectrum-filtering techniques 400 350 300 250 200 150 100 50 5513.235

4335.381

3157.527

1979.672

882.126

–1.264

–938.194

–2116.048

–3374.211

–4659.143

–5944.075

–7095.160

0

Fig. 2.2.1. Histogram of bispectrum real part values for signal-absence case, 𝑀 = 1.

400 350 300 250 200 150 100 50 525.153

389.616

250.415

114.877

–20.660

–159.861

–310.051

–460.241

–610.432

–760.622

–910.812

–1071.992

–1229.509

0

Fig. 2.2.2. Histogram of bispectrum real part values for signal-absence case, 𝑀 = 50.

tion in (1.4.1) can be written as [45] (𝑚)

𝑝

+ 𝑆(𝑝)𝑆∗ (𝑝 + 𝑞)𝑁(𝑚) (𝑞)𝑒𝑗𝜏

(𝑚)

𝑝

+ 𝑆(𝑝)𝑁(𝑚) (𝑞)𝑁(𝑚)∗ (𝑝 + 𝑞)𝑒−𝑗𝜏

(𝑚)∗ 𝐵(𝑚) (𝑝 + 𝑞)𝑒−𝑗𝜏 𝑛 (𝑝, 𝑞) = 𝑆(𝑝)𝑆(𝑞)𝑁

+ 𝑆(𝑞)𝑆∗ (𝑝 + 𝑞)𝑁(𝑚) (𝑝)𝑒𝑗𝜏

+ 𝑆(𝑞)𝑁(𝑚) (𝑝)𝑁(𝑚)∗ (𝑝 + 𝑞)𝑒−𝑗𝜏

(𝑚)

𝑞

(𝑚)

𝑞 (𝑚)

𝑝

+ 𝑆∗ (𝑝 + 𝑞)𝑁(𝑚) (𝑝)𝑁(𝑚) (𝑞)𝑒𝑗𝜏

(𝑚)

(𝑝+𝑞)

+ 𝑁(𝑚) (𝑝)𝑁(𝑚) (𝑞)𝑁(𝑚)∗ (𝑝 + 𝑞) . (2.2.1)

2.2 Statistical properties of bispectrum estimate contaminated by noise

| 65

Consider the last term 𝑁(𝑚) (𝑝)𝑁(𝑚) (𝑞)𝑁(𝑚)∗ (𝑝 + 𝑞) in (2.2.1) to assess its statisti(𝑚) (𝑚) cal properties. Let us use the following notations: 𝑁(𝑚) (𝑝) = 𝑁Re (𝑝) + 𝑗𝑁Im (𝑝), (𝑚) (𝑚) (𝑚) (𝑚) (𝑚) (𝑚)∗ 𝑁 (𝑞) = 𝑁Re (𝑞) + 𝑗𝑁Im (𝑞), and 𝑁 (𝑝 + 𝑞) = 𝑁Re (𝑝 + 𝑞) − 𝑗𝑁Im (𝑝 + 𝑞), where all Re and Im noise components are zero mean and Gaussian. As a result, we obtain (𝑚) (𝑚) (𝑚) Re{𝑁(𝑚) (𝑝)𝑁(𝑚) (𝑞)𝑁(𝑚)∗ (𝑝 + 𝑞)} = 𝑁Re (𝑝)𝑁Re (𝑞)𝑁Re (𝑝 + 𝑞) (𝑚) (𝑚) (𝑚) (𝑝)𝑁Im (𝑞)𝑁Im (𝑝 + 𝑞) + 𝑁Re (𝑚) (𝑚) (𝑚) (𝑝)𝑁Re (𝑞)𝑁Im (𝑝 + 𝑞) + 𝑁Im (𝑚) (𝑚) (𝑚) (𝑝)𝑁Im (𝑞)𝑁Re (𝑝 + 𝑞) , − 𝑁Im

(2.2.2)

(𝑚) (𝑚) (𝑚) Im{𝑁(𝑚) (𝑝)𝑁(𝑚) (𝑞)𝑁(𝑚)∗ (𝑝 + 𝑞)} = 𝑁Im (𝑝)𝑁Im (𝑞)𝑁Im (𝑝 + 𝑞) (𝑚) (𝑚) (𝑚) + 𝑁Im (𝑝)𝑁Re (𝑞)𝑁Re (𝑝 + 𝑞) (𝑚) (𝑚) (𝑚) (𝑝)𝑁Im (𝑞)𝑁Re (𝑝 + 𝑞) + 𝑁Re (𝑚) (𝑚) (𝑚) (𝑝)𝑁Re (𝑞)𝑁Im (𝑝 + 𝑞) . − 𝑁Re

(2.2.3)

Both Re and Im parts in (2.2.2) and (2.2.3) are represented as sums of four terms that are the products of three zero-mean Gaussian random variables. Our attempts to analytically evaluate the PDF of the eight triple product random values like (𝑚) (𝑚) (𝑚) 𝑁Re/Im (𝑝)𝑁Re/Im (𝑞)𝑁Re/Im (𝑝 + 𝑞) failed and we had to employ numerical simulation for this purpose. It has been clearly demonstrated that the PDF of such random variables is non-Gaussian. An example of the histogram computed in [45] is shown in Figure 2.2.3. The estimated kurtosis values are within the limits, about 7. . . 7.5, and this also strongly evidences in favor of non-Gaussianity of PDF.

400 300 200 100 0 –10 –8 –6 –4 –2 0

2

4

6

8

Fig. 2.2.3. Histogram of the noise terms contained in ((2.2.2)) and (2.2.3).

Both equations (2.2.2) and (2.2.3) contain the sums of four random variables with non-Gaussian distributions and the Re and Im components of the term 𝑁(𝑚) (𝑝)𝑁(𝑚) (𝑞)𝑁(𝑚)∗ (𝑝 + 𝑞) are characterized by non-Gaussian PDFs. According to our computer simulations [45], the estimated kurtosis values for Re and Im com-

66 | 2 Unknown noisy signal shape estimation by bispectrum-filtering techniques ponents of this term are within the limits of 2. . . 5, that is, they are positive, and this confirms aforementioned properties of its PDF. Moreover, let us demonstrate that the fourth, fifth and sixth terms in (2.2.1) are also characterized by non-Gaussian behavior of Re and Im part distributions. For this purpose let us analyze in detail one of these terms that includes the product of two (𝑚) noise DFTs, for example, the fourth term 𝑆(𝑝)𝑁(𝑚) (𝑞)𝑁(𝑚)∗ (𝑝 + 𝑞)𝑒−𝑗𝜏 𝑝 . Its Re and Im parts can be expressed, respectively, as [45]

Re[𝑆(𝑝)𝑁(𝑚) (𝑞)𝑁(𝑚)∗ (𝑝 + 𝑞)𝑒−𝑗𝜏

(𝑚)

𝑝

]

(𝑚) (𝑚) = 𝑁Re (𝑞)𝑁Re (𝑝 + 𝑞){𝑆Re (𝑝) cos 𝜏(𝑚) 𝑝 + 𝑆Im (𝑝) sin 𝜏(𝑚) 𝑝} (𝑚) (𝑚) (𝑞)𝑁Im (𝑝 + 𝑞){𝑆Re (𝑝) cos 𝜏(𝑚) 𝑝 + 𝑆Im (𝑝) sin 𝜏(𝑚) 𝑝} + 𝑁Im (𝑚) (𝑚) (𝑞)𝑁Im (𝑝 + 𝑞){𝑆Re (𝑝) sin 𝜏(𝑚) 𝑝 − 𝑆Im (𝑝) cos 𝜏(𝑚) 𝑝} + 𝑁Re (𝑚) (𝑚) (𝑞)𝑁Re (𝑝 + 𝑞){𝑆Re (𝑝) sin 𝜏(𝑚) 𝑝 − 𝑆Im (𝑝) cos 𝜏(𝑚) 𝑝} , + 𝑁Im

Im[𝑆(𝑝)𝑁(𝑚) (𝑞)𝑁(𝑚)∗ (𝑝 + 𝑞)𝑒−𝑗𝜏

(𝑚)

𝑝

(2.2.4a)

]

(𝑚) (𝑚) = 𝑁Re (𝑞)𝑁Re (𝑝 + 𝑞){𝑆Im (𝑝) cos 𝜏(𝑚) 𝑝 − 𝑆Re (𝑝) sin 𝜏(𝑚) 𝑝} (𝑚) (𝑚) + 𝑁Im (𝑞)𝑁Im (𝑝 + 𝑞){𝑆Im (𝑝) cos 𝜏(𝑚) 𝑝 − 𝑆Re (𝑝) sin 𝜏(𝑚) 𝑝} (𝑚) (𝑚) (𝑞)𝑁Re (𝑝 + 𝑞){𝑆Im (𝑝) sin 𝜏(𝑚) 𝑝 + 𝑆Re (𝑝) cos 𝜏(𝑚) 𝑝} − 𝑁Im (𝑚) (𝑚) (𝑞)𝑁Im (𝑝 + 𝑞){𝑆Im (𝑝) sin 𝜏(𝑚) 𝑝 + 𝑆Re (𝑝) cos 𝜏(𝑚) 𝑝} , + 𝑁Re (𝑚)

(2.2.4b)

(𝑚)

where 𝑆Re (. . .) and 𝑆Im (. . .) are the Re and Im parts of the signal DFT component, respectively. Now let us study the distributions related to both Re (2.2.4a) and Im (2.2.4b) components. For this objective we consider two PDFs 𝑓1 (𝑥) and 𝑓1 (𝑦) referred to two independent Gaussian variables 𝑥 and 𝑦 and expressed as

𝑓1 (𝑥) =

1 𝑥2 ⋅ exp (− 2 ) , 2𝜎𝑋 √2𝜋 ⋅ 𝜎𝑋

𝑓1 (𝑦) =

𝑦2 1 ⋅ exp (− 2 ) , (2.2.5) 2𝜎𝑌 √2𝜋 ⋅ 𝜎𝑌

where 𝜎𝑋 and 𝜎𝑌 are the standard deviations of the corresponding random variables 𝑥 and 𝑦. The PDF 𝑔(𝑧) of the product 𝑥𝑦 = 𝑧 can be written in the form [55] as ∞

𝑔(𝑧) = ∫ 𝑓1 (𝑥) ⋅ 𝑓2 (𝑦) −∞

𝑑𝑥 . |𝑥|

(2.2.6)

2.3 Novel techniques developed for improving noisy bispectrum estimates | 67

After mathematical transformations, the formula (2.2.6) can be represented finally as

1 𝑧 𝑔(𝑧) = 𝐶 ⋅ √ ⋅ exp (− ), 𝑧 𝜎𝑋 𝜎𝑌

(2.2.7)

where 𝐶 = (1/√2𝜋𝜎𝑋 𝜎𝑌 ). (𝑚) (𝑚) It is seen from the formula (2.2.7) that the PDF of the product like 𝑁Re (𝑞)𝑁Re (𝑝+ 𝑞) are of non-Gaussian state and heavy-tailed. Both (2.2.4a) and (2.2.4b) contain four such terms multiplied by the corresponding factors like [𝑆Re (𝑝) cos 𝜏(𝑚) 𝑝 + 𝑆Im (𝑝) sin 𝜏(𝑚) 𝑝] that do not change the form of PDF (2.2.7). Although four random (𝑚) (𝑚) variables like 𝑁Re (𝑞)𝑁Re (𝑝 + 𝑞){𝑆Re (𝑝) cos 𝜏(𝑚) 𝑝 + 𝑆Im (𝑝) sin 𝜏(𝑚) 𝑝} are summed up, the PDFs of the fourth, fifth and sixth terms in equation (2.2.1) are still characterized by non-Gaussian distributions of Re and Im parts. The first, second and third terms in (2.2.1) possess complex-valued Gaussian behavior. However, due to non-Gaussianity of the latter four terms, the PDFs of Re and Im parts of 𝐵(𝑚) 𝑛 (𝑝, 𝑞) are, in general, non-Gaussian. Only if the input SNR is large, that is, if |𝑆(..)| ≫ 𝜎𝐵𝑁 (where 𝜎𝐵𝑁 is the Re or Im root mean square of 𝐵(𝑚) 𝑛 (. . .)), the PDF of Re or Im parts of 𝐵(𝑚) (. . .) tends to Gaussian shape. 𝑛 It should be stressed that under such conditions conventional bispectrum estimation techniques [2] based on ensemble averaging procedure (1.3.6) operate in optimal manner. However, in the case of low input SNR, as it can be seen from the considerations above, the term 𝐵(𝑚) 𝑛 (𝑝, 𝑞) in (2.2.1) becomes non-Gaussian and this provokes considerable errors in bispectral estimates. Note that this peculiarity has not been discussed in literature and was first studied in [45]. Therefore, due to non-Gaussianity valid for low SNR and nonstationary state of the real and imaginary bispectrum estimate parts, other methods for enhancement of noisy bispectrum estimates should be used and they can be expected to be more efficient in the sense of obtaining better bispectrum estimator performance and, consequently, better performance of signal reconstruction systems. Several novel approaches developed in [44–48] and dedicated to improving bispectrum estimates will be described in the next Subsection.

2.3 Novel techniques developed for improving noisy bispectrum estimates First, we begin with describing an approach based on processing the real and imaginary parts of bispectrum estimates by using vector filters [44]. The motivation for using vector filters is that they can process multichannel data and the real and imaginary bispectrum estimate parts can be considered as two-channel 2-D data (digital images).

68 | 2 Unknown noisy signal shape estimation by bispectrum-filtering techniques Since statistical characteristics of noise in (2.2.1) depend on original signal Fourier spectrum – usually supposed to be unknown – then it is difficult to apply the vector filters that commonly take into account statistical characteristics of noise in processed data. Therefore, at our disposal, we only have quite a limited set of vector filters that can be applied. Among them we have considered the standard vector median filter (denoted as VMF) [57]. Then the vector sample is x𝑖 = {Re(𝐵̂̇ 𝑥 (𝑝𝑙 , 𝑞𝑗 )); Im(𝐵̂̇ 𝑥 (𝑝𝑙 , 𝑞𝑗 ))}, 𝑖 = 1, . . ., 𝑁𝑥𝑁 and 𝑝𝑙 varies from 𝑝−(𝑁−1)/2 to 𝑝+(𝑁−1)/2, 𝑞𝑗 varies from 𝑞−(𝑁−1)/2 to 𝑞 − (𝑁 + 1)/2 for the scanning 2-D window of size 𝑁 × 𝑁 centered on the (𝑝, 𝑞)th bispectrum estimate sample. Two types of directional filters have been used in our experiments [44]. The choice of directional vector filters is motivated by desire to preserve phase information in bispectral data and, as known, directional vector filters often perform this task (in the sense of color preservation in RGB images) better than conventional vector filters [56]. The basic idea of vector directional filters is to order vector samples in accordance to the distance measure that is equal to the angle between the vectors in the scanning window. According to this definition, vector directional median filter can be represented as 𝑁

𝑁

𝑘=1

𝑘=1

xvdm = { ∑ 𝐴 (xvdm , x𝑘 ) ≤ ∑ 𝐴 (x𝑚 , x𝑘 ) ; 𝑚 = 1, 𝑁; vdm ∈ 1, 𝑁} ,

(2.3.1)

where 𝐴(x𝑖 , x𝑘 ) denotes the absolute value of angle between the vectors x𝑖 and x𝑘 . In order to provide computational efficiency improvement of directional ordering of vector samples, we propose using a distance measure that is proportional, but not equal, to the angles between vector samples. Such measure can be calculated as a distance between vectors normalized on their lengths as: 𝑁 󵄩 󵄩󵄩 x x 󵄩󵄩󵄩 𝑝ℎ 𝐷𝑘,𝑚 = ∑ 󵄩󵄩󵄩 𝑘 − 𝑚 󵄩󵄩󵄩 , 󵄩 |x𝑚 | 󵄩󵄩 𝑚=1 󵄩 |x𝑘 |

(2.3.2)

where ‖⋅‖ is either 𝐿 1 or 𝐿22 norm, |x| denotes the length of vector x. Despite the fact that vector directional median filters use only directional information for ordering vector samples, the data processing also changes the amplitude of filtered vectors. Assuming the condition that filter output should not change the amplitude of central window sample and using the distance measure (2.3.2) for ordering, we propose the following definition of median phase filter (MPF)

xmpf = |x𝑐 | ⋅

x𝑝 |x𝑝 |

,

(2.3.3)

where x𝑐 is the window central sample; x𝑝 corresponds to vector median calculated from direction ordered samples as 𝑝ℎ

𝑝ℎ

x𝑝 = {𝐷𝑝,𝑘 ≤ 𝐷𝑚,𝑘 ; 𝑚 = 1, 𝑁; 𝑝 ∈ 1, 𝑁} .

(2.3.4)

2.3 Novel techniques developed for improving noisy bispectrum estimates

| 69

Applying the same approach to vector 𝛼-trimmed filter [57] we can define 𝛼-trimmed phase filter (TPF) as

xapf = |x𝑐 | ⋅

x𝑝𝑎

|x𝑝𝑎 |

,

(2.3.5)

where x𝑝𝑎 is the 𝛼-trimmed mean output value computed as

x𝑝𝑎 =

𝑁(1−𝛼)

1 ⋅ ∑ X (𝑘) . 𝑁(1 − 𝛼) 𝑘=1 𝑝

(2.3.6) 𝑝ℎ

X𝑝(𝑘) in (2.3.6) is the 𝑘th vector sample ordered according to the distance 𝐷𝑘,𝑚 (2.3.2). A parameter 𝛼 permits to vary the filter noise suppression and robust properties similarly to the vector 𝛼-trimmed filter. Below we consider the case when 𝛼 = 0.5. It is worth noting that for all aforementioned vector filters their performance depends on the scanning window size and the selected norm denoted as 𝐿 1 and 𝐿22 . For quantitative performance evaluation of the suggested techniques, the fluctuation variance and MS bias computed in the form of (1.5.7) and (2.1.3), respectively, have been used as quality criterions. Note that the signal shape estimates reconstructed by noisy bispectrum estimates are usually biased. The ideal main objective is to minimize both fluctuation variance and MS bias at the same time or, at least, to reduce their contributions maximally. However, commonly this is a conflicting problem. Table 2.3.1 contains the results computed for a wide range of input SNR variation [44]. All three vector filters have the same scanning WS of 5 × 5 samples. The test signal is modeled by usage of two short pulses with rectangular shapes and different amplitudes. In our simulations, the observed process contains 𝐼 = 256 samples, the realization number 𝑀 = 256, the experiment repetition number 𝐾 = 30. Computer simulation results given in Table 2.3.1 demonstrate the following. (1) The application of the considered vector filters to bispectrum estimation can lead to both improvement and worsening of the reconstruction system performance. In 2 particular, the VMF application commonly results in larger output variance 𝜎̄ out than for the conventional BLW method. At the same time, the bispectrum process2 than the conventional BLW technique. ing by VMF can produce smaller 𝛿out (2) The results for the MPF (𝐿 1 norm) application cases are almost always better than 2 2 for the BLW technique: both smaller fluctuation error 𝜎̄out , and MS bias 𝛿out are provided. In general, the use of 𝐿 1 norm for the MPF and TPF produces better results than in the case of 𝐿22 norm used. This is, probably, because the vector filters based on 𝐿 1 norm distort details in bispectrum real and imaginary parts to a lesser degree. Besides, its usage in the considered case can be motivated by the fact that noise in bispectrum estimations can be non-Gaussian (see previous Subsection 2.2). (3) TPF possesses “intermediate place” properties between VMF and MPF. The application of TPF to the bispectrum estimation results in some improvement of signal

70 | 2 Unknown noisy signal shape estimation by bispectrum-filtering techniques Table 2.3.1. Computer simulation results [44]. Filter type

Input SNR

Output variance

MS Bias

VMF, norm 𝐿 1

0.25 0.5 1.0 2.0 5.0

0.0724 0.0285 0.0132 0.0072 0.0039

0.0633 0.0226 0.0170 0.0129 0.0104

MPF, norm 𝐿 1

0.25 0.5 1.0 2.0 5.0

0.0257 0.0094 0.0046 0.0031 0.0030

0.0298 0.0256 0.0225 0.0215 0.0111

0.25 0.5 1.0 2.0 5.0

0.0371 0.0174 0.0114 0.0031 0.0030

0.0573 0.0391 0.0307 0.0173 0.0162

0.25 0.5 1.0 2.0 5.0

0.0280 0.0089 0.0039 0.0020 0.0015

0.0373 0.0228 0.0195 0.0178 0.0174

0.25 0.5 1.0 2.0 5.0

0.0314 0.0119 0.0061 0.0045 0.0026

0.0417 0.0317 0.0213 0.0164 0.0107

0.25 0.5 1.0 2.0 5.0

0.0343 0.0152 0.0077 0.0054 0.0022

0.0632 0.0381 0.0243 0.0155 0.0102

MPF, norm 𝐿22

TPF, norm 𝐿 1

TPF, norm 𝐿22

BLW

processing system performance in comparison to the BLW technique for the cases of low input SNR. Summarizing the results obtained for vector filters in combined bispectrum-filtering strategy, two conclusions can be drawn. First, there is no improvement due to using vector filters in comparison to separate processing of bispectrum real and imaginary components by nonlinear filters. This is not surprising since vector filters produce benefits in comparison to their scalar counterparts if there is essential correlation of multichannel data components [56, 57]. However, bispectrum real and imaginary components are not strongly correlated. Second, the obtained results indicate that nonadaptive filters (either scalar or vector) are unable to produce stable improvement of combined bispectrum-filtering processing in comparison to the conventional BLW technique. Based on the conclusions according to which the real and imaginary bispectrum components become non-Gaussian and nonstationary state, it can be expected that

2.3 Novel techniques developed for improving noisy bispectrum estimates | 71

this peculiarity is also valid for the real and imaginary parts of signal Fourier spectrum estimates recovered from bispectrum. The corresponding investigations have been carried out in [46]. From the earlier experience, it can be expected that the statistical properties of leaked noise depend on many factors like an original signal shape (unknown for our case), the number of realizations, variance of input noise, and so on. Based on the results of these investigations, the selection of an appropriate filtering technique becomes more purposeful and effective. It was demonstrated in [46] that the modified adaptive discrete cosine transform (DCT) based filters can be rather successfully used for improving the signal Fourier spectrum estimate recovered from bispectrum. Besides, we have come to the conclusion that it is necessary to study the bispectrum-based processing performance for a wider set of test signals. This is needed to be sure that the designed methods are effective for different characteristics of input signals. Three test signals 𝑠(𝑖) in the form of two pulses with rectangular shapes, various lengths Δ𝑡1,2,3 , different amplitudes of 𝐴 1 = 2 and 𝐴 2 = 6 and the mutual pulse shift that was the same for all three test signals and equal to Δ𝑡12 = 5 samples have been studied in simulations [46] (see Figures 2.3.1 (a), 2.3.2 (a) and 2.3.3 (a)). The pulse length Δ𝑡 has been varied. It was equal to Δ𝑡1 = 3 (signal #1), Δ𝑡2 = 7 (signal #2) and Δ𝑡3 = 11 (signal #3) samples. The test signal powers 𝑃𝑠 are of the values of 0.46, 1.06 and 1.60, respectively. 6

6

x(m)(i)

s(i) 4 4 2 0

2

i 0

–2

i

0 0

(a)

50

100

150

200

250

0

50

100

150

200

250

(b)

Fig. 2.3.1. The noise-free test signal #1 (Δ𝑡1 = 3 samples) (a) and a noisy realization 𝑥𝑚 (𝑖), SNRinp = 0.46 (b).

The original noise-free test signals were modeled as the sequences of nonnegative real values generated within the array of 𝐼 = 256 samples. We have simulated 𝑀 = 200 realizations by 200 Monte Carlo runs performed for each separate 𝑘th statistical experiment, 𝑘 = 1, . . ., 𝐾. The corresponding noisy and randomly shifted real-

72 | 2 Unknown noisy signal shape estimation by bispectrum-filtering techniques 6

6

s(i)

x(m)(i)

4

4

2 2

0 i –2

0 0

50

100

150

200

i

250

(a)

50

100

150

200

250

(b)

Fig. 2.3.2. The noise-free test signal #2 (Δ𝑡2 = 7 samples) (a) and a noisy realization 𝑥𝑚 (𝑖), SNRinp = 1.05 (b).

6 x(m)(i)

6

s(i)

4

4

2 2

0 i –2

0 0

50

100

150

(a)

200

250

0

i 0

50

100

150

200

250

(b)

Fig. 2.3.3. The noise-free test signal #3 (Δ𝑡3 = 11 samples) (a) and a noisy realizations 𝑥𝑚 (𝑖), SNRinp = 1.60 (b).

izations 𝑥𝑚 (𝑖) for the test signals obtained for different input SNRs (SNRinp ) are shown in Figures 2.3.1 (b), 2.3.2 (b) and 2.3.3 (b), respectively. As the first step in analysis of statistical characteristics of noise contained in

Re{𝑆̂̇ bisp (𝑟)} and Im{𝑆̂̇ bisp (𝑟)} parts, the following variance estimates were computed for noisy signal Fourier spectra recovered from bispectrum as

𝜎2ReEns (𝑟)

2

1 𝐾 = ∑ [Re {𝑆̂̇𝑘Bisp (𝑟)} − Re {𝑆̂̇Bisp (𝑟)}] , 𝐾 − 1 𝑘=1

𝜎2ImEns (𝑟) =

(2.3.7a)

2

1 𝐾 ∑ [Im {𝑆̂̇𝑘bisp (𝑟)} − Im {𝑆̂̇bisp (𝑟)}] , 𝐾 − 1 𝑘=1

(2.3.7b)

2.3 Novel techniques developed for improving noisy bispectrum estimates

| 73

Re ŜBisp(r)

150

σ–2Re Ens(r)

100

50

0

50

100

150

200

250

r

(a) 1 W(r) 0

Fig. 2.3.4. Signal #1 (Δ𝑡 = 3 samples):

50

100

150

200

250

r

Re{𝑆̂̇𝑘bisp (𝑟)} and 𝜎2ReEns (𝑟) (a) and the W-test results evaluated for SNRinp = 0.46 (b).

(b)

where

1 𝐾 Re{𝑆̂̇ bisp (𝑟)} = ∑ Re{𝑆̂̇𝑘bisp (𝑟)} , 𝐾 𝑘=1

1 𝐾 Im{𝑆̂̇bisp (𝑟)} = ∑ Im{𝑆̂̇ 𝑘bisp (𝑟)} . 𝐾 𝑘=1

As seen, the estimates have been evaluated for different frequencies defined by their indices 𝑟. In order to determine whether or not leaked noise obeys Gaussian distribution, the analysis of the functions Re{𝑆̂̇𝑘bisp (𝑟)} and Im{𝑆̂̇ 𝑘bisp (𝑟)} has been performed using W-criterion of Wilcoxon (W-test) [55] for each 𝑟th frequency (𝑟 = 0, 1, . . ., 𝐼 − 1) and for 𝐾 = 30 Monte Carlo runs. 2 The plots of realizations of Re{𝑆̂̇ 𝑘bisp (𝑟)} and variance 𝜎ReEns (𝑟) (2.3.7a) are demonstrated in Figures 2.3.4 (a), 2.3.6 (a) and 2.3.8 (a) for all three studied test signals. The plots of realizations of Im{𝑆̂̇ 𝑘bisp (𝑟)} and variance 𝜎ImEns (𝑟) (2.3.7b) are represented in Figures 2.3.5 (a), 2.3.7 (a) and 2.3.9 (a), respectively. The W-test results are illustrated in Figures 2.3.4 (b)–2.3.9 (b). For the frequencies for which noise has been decided to obey Gaussian distribution, the condition 𝑊(𝑟) = 1 must be performed. In the opposite case, 𝑊(𝑟) = 0. Figures 2.3.4 and 2.3.5 correspond to the test signal #1 (Figure 2.3.1), Figures 2.3.6 and 2.3.7 relate to the test signal #2 (Figure 2.3.2) and Figures 2.3.8 and 2.3.9 refer to the test signal #3 (Figure 2.3.3). The analysis of the plots demonstrated in Figures 2.3.4–2.3.9 shows that noise contribution is of nonstationary state in the sense that noise variance obviously depends on frequency. Furthermore, Gaussian distribution is observed mainly in low frequency domain (the frequency sample number 𝑟 = 129 corresponds to zero frequency in Figures 2.3.4 (b)–2.3.9 (b)). 2

74 | 2 Unknown noisy signal shape estimation by bispectrum-filtering techniques

60 40 20 0 Im ŜBisp(r)

–20

σ–2Im Ens (r)

50

100

150

200

250

r

(a) 1 W(r) 0

Fig. 2.3.5. Signal #1 (Δ𝑡 = 3): Im{𝑆̂̇𝑘bisp (𝑟)} and

50

100

150

200

250

r

(b)

𝜎2ImEns (𝑟) (a) and the W-test results evaluated for SNRinp = 0.46 (b).

60 Re ŜBisp(r) σ–2Re Ens(r)

40

20

0 50

100

150

200

250

r

(a) 1 W(r) 0 (b)

Fig. 2.3.6. Signal #2 (Δ𝑡 = 7): 𝑅𝑡{𝑆̂̇ 𝑘bisp (𝑟)} and

50

100

150

200

250

r

𝜎2ReEns (𝑟) (a) and the W-test results evaluated for SNRinp = 1.05 (b).

Moreover, it is seen from the W-test results (see Figures 2.3.4 (b)–2.3.9 (b)) that the 𝐼 number of the frequencies that obey Gaussian distribution 𝑁𝑊 = ∑𝑟=0 𝑊(𝑟) depends on both signal length and SNRinp . As can be seen from Table 2.3.2, the values 𝑁𝑊 tend to decrease when SNRinp becomes smaller, that is, the noise component becomes “more of non-Gaussian behavior”. Note that the same tendency to declination from Gaussianity has been observed for the real and imaginary parts of bispectrum estimates [45]. Therefore, the complex-valued Fourier spectrum recovered from the bispectrum estimate is mainly contaminated by nonstationary and non-Gaussian noise. This conclusion gives us an

2.3 Novel techniques developed for improving noisy bispectrum estimates

| 75

30 20 10 0 –10 –20

Im ŜBisp(r) σ–2Im Ens (r)

–30 0

50

100

150

200

250

r

(a) 1 W(r) 0

Fig. 2.3.7. Signal #2 (Δ𝑡 = 7 samples):

50

100

150

200

250

r

Im{𝑆̂̇𝑘bisp (𝑟)} and 𝜎2ImEns (𝑟) (a) and the W-test results obtained for SNRinp = 1.05 (b).

(b)

Re ŜBisp(r)

80

σ–2Re Ens(r)

60 40 20 0 50

100

150

200

250

r

(a) 1 W(r) 0 (b)

Fig. 2.3.8. Signal #3 (Δ𝑡 = 11 samples):

50

100

150

200

250

r

Re{𝑆̂̇𝑘bisp (𝑟)} and 𝜎2ReEns (𝑟) (a) and the W-test for SNRinp = 1.60 (b).

opportunity to restrict the class of filters that are applicable for filtering the considered processes. The aforementioned properties (possible non-Gaussianity) of noise show that it is not worth applying conventional linear filters for processing Re{𝑆̂̇bisp (𝑟)} and

Im{𝑆̂̇ bisp (𝑟)} parts. At the same time, for numerous nonlinear filters, there is always a problem which among them to select and with what parameters, for example, to scan the window size [53]. Therefore, here we run into not a typical but rather a complicated situation. Noise is not purely additive and not purely multiplicative, but it is frequency dependent.

76 | 2 Unknown noisy signal shape estimation by bispectrum-filtering techniques 40 Im ŜBisp(r)

30

σ–2Im Ens (r)

20 10 0 –10 –20 –30 –40 50

100

150

200

250

r

(a) 1 W(r) 0 (b)

Fig. 2.3.9. Signal #3 (Δ𝑡 = 11 samples):

50

100

150

200

250

r

Im{𝑆̂̇ 𝑘bisp (𝑟)} and 𝜎2ImEns (𝑟) (a) and the W-test results evaluated for SNRinp = 1.60 (b).

There are probably several ways out and below we consider only one. The basic attention is paid to the DCT-based filtering techniques. Note that the filters based on orthogonal transforms such as wavelets or DCT have been originally designed for removal of additive Gaussian noise with a priori known variance that is constant for the entire processed signal [58, 59]. However, later DCT-based filter modifications for data-dependent noise suppression have been proposed in [59]. Moreover, in [60] it has been demonstrated that the filtering procedure based on orthogonal transform can be rather successfully applied in the case a noise PDF is not strictly Gaussian but quite close to it. In particular, electromyographic noise (that is non-Gaussian) was removed from electrocardiograms by means of wavelet-based denoising [60]. This fact was selected as the basis and motivation for our trial to apply DCT-based filtering for bispectrum-based signal processing [46]. Note that below we will consider spatially invariant DCT filtering in blocks which is more time consuming but more efficient than filtering in nonoverlapping blocks. For applying DCT-based filters, one should a priori know noise variance or standard deviation for threshold setting. Since in our case, noise variance is supposed to be a priori unknown, we have to somehow estimate it. This can be accomplished in two 2 ways. First, one can estimate some average noise variance 𝜎̂ent or standard deviation 𝜎̂ ent for the entire signal to be processed and use this estimate for setting a constant threshold for all blocks. This type of DCT filtering will be further denoted as DCTc. Another approach is to introduce the following modification into the DCT-filter. It makes it locally adaptive. Let us obtain a local estimate of noise variance for each position of a block and then use it for the threshold setting. One way to obtain such a local estimate for each block is [60]

𝜎̂ = 1.483 ⋅ med {|𝑊(𝑙)|} ,

𝑙 = 1, . . ., 𝑁𝑏 ,

(2.3.8)

2.3 Novel techniques developed for improving noisy bispectrum estimates | 77

Table 2.3.2. Value 𝑁𝑤 as a function of signal length and SNRinp . Signal #

Signal #1 (Δ𝑡 = 3 samples)

𝑁

1.53 101

0.92 83

0.46 55

0.23 43

0.15 37

𝑁

58

49

54

38

48

SNRinp 𝑊(Re{𝑆̂̇bisp (𝑟)}) 𝑊(Im{𝑆̂̇bisp (𝑟)})

Signal #

Signal #2 (Δ𝑡 = 7 samples)

𝑁

3.50 179

2.10 129

1.05 113

0.53 77

0.35 74

𝑁

170

134

106

92

82

SNRinp 𝑊(Re{𝑆̂̇bisp (𝑟)}) 𝑊(Im{𝑆̂̇bisp (𝑟)})

Signal #

Signal #3 (Δ𝑡 = 11 samples)

𝑁

3.20 189

1.60 155

0.80 119

0.53 125

0.32 96

𝑁

181

144

117

104

80

SNRinp 𝑊(Re{𝑆̇̂bisp (𝑟)}) 𝑊(Im{𝑆̂̇bisp (𝑟)})

where med {. . .} means the sample median; 𝑊(𝑙) is the 𝑙th DCT spectral coefficient; ̂ is evaluated for the 𝑁𝑏 is the block size. For a given 𝑟 the corresponding value 𝜎(𝑟) block that includes the data samples with the indices from 𝑟 − 𝑁𝑏 /2 + 1 to 𝑟 − 𝑁𝑏 /2. This locally adaptive type of DCT filtering will be referred to as DCTla. On one side, the accuracy of the estimate (2.3.8), in general, reduces with decreasing 𝑁𝑏 . On the other hand, if noise is essentially of nonstationary behavior, the esti2 mate 𝜎̂ent can considerably differ from the values of local variance of noise. Recall that DCT filtering is applied separately for processing the real and imaginary parts of signal Fourier spectrum recovered from bispectrum. Hard thresholding [58–60] has been employed. For DCTc the threshold was set as 𝛽𝜎̂ent , and for DCTla ̂ individually for each block. Recall that in DCT and the threshold was set as 𝛽𝜎(𝑟) wavelet-based filtering the factor 𝛽 determines the denoising algorithm properties and it is commonly recommended to apply 𝛽 varying within the limits of 2.0. . . 4.0. Increasing of 𝛽 commonly leads to better noise suppression but at the same time it results in worse preservation of details and discontinuities. A particular example of computation Im{𝑆̂̇ bisp (𝑟)} represented in Figure 2.3.10 [46] ̂ correlate with intensity of noise fluctuations. demonstrates that the estimates 𝜎(𝑟) It is seen from Figure 2.3.10 that the most intensive noise fluctuations are observed in the neighborhoods of 𝑟 equal to 47 and 210 as well as in the neighborhoods of 𝑟 equal ̂ are the largest just in those neighborto 17 and 240 samples. And the estimates 𝜎(𝑟) ̂ = 𝜎(257 ̂ − 𝑟) for 𝐼 = 256. hoods. Note that 𝜎(𝑟) One more question in DCT-based filtering is the selection of the block size 𝑁𝑏 . In our simulations [46] we used two values, namely, 𝑁𝑏 = 16 and 𝑁𝑏 = 32. The set of parameters (1.5.7–1.5.11) is used for evaluating the proposed technique performance.

78 | 2 Unknown noisy signal shape estimation by bispectrum-filtering techniques 20 Im{s̑Bisp (r)} σ̑(r)

15 10 5 0 –5 –10 –15 –20 50

100

150

200

250 r

Fig. 2.3.10. Noisy realization of Im{𝑆̂̇ bisp (𝑟)} for ̂ . the test signal #1 and the local estimates 𝜎(𝑟)

The numerical simulation results for all three aforementioned test signals and for several values of SNRinp are presented in Tables 2.3.3, 2.3.4, and 2.3.5, respectively. The filter MSE performance criterion 𝜀 for the conventional bispectrum technique is denoted as 𝜀CONV and for the proposed bispectrum-filtering method as 𝜀DCT16 and 𝜀DCT32 depending upon 𝑁𝑏 . The analysis of the data in Tables 2.3.3–2.3.5 allows us to conclude the following. (1) The application of the DCT-based filtering for different cases, that is, different test signals and SNRinp values produces better SNRout and, respectively, larger 𝜀 in comparison to the conventional bispectrum-based data processing technique [10]. (2) The maximal benefit provided by combined bispectrum-filtering techniques is observed for rather small SNRinp and this peculiarity is very promising for practice. In such cases SNRout for the proposed combined techniques employing the DCTbased filtering can be more than six times larger than SNRout for the conventional bispectrum-based method. (3) The results for 𝑁𝑏 = 16 and 𝑁𝑏 = 32 are practically the same. Because of this, it seems reasonable to apply DCT-based filters with 𝑁𝑏 = 16 since, in this case, data processing can be performed faster. (4) The DCT-based filtering with local adaptation of the set threshold produces better results than the DCT-based filtering with the fixed threshold (see data in Table 2.3.3). Similar tendencies have also been observed for other test signals. (5) For the considered application, setting the parameter 𝛽 equal to 4 is practically always the optimum selection. This is confirmed by data presented in Tables 2.3.4 and 2.3.5 where the results for three different values of 𝛽 are presented. One more advantage of the DCT-based filtering application within the strategy of the combined bispectrum-filtering technique is that it provides improvement of the signal waveform estimation irrespectively to what this waveform is. Note that for the bispectrum-filtering techniques considered in [42] the application of some linear or nonlinear filters led to improvement of the signal waveform estimation while the use of some other filters or the same filters but with other scanning windows resulted in degradation of such estimates. And the problem was to ensure robustness of bispectrum-fil-

2.3 Novel techniques developed for improving noisy bispectrum estimates

| 79

Table 2.3.3. Simulation results for the test signal #1 using constant threshold with 𝛽 = 4. SNRinp

𝜀CONV 𝜀DCT c 16 𝜀DCT la16 𝜀DCT c 32 𝜀DCT la32

1.53

0.92

0.46

0.15

0.09

9.07 9.63 10.38 9.71 10.35

12.80 14.18 15.29 14.40 15.28

16.80 21.68 23.50 22.30 23.53

14.17 48.93 52.18 50.77 56.89

11.01 65.31 65.01 71.66 74.47

Table 2.3.4. Simulation results for the test signal #2 and different threshold values. SNRinp

𝛽

3.50

2.10

1.05

0.53

0.21

𝜀CONV

–

9.42

11.25

16.30

14.75

10.72

𝜀DCT la16

2.7 3.2 4.0

10.75 10.92 11.24

13.52 13.73 14.13

24.53 25.25 26.12

31.68 33.40 35.28

56.43 61.51 65.05

𝜀DCT la32

2.7 3.2 4.0

10.71 10.90 11.15

13.38 13.57 13.86

24.07 24.80 25.45

31.19 32.75 34.50

60.83 66.00 67.18

Table 2.3.5. Simulation results for the test signal #3 and different threshold values. SNRinp

𝛽

3.20

1.60

0.80

0.53

0.32

𝜀CONV

–

17.61

16.12

15.84

13.57

10.35

𝜀DCTla16

2.7 3.2 4.0

23.24 23.74 24.14

24.90 25.46 26.06

37.94 39.79 41.31

46.15 49.22 50.42

50.31 54.23 55.73

𝜀DCTla32

2.7 3.2 4.0

23.03 23.23 23.34

24.47 25.14 25.85

38.33 40.85 42.01

47.02 50.27 51.67

53.70 58.12 59.58

tering techniques in a wide sense, that is, to provide enhancement of reconstruction for a wide set of unknown signal waveforms and ranges of SNRinp variations. In order to demonstrate the performance of the DCT-based filtering, the plot of

Im{𝑆̂̇ bisp (𝑟)} for one experiment before filtering (dotted curve) and the corresponding plot after DCT-based filter (DCTla, 𝑁𝑏 = 16, 𝛽 = 4, solid curve) are shown in Figure 2.3.11. As seen from Figure 2.3.11, the noise is considerably suppressed and the information features are preserved well. This leads to improving the signal waveform reconstruction. To prove this, the signal reconstructed by the conventional bispectrum technique [10] is given in Figure 2.3.12. The signal waveform reconstructed by the proposed DCT-based bispectrum-filtering method is represented in Figure 2.3.13. Obviously, the

80 | 2 Unknown noisy signal shape estimation by bispectrum-filtering techniques 20

before filtering after DCT-based filtering

15 10 5 0 –5

Fig. 2.3.11. Im{𝑆̂̇bisp (𝑟)} for one experiment before filtering (dashed line) and after DCTbased filter with 𝑁𝑏 = 16 (solid line), the test signal #1.

–10 –15 –20 50

100

150

200

250 r

residual noise fluctuations in this plot are noticeably less intensive and two rectangular shape pulses can be more easily identified. Note that residual distortions in the central part of the plot can be explained by phase wrappings. Thus, the main attractive benefits obtained with using the DCT-based filtering for bispectrum-based signal reconstruction [46] are the following. First, the best performance improvement is reached for small input SNR values. Second, for a rather wide set of the test signals that participated in computer simulation the performance improvement is provided without solving the task of proper (optimal) selection of the filter type and/or filter parameters. Finally, it has been demonstrated that the DCTbased filtering can be modified to perform well enough in case of nonstationary noise whose variance is not a constant value for the entire observed process to be filtered. A robust and adaptive technique for improving the accuracy of noisy bispectrum estimated has also been proposed in our paper [47]. Several different approaches for processing noisy bispectrum data have been considered and compared. They are the following. Recall that the conventional bispectrum estimation [2] is based on ensemble averaging of sample bispectrum estimates related to an ensemble of 𝑀 realizations ⌢

(see 1.3.6) to obtain a smoothed estimate 𝐵̇ 𝑋 (𝑝, 𝑞) in the form of

𝐵̂̇ 𝑋 = 𝐵̂̇ mean (𝑝, 𝑞) = ⟨𝐵̂̇ (𝑚) (𝑝, 𝑞)⟩

𝑀

,

(2.3.9)

where ⟨. . .⟩𝑀 denotes the ensemble averaging by 𝑀 realizations. However, bispectrum estimates can also be derived as [45]

̂ (𝑝, 𝑞) , ̂ (𝑝, 𝑞) + 𝑗𝑖 med 𝐵̂̇ med (𝑝, 𝑞) = 𝑟med where

̂ (𝑝, 𝑞) = med𝑚 {Re [𝐵̂̇ (𝑚) (𝑝, 𝑞)] , 𝑚 ∈ [1; 𝑀]} , 𝑟med ̂ (𝑝, 𝑞) = med𝑚 {Im [𝐵̂̇ (𝑚) (𝑝, 𝑞)] , 𝑚 ∈ [1; 𝑀]} . 𝑖med

(2.3.10)

2.3 Novel techniques developed for improving noisy bispectrum estimates

| 81

Ŝ(i)

6

4 2

0 i 0

50

100

150

200

250

Fig. 2.3.12. Signal reconstructed by the conventional BLW technique for SNRinp = 0.15 (test signal #1).

Ŝ(i) 6

4

2

0

i 0

50

100

150

200

250

Fig. 2.3.13. Signal waveform reconstructed by the proposed bispectrum-filtering method for SNRinp = 0.15 (test signal #1).

Similarly to sample median, any other estimate can be applied. For example, the bispectrum estimate (BE) obtained using the Hodges–Lehmann [53] estimate can be written as ̂ (𝑝, 𝑞) , ̂ (𝑝, 𝑞) + 𝑗𝑖 HL (2.3.11) 𝐵̂̇ HL (𝑝, 𝑞) = 𝑟HL where

̂ (𝑝, 𝑞) = med 𝑟HL {Re[𝐵̂̇ (𝑚) (𝑝, 𝑞)], 𝑚 ∈ [1; 𝑀]} ; 𝑚 ⌢

⌢

(Re[𝐵̇ (𝑚) (𝑝, 𝑞)] + Re[𝐵̇ (𝑀+1−𝑚) (𝑝, 𝑞)])/2, 𝑚 ∈ [1; 𝑀/2]} and

̂ (𝑝, 𝑞) = med{Im[𝐵̂̇ (𝑚) (𝑝, 𝑞)], 𝑚 ∈ [1; 𝑀]}; 𝑖HL 𝑚 ⌢

⌢

(Im[𝐵̇ (𝑚) (𝑝, 𝑞)] + Im[𝐵̇ (𝑀+1−𝑚) (𝑝, 𝑞)])/2, 𝑚 ∈ [1; 𝑀/2]} , ⌢

and the notation 𝐵̇ (𝑚) (𝑝, 𝑞) is used for the 𝑚th order statistic in the processed data sample [53]. Recently, we have introduced a novel, simple and efficient robust adaptive estimate [61]. This estimate is worth applying for processing samples of data having heavy

82 | 2 Unknown noisy signal shape estimation by bispectrum-filtering techniques tailed PDFs and symmetrical with respect to their mean. For bispectrum-based signal processing the algorithm [47] is

𝐵̂̇ (𝑝, 𝑞), if 𝐾𝑃 ∈ [𝜓; +∞) , 𝐵̂̇ adapt (𝑝, 𝑞) = { ̂ HL 𝐵̇ med (𝑝, 𝑞), if 𝐾𝑃 ∈ (0; 𝜓)

(2.3.12)

where 𝐾𝑝 is the robust estimate of PDF kurtosis obtained on basis of percentiles (𝐾𝑝 =

0, 5(𝑋75 − 𝑋25 )/(𝑋90 − 𝑋10 ), where 𝑋𝑞 denotes the 𝑞th percentile); 𝜓 is the threshold and its quasi optimal value is equal to 0.2. The parameter 𝐾𝑝 in (2.3.12) serves as adaptation parameter for hard switching of sample median and the Hodges–Lehmann estimate (see details in [61]). As a test signal, we have used two rectangular shape pulses with the amplitudes of 2 and 6. The lengths of both pulses 𝛿𝑝 were equal, but three different values of 𝛿𝑝 have been considered in our computer simulations: namely, 3, 7, and 11 samples. The distance between two rectangular pulses was the same in all computer experiments and equal to 5 samples. Such test signals have been used to model different radar range profiles and to examine the applicability of the considered methods for processing various types of signals [47]. The performances of the following six bispectrum-based methods have been studied and compared in [47] by using: (1) Conventional method of signal waveform reconstruction [2, 10] by using (2.3.9) (Mean). (2) Method of signal waveform estimation that uses the sample median (2.3.10) for obtaining the bispectrum estimate for 𝑀 realizations (Median). (3) The proposed method based on adaptive robust estimate (2.3.10) and (2.3.11) (AE). (4) Combined method on basis of (2.3.9) and post-processing of Re and Im parts of 𝑓𝑖𝑙𝑡 𝑆̂̇bisp (𝑟)by the DCT-based filter described above (Mean+DCT), the recommended value 𝛽 = 4 was used in our simulations.

(5) Combined technique that employs bispectrum estimate (2.3.10) and DCT-based 𝑓𝑖𝑙𝑡

filtering of of Re and Im components of 𝑆̂̇bisp (𝑟) (Median+DCT). (6) Adaptive combined method by using (2.3.11) and (2.3.12) followed by post-process𝑓𝑖𝑙𝑡

ing of Re and Im components of 𝑆̂̇bisp (𝑟) by DCT-based filter (AE+DCT). Thus, for the last two methods we have combined two approaches for improving the bispectrum estimate, namely, its forming using robust estimates and the obtained estimate filtering. 2 The MSE 𝜎out values (denoted as the output variance in the graphs) as functions of input noise variance 𝜎𝑛2 are shown in Figures 2.3.14–2.3.17. The plots in Figures 2.3.14– 2.3.16 correspond to realization number of 𝑀 = 64 and the plots in Figure 2.3.17 are related to 𝑀 = 128. Analysis of the plots in Figures 2.3.11– 2.3.14 allows concluding the following.

2.3 Novel techniques developed for improving noisy bispectrum estimates |

Output variance

100

Mean Median AE Median+DCT Mean+DCT AE+DCT

10–1

0

Output variance

100

1

2 3 4 Input noise variance

5

Fig. 2.3.14. Output MSE as a function of the input noise variance, 𝛿𝑝 = 3, 𝑀 = 64.

3

Fig. 2.3.15. Output MSE as a function of the input noise variance, 𝛿𝑝 = 7, 𝑀 = 64.

Mean Median AE Mean+DCT AE+DCT Median+DCT

10–1

0

100

Output variance

83

0.5

1 1.5 2 2.5 Input noise variance

Mean Median AE Mean+DCT Median+DCT AE+DCT

10–1

0

1

2 3 4 Input noise variance

5

Fig. 2.3.16. Output MSE as a function of the input noise variance, 𝛿𝑝 = 11, 𝑀 = 64.

(1) The use of the adaptive robust estimate (2.3.11) and (2.3.12) provides “robustness” of bispectrum-based processing (Method 3 in the above list) in a wide sense; for small 𝜎𝑛2 it performs like conventional Method 1 while for large 𝜎𝑛2 (small input

84 | 2 Unknown noisy signal shape estimation by bispectrum-filtering techniques 100

Output variance

Mean Median AE Median+DCT Mean+DCT AE+DCT

10–1

0

1

2

3

Input noise variance

(2)

(3)

(4)

(5)

(6)

4

5

Fig. 2.3.17. Output MSE as a function of the input noise variance, 𝛿𝑝 = 7, 𝑀 = 128.

SNR) Method 3 has the same performance as Method 2 based on the median estimate. The use of the robust estimates (methods 2 and 3) instead of conventional estimate 2 (2.3.9) produces sufficient reduction of 𝜎 out for large 𝜎𝑛2 (small input SNRs); the val2 ues of 𝜎out for Methods 2 and 3 can be up to 30% smaller than for the conventional bispectrum-based Method 1; this property is important for practical applications. The DCT-based filtering (Methods 4–6) provides even more radical reduction of 𝜎2out in comparison to Methods 1–3 and this improvement of performance is the largest for small input SNRs. The performance of Method 6 is either the best or close to the best reachable among the considered methods for entire range of 𝜎𝑛2 variation and for all test signals. As can be expected, the use of a larger number of accumulated realizations 𝑀 leads to a better performance (compare the corresponding plots in Figures 2.3.17 and 2.3.15) but in practice the number of observed realizations is limited. All the tendencies observed for smaller 𝑀 (see the items (1)–(4)) are also valid for larger 𝑀.

Thus, the main benefits of the adaptive methods proposed in [47] are their applicability and high performance for a wide range of input SNRs (especially for the situations when input SNR is low) and different a priori unknown signal waveforms. Another combined bispectrum-filtering technique proposed in paper [48] is based on the 2-D DCT data processing, evaluation of the optimal parameters of the corresponding filters, estimation of the bounds that can be reached in the improvement of the output SNR and analysis of the conditions under which maximum improvement of SNR takes place. According to results obtained for several bispectrum-based approaches considered above in this Chapter, the separate filtering of the real and imaginary parts of

2.3 Novel techniques developed for improving noisy bispectrum estimates |

85

·105 3 2 1 60 0 20

20 40

⌢

̇ 𝑞)} as a function of Fig. 2.3.18. Noisy Re{𝐵(𝑝, two frequencies computed for Δ𝑡 = 7, SNRinp = 0.35.

40 60

·104 2 0 60

–2 20

40

⌢

̇ 𝑞)} as a function of Fig. 2.3.19. Noisy Im{𝐵(𝑝, two frequencies computed for Δ𝑡 = 7, SNRinp = 0.35.

40 20 60

bispectrum estimate provides considerably better results compared to other bispectrum-based signal reconstruction approaches. However, the following question arises: what filters are reasonable to employ in this case? The point is that the noise leaking to the bispectrum estimate is neither of additive nor of multiplicative nature in a strict sense. More specifically, this noise possesses signal (bispectrum) depending properties (see [45]). Note that noise leaks into bispectrum domain from the input noisy signal and is subjected to several nonlinear transformations. Characteristics of the noise in the bispectrum domain depend not only on the variance of input noise but on the shape of the signal Fourier spectrum. For illustration of noise distortions leaked in bispectrum estimates, the graphs ⌢

⌢

̇ 𝑞)} and Im{𝐵(𝑝, ̇ 𝑞)} corrupted by noise are plotted in Figof the functions Re{𝐵(𝑝, ures 2.3.18 and 2.3.19, respectively, for a single pulse signal of rectangular shape and pulse length of Δ𝑡 = 7. Only a quarter of the bifrequency plane is shown in these Figures due to the symmetry property of bispectrum (1.2.10) [2]. It has been demonstrated in [62] that for the majority of frequencies in a bifrequency plane, induced noise possesses non-Gaussian PDF with zero mean and unknown variance. Such properties of noise cause problems in selecting a proper filter for smoothing ⌢

⌢

̇ 𝑞)} and Im{𝐵(𝑝, ̇ 𝑞)}. One more peculiarity is that the behavior of 2-D arrays Re{𝐵(𝑝, information component of these functions is a priori unknown. Thus, it is difficult to

86 | 2 Unknown noisy signal shape estimation by bispectrum-filtering techniques recommend some nonlinear nonadaptive filter [53] and to properly select its parameters (scanning window size, trimming factors for 𝛼-trimmed filters, etc.). It is known (see [59, 63]) that the DCT-based filtering of images is commonly carried out by square shape blocks. Processing with both overlapping or nonoverlapping blocks is possible. The best performance of the algorithm can be obtained for full overlapping of blocks. But this requires more time for computations. Full overlapping means that each consequent block is shifted by only one sample with respect to the previous one. Respectively, the worst performance but minimal time consumption due to sufficient decreasing of summation and multiplication operations, is observed for nonoverlapping blocks. This mode corresponds to the block mutual shift equal to the block size. However, the so-called blocking artifacts can arise in the case of nonoverlapping block filtering [59]. For noise variance estimation that is usually necessary for operation of the DCTbased filters, there exist two possible approaches. According to the first one, the averaged variance of the observed process can be defined for the total process and used for calculating the constant-valued threshold for all blocks. The second alternative way is to estimate local variance inside each block and to calculate on its basis the local threshold value for each given block. This makes the DCT-based filtering locally adaptive. It was demonstrated in [46] that the DCT-based filtering with adaptively adjusted threshold is a more efficient procedure. Let us modify the 1-D adaptive DCT-based filtering technique to the case of processing 2-D arrays of bispectrum samples. Similarly to the 1-D signal case, let us estimate the standard deviation (SD) for each 2-D block as ⌢ 𝜎𝑝,𝑞 = 1, 483 ⋅ med{|𝑊𝑝𝑞 (𝑥, 𝑦)|} , (2.3.13) where med {. . .} is the median value of the sample; 𝑊𝑝𝑞 (𝑥, 𝑦) denotes the DCT coefficients for the block with coordinates 𝑝0 = 𝑝−𝑁𝑏 /2+1, 𝑝−𝑁𝑏 /2+2, . . ., 𝑝+𝑁𝑏 /2, 𝑞0 = 𝑞 − 𝑁𝑏 /2 + 1, 𝑞 − 𝑁𝑏 /2 + 2, . . ., 𝑞 + 𝑁𝑏 /2; 𝑁𝑏 is the block side size (each 2-D block contains 𝑁𝑏 × 𝑁𝑏 samples). Thus, the proposed in [48] filtering procedure includes the following steps: – calculation of DCT spectral samples of the real and imaginary parts of the bispec⌢

– –

– – –

̇ 𝑞) containing 𝑁𝑏 × 𝑁𝑏 samples in each block; trum estimate 𝐵(𝑝, estimation of the local SD according to (2.3.13); ⌢ calculation of threshold 𝛽⋅ 𝜎𝑝,𝑞 , where 𝛽 defines smoothing properties of the filter. Usually the latter value is within the interval 2. . .4 [58–60, 63]. Note that noise filtering improves with 𝛽 increasing but it leads to worse detail preserving; ⌢ zeroing of the samples with absolute values are smaller than the threshold 𝛽⋅ 𝜎𝑝,𝑞 ; carrying out inverse DCT; joint processing (averaging) of block outputs for all samples if overlapping blocks are used.

2.3 Novel techniques developed for improving noisy bispectrum estimates |

87

The following four different methods were studied in computer simulations performed and discussed in [48]: (1) the traditional bispectrum-based signal reconstruction technique [2, 10] (Technique #1); (2) the combined bispectrum-filtering signal reconstruction technique with filtering ⌢

̇ 𝑞) by the standard median filter with the sliding window size of 5 × 5 [53] of 𝐵(𝑝, (Technique #2). In fact, this technique has been analyzed earlier in Subsection 2.2 and below it is used for comparison purposes. ⌢

̇ 𝑞) by the (3) the combined bispectrum-filtering technique with filtering of 𝐵(𝑝, 𝛼-trimmed filter with the sliding window size of 5 × 5 and trimming of 5 maximum and 5 minimum values [53] (Technique #3). ⌢

̇ 𝑞) (Technique #4). (4) the proposed technique with the DCT-based filtering of 𝐵(𝑝, For the latter Technique the block size was of 8×8 pixels and three different values of the parameter 𝛽 have been considered. Filtering was carried out both with partial and full block overlapping. Note that the best results have been obtained for full overlapping of blocks. In case of partial overlapping with the shift equal to 𝑁𝑏 /2, the provided 𝜀 values were smaller by 3. . . 15% than in a case of full overlapping. Therefore, below, only the data for DCT-based filtering with full overlapping are presented. Three test signals of the same type (denoted below as signals ##1, 2, and 3) containing two positive-valued pulses of rectangular shape and amplitudes of 2 and 6 were used for computer simulation. The pulse lengths were selected equal to Δ𝑡1,2,3 = 3, 7, and 11 samples for the test signals ##1, 2, and 3, respectively. The interval between two pulses was fixed and it was equal to 5 samples. The total length of each sequence is of 𝐼 = 256 samples. The results of computer simulations obtained in [48] are given in Tables 2.3.6– 2.3.8. The nonadaptive Techniques ##2 and 3 comparing to Technique #1 can provide both improving and worsening of the SNRout (increasing or decreasing of 𝜀). This depends on the test signal parameters and SNRinp . For example, for the test signal #1, Technique #2 occurs to be very effective for SNRinp = 1.53. But the same Technique #2 is characterized by the minimal 𝜀 among the considered bispectrum-filtering methods for SNRinp = 0.15 (see Table 2.3.6). Moreover, the use of nonadaptive Techniques ##2 and 3 can even lead to 𝜀 decreasing in comparison to traditional bispectrum Technique #1. For instance, this happens for the test signal #2 when SNRinp are rather large (3.50 and 2.10, see Table 2.3.7). These results confirm the existence of problems in the selection of proper nonadaptive filters to be applied within the combined bispectrum-filtering framework. Therefore, practical use of Techniques ##2 and 3 is limited. Technique #4 outperforms Techniques 2 and 3 in most practically important cases. The exclusion is the signal #1 for which Techniques ##2 and 3 can provide larger val-

88 | 2 Unknown noisy signal shape estimation by bispectrum-filtering techniques Table 2.3.6. The values 𝜀 obtained for the test signal #1. SNRinp Technique # 1 2 3 4

𝛽 = 2.7 𝛽 = 3.2 𝛽 = 3.7

1.53

0.92

0.46

0.23

0.15

9.07 26.11 13.74 11.22 11.71 12.02

12.80 33.04 17.78 15.19 19.11 14.34

16.80 34.45 27.42 24.93 27.99 28.19

17.07 39.54 38.58 42.67 45.77 51.10

14.17 46.15 62.72 68.55 66.38 89.25

Table 2.3.7. The values 𝜀 obtained for the test signal #2. SNRinp Technique # 1 2 3 4

𝛽 = 2.7 𝛽 = 3.2 𝛽 = 3.7

3.50

2.10

1.05

0.53

0.35

9.42 5.99 5.91 10.19 11.06 11.54

11.25 9.80 10.34 14.34 14.05 13.93

16.30 17.73 20.46 27.17 27.84 28.39

14.75 28.36 34.09 48.25 60.85 54.95

13.50 32.38 37.97 62.44 72.23 64.35

Table 2.3.8. The values 𝜀 obtained for the test signal #3. SNRinp Technique # 1 2 3 4

𝛽 = 2.7 𝛽 = 3.2 𝛽 = 3.7

3.20

1.60

0.80

0.53

0.32

17.61 18.88 13.53 19.72 20.82 21.53

16.12 27.04 22.20 30.50 30.94 32.08

15.84 35.92 33.65 44.42 42.92 44.61

13.57 38.26 38.08 50.00 54.95 53.71

10.35 33.87 36.60 47.16 51.80 53.51

ues of 𝜀 for relatively large input SNRinp . The latter peculiarity can be explained by smoothness of the processed 3-D function. Note that the proposed adaptive Technique #4 is the most effective (among the considered techniques) in cases of low SNRinp for all test signals. The optimal value of the parameter 𝛽 is recommended to be within interval from 3.2 to 3.7. Visual comparison of the real and imaginary parts of the bispectrum estimate before (Figures 2.3.15 and 2.3.16) and after (Figures 2.3.17 and 2.3.18) filtering allows imagining the efficiency of the proposed Technique #4. It is evident that the use of the proposed adaptive 2-D DCT-based filter provides noise suppression in the real and imaginary parts of bispectrum estimate.

2.3 Novel techniques developed for improving noisy bispectrum estimates | 89

·105 2 1 ⌢

60

0

̇ 𝑞)𝑓𝑖𝑙𝑡 } processed by adaptive Fig. 2.3.20. Re{𝐵(𝑝, DCT-based filter for Δ𝑡 = 7, SNRinp = 0.35, 𝛽=3.7.

40 20

20

40

60

·104 2 0 –2

7 6

̇ 𝑞)𝑓𝑖𝑙𝑡 } processed by adapFig. 2.3.21. Im{𝐵(𝑝, tive DCT-based filter for Δ𝑡 = 7, SNRinp = 0.35, 𝛽=3.7.

i

Fig. 2.3.22. The reconstructed signal 𝑠(𝑖) obtained by Technique #1, Δ𝑡 = 7, SNRinp = 0.35.

40 20

40

⌢

60 20 60

S(i)

5 4 3 2 1 0 –1 50

100

150

200

250

⌢

Considerable residual fluctuations are observed in the signal reconstructed by the conventional bispectrum-based Technique #1 (see Figure 2.3.22). Technique #3 (see Figure 2.3.23) causes rather large distortions but, at the same time, it provides better noise suppression than Technique #1. In turn, the proposed Technique #4 provides only negligible noise presence and small distortions in the reconstructed signal (see Figure 2.3.24). The presented results demonstrate good performance of the proposed technique [48] within the total range of the considered input SNRs and test signals. This

90 | 2 Unknown noisy signal shape estimation by bispectrum-filtering techniques 7

S(i)

6 5 4 3 2 1 0 i

–1 50 7 6

150

250

⌢

Fig. 2.3.23. The reconstructed signal 𝑠(𝑖) obtained by Technique #3, Δ𝑡 = 7, SNRinp = 0.35.

S(i)

5 4 3 2 1 ⌢

0

i

–1 50

150

250

Fig. 2.3.24. The reconstructed signal 𝑠(𝑖) obtained by Technique #4, Δ𝑡 = 7, SNRinp = 0.35, 𝛽=3.7.

technique provides improvement of the SNRout by 8 dB comparing the conventional bispectrum technique [10] in the most important cases of low input SNRs. As it was supposed, the use of the nonadaptive algorithms seems to be unsuitable in general. However, in some cases they are able to give good results (see, e.g., the data in Table 2.3.6 obtained by Technique 2). In turn, adaptive procedures are able to change their parameters depending on the situation at hand (behavior of the processed 2-D functions and noise level). The comparison of the values of 𝜀 has been also carried out for the proposed Technique #4 and the techniques earlier developed in [46]. The results are approximately of the same value for rather large input SNRs. However, the proposed Technique #4 provides the values 𝜀 that are larger by approximately 10. . . 20% than the adaptive technique considered in [46]. Recall that the difference is in using either the 2-D DCTbased filtering for Technique #4 or the 1-D DCT-based filtering for the methods in [46]. Certainly, the 2-D DCT-based filtering requires larger processing time. But within entire framework of the combined bispectrum-filtering processing it does not lead to considerable increasing of total computation time.

2.3 Novel techniques developed for improving noisy bispectrum estimates

| 91

The latter numerical simulation results show that the use of adaptive filtering within the combined bispectrum-filtering approach leads to obvious benefits. However, the use of the designed local adaptive DCT-based filters is not the only opportunity. Thus, in this Section we consider and carry out brief performance analysis for three other techniques. First, in the paper [64] a method of 2-D adaptive filtering based on the so-called Z-parameter was designed. This method (further Technique #5) is based on the use of preliminary nonlinear filter, calculation of Z-parameter for each sliding window position, and hard-switching between two or several nonlinear filter outputs according to results of Z-parameter comparison to the corresponding threshold. One advantage of this filter is that noise type (additive, multiplicative, mixed) should not be known in advance. For the considered application, we used the 𝛼-trimmed filter with the sliding window size of 5 × 5 and trimming of 5 maximum and 5 minimum values as a preliminary filter. Besides, this filter as well as the standard 5 × 5 median filter were used as component filters (switching between their outputs was performed). The threshold was equal to 0.4. Second, we considered an opportunity of applying the standard sigma filter [65] where, for each given sliding window position, a local standard deviation is estimated according to (2.3.13). Then, using this estimate, 2𝜎-neighborhood is determined and averaging of pixel values that belong to this neighborhood for a given sliding window position is carried out. This method is referred to below as Technique #6. Third, it is possible to apply a two-stage filtering procedure. At the first stage, Technique #4 (𝛽 = 3.2) is applied and then its output is subjected to post-processing by the 3 × 3 center weighted median filter with the center weight 3. The goal of such postprocessing is to remove spiky values retained by the DCT-based filter. This approach is noted as Technique #7. The obtained simulation results for Techniques #5, 6, and 7 for all three test signals are presented below in Tables 2.3.9, 2.3.10, and 2.3.11, respectively. Comparing the data in Tables 2.3.9, 2.3.10, and 2.3.11 to the corresponding data in Tables 2.3.6, 2.3.7, and 2.3.8 as well as analyzing these data within each Table, it is possible to conclude the following. Technique #6 is able to produce some performance improvement in comparison to the conventional bispectrum method (BLW) (Technique #1) but among the considered combined bispectrum-filtering techniques, Technique #6 is surely not the best. The use of an adaptive nonlinear filter (TechTable 2.3.9. The values 𝜀 obtained for the test signal #1. SNRinp Technique # 5 6 7

1.53

0.92

0.46

0.23

0.15

20.34 14.26 10.97

22.77 14.06 17.99

29.31 26.27 36.41

42.87 30.73 40.31

49.92 35.13 61.96

92 | 2 Unknown noisy signal shape estimation by bispectrum-filtering techniques Table 2.3.10. The values 𝜀 obtained for the test signal #2. SNRinp Technique # 5 6 7

3.50

2.10

1.05

0.53

0.35

33.55 12.71 11.61

38.12 19.57 14.64

40.45 34.26 28.71

45.17 29.14 60.84

43.82 28.14 73.16

Table 2.3.11. The values 𝜀 obtained for the test signal #3. SNRinp Technique # 5 6 7

3.20

1.60

0.80

0.53

0.32

17.20 9.83 21.05

24.78 21.92 31.39

39.13 29.77 41.93

46.08 27.80 55.37

39.52 21.61 52.51

nique #5) instead of nonadaptive ones (Technique #2 and Technique #3) can be expedient. The Technique #5 either provides the largest values of 𝜀 (like, e.g., for the test signals #2 and #3, see data in Tables 2.3.10 and 2.3.11 and compare them to the corresponding data in Tables 2.3.7 and 2.3.8) or appropriately large values of 𝜀. The use of Technique #7 instead of Technique #4 does not seem reasonable. Really, in the majority of the considered situations Technique #7 produces practically the same values of 𝜀 as Technique #4 does. Since Technique #7 is slightly more complicated than Technique #4, there are no reasons to apply Technique #7. Thus, the developed approach (Technique #4) provides the best signal reconstruction system performance compared to the nonadaptive techniques in most cases. It is shown that the maximum improvement has been obtained for small input SNR that seems to be very important in many practical applications. The novelty of the approach proposed in [48] is in the local estimation of standard deviations inside each block as well as in the corresponding calculation of hard threshold for DCT-based filter.

2.4 Adaptive 1-D filtering applied for bispectrum-based signal reconstruction A detailed performance study of both nonadaptive and adaptive filtering based on socalled Z-parameter and LPA-ICI in bispectral-based signal reconstruction in noise was performed in our paper [49]. Below, we describe the results obtained with application of 1-D adaptive filtering of the data in a bispectrum-based signal reconstruction system.

2.4 Adaptive 1-D filtering applied for bispectrum-based signal reconstruction

|

93

The 1-D local adaptive filters (LAFs) perform in such a manner that for each estimation sample the window size is selected to suppress the random noise as much as possible and to preserve the signal features. One additional motivation to use LAFs [64, 66–69] is their ability to operate under a priori unknown noise variance and/or in the cases of nonstationary noise. This is important since the noise components in the Fourier spectrum real and imaginary parts recovered by bispectrum estimation have frequency-dependent statistical properties (see previous Subsection). A LAF described in [64, 66] is a hard switching filter which output 𝑈𝑍 (𝑟) is computed as

𝑈𝑓1 (𝑟) if |𝑍(𝑟)| ≥ 𝑎2 { { { 𝑈𝑍 (𝑟) = {𝑈𝑓2 (𝑟) if 𝑎1 ≤ |𝑍(𝑟)| < 𝑎2 , { { 𝑓3 𝑈 (𝑟) if |𝑍(𝑟)| < 𝑎1 {

(2.4.1)

where 𝑈𝑓1 (𝑟), 𝑈𝑓2 (𝑟) and 𝑈𝑓3 (𝑟) are the outputs of the three nonadaptive filters 𝑓1 , 𝑓2 and 𝑓3 , and 𝑎1 and 𝑎2 are the thresholds (see the block scheme in Figure 2.4.1). Z-parameter in (2.4.1) is computed as 𝑟+(𝑁 −1)/2

𝑍(𝑟) =

𝑝 (𝑈𝑓 (𝑗) − 𝑈(𝑗)) ∑𝑗=𝑟−(𝑁 𝑝 −1)/2

𝑟+(𝑁 −1)/2

𝑝 ∑𝑗=𝑟−(𝑁 |𝑈𝑓 (𝑗) − 𝑈(𝑗)| 𝑝 −1)/2

,

(2.4.2)

where 𝑈(𝑗) denotes the input signal, 𝑈𝑓 (𝑗) is the output of the nonlinear prefilter [66], for example, the 𝛼-trimmed or Wilcoxon filter. The 𝛼-trimmed filter with 𝑁𝑝 = 9 and the trimming parameter 𝑁𝛼 = 2 has been employed in [66], where 𝑁𝛼 is the number of the values rejected after sample data sorting in the scanning window.

a1 U

Nonlinear prefilter

U

f

Calculation of Z-parameter

f1

Filter f1

Filter f2

Filter f3

U

Uf2

Z

a2

Comparison of |Z| to thresholds Decision on filter selection

Switch

Uf3

Fig. 2.4.1. Scheme of local adaptive filters based on Z-parameter.

Uz

94 | 2 Unknown noisy signal shape estimation by bispectrum-filtering techniques Z-parameter values in (2.4.2) are quite close to zero if a current 𝑟th sample belongs to a constant or linearly changing fragment of the input signal 𝑈(𝑗). Then, if |𝑍(𝑟)| < 𝑎1 , a filter that possesses good noise suppression properties (𝑓3 ) should be applied. On the contrary, large absolute values of Z-parameter (|𝑍(𝑟)| ≥ 𝑎2 ) are observed for fragments containing discontinuities for which one needs to apply a signal shape preserving filter (𝑓1 ). Note that the maximal value of |𝑍(𝑟)| is equal to unity. For intermediate values of |𝑍(𝑟)|, 𝑎1 ≤ |𝑍(𝑟)| ≤ 𝑎2 , the output of the local adaptive filter is assigned to the output of 𝑓2 . In general, the performance of the LAF from [66] depends on the set of filters 𝑓1 , 𝑓2 and 𝑓3 as well as on thresholds 𝑎1 and 𝑎2 that are used. In what follows, we consider possible sets of filters 𝑓1 , 𝑓2 and 𝑓3 . As for the thresholds, they are fixed (𝑎1 = 0.2 and 𝑎2 = 0.4) according to the recommendations given in [66]. Two typical LAFs given by (2.4.1) and (2.4.2) have been considered in [49]. The first one is nonlinear LAF (ADAPT-NLIN) for which the following switch filters have been used: a median filter with a scanning window size of 5 samples (𝑓1 ), a 𝛼-trimmed filter with 𝑁 = 9, 𝑁𝛼 = 2 (𝑓2 ), and a 𝛼-trimmed filter with 𝑁 = 13, 𝑁𝛼 = 3 (𝑓3 ). The second one is linear LAF (ADAPT-LIN) that consists of the following switch filters: a mean filter with a window size of 𝑁 = 5 samples (𝑓1 ), a mean filter with 𝑁 = 9 (𝑓2 ), and a mean filter with 𝑁 = 13 (𝑓3 ). Another approach to adaptive filtering is the local polynomial approximation (LPA) with use of the intersection of confidence intervals (ICI) rule proposed in [67– 72]. The LPA is a tool for linear filter design. In particular, the zero-order polynomial approximation is used in order to obtain the scanning window mean filters. LPA also allows one to apply higher-order polynomial approximations, which can be useful provided that the filtered signal is smooth enough. We will only pay attention to the second-order polynomial approximation. The ICI rule is an algorithm for adaptive window size selection. The idea of this approach is as follows. The algorithm searches for a largest local vicinity of the point of estimation where the LPA assumption fits well to the data. The estimates 𝑈̂ ℎ (𝑟) (the filtered estimates of the bispectrum real Re{𝑆̂bisp (𝑟)} or imaginary Im{𝑆̂bisp (𝑟)} parts for the case considered) are calculated for a grid of window sizes (scales) ℎ ∈ 𝐻 = {ℎ1 , ℎ2 , . . ., ℎ𝐽 }, where ℎ1 < ℎ2 < . . . < ℎ𝐽 . The adaptive scale is defined as the largest ℎ+ of those windows in the set 𝐻 whose estimate does not differ significantly from the estimators corresponding to the smaller window sizes. More precisely, a sequence of confidence intervals 𝐷𝑗 = [𝑈̂ ℎ𝑗 (𝑟) − 𝛤𝜎𝑈̂ ℎ , 𝑈̂ ℎ𝑗 (𝑟) + 𝛤𝜎𝑈̂ ℎ ] is determined, where 𝛤 > 𝑗 𝑗 0 is a threshold parameter and 𝜎𝑈̂ ℎ is a standard deviation of estimate. The ICI rule 𝑗

𝑗

can be stated as follows: consider the intersection of confidence intervals 𝐼𝑗 = ⋂𝑖=1 𝐷𝑖 and let 𝑗+ be the largest of the indices 𝑗 for which 𝐼𝑗 is nonempty. Then the optimal scale ℎ+ is defined as ℎ+ = ℎ𝑗+ and, as a result, the optimal scale estimate is equal to

𝑈̂ ℎ+ (𝑟).

2.4 Adaptive 1-D filtering applied for bispectrum-based signal reconstruction

|

95

Theoretical analysis provided in [71] shows that this adaptive scale gives the best possible point-wise mean-squared error. In practice this means that adaptively, for every sample, ICI allows the maximum degree of smoothing, stopping before oversmoothing begins [70]. The threshold parameter 𝛤 is a key element of the algorithm. Too large a value of this parameter leads to signal over-smoothing and too small a value leads to undersmoothing. A reasonable value to preserve the signal and remove the noise as much as possible is somewhere in between. Optimal values of 𝛤 can be derived from some heuristic and theoretical considerations (see, e.g. [67, 68, 70, 72]). We prefer to treat the threshold 𝛤 as a fixed design parameter in our investigations. This approach has been developed for linear estimates [67, 68, 70] as well as for nonlinear (median) filters [69]. For filtering, we use symmetric 𝑈ℎsym and nonsymmetric (left 𝑈ℎleft and right 𝑈ℎright ) windowed estimates, where ℎsym , ℎleft and ℎright are the window sizes of the corresponding estimates. The ICI rule gives the adaptive window size for each of these estimates. Let us denote the corresponding optimal estimates by 𝑈̂ ℎ+sym (𝑟), 𝑈̂ ℎ+ (𝑟) and 𝑈̂ ℎ+ (𝑟). The final estimate is calculated as the weighted mean left

right

̂ produced in the form of with the combined final LPA estimate 𝑈(𝑟) ̂ 𝑈(𝑟) = 𝜆 left (𝑟)𝑈ℎ+ (𝑟) + 𝜆 right (𝑟)𝑈ℎ+ (𝑟) + 𝜆 sym (𝑟)𝑈ℎ+sym (𝑟) , left

𝜆 left (𝑟) = 𝜆 right (𝑟) = 𝜆 sym (𝑟) =

right

−2 𝜎left (𝑟) +

−2 𝜎left (𝑟) +

−2 𝜎left (𝑟) +

−2 (𝑟) 𝜎left −2 𝜎right (𝑟) + −2 𝜎right (𝑟) −2 𝜎right (𝑟) + −2 𝜎sym (𝑟) −2 𝜎right (𝑟) +

−2 (𝑟) 𝜎sym

−2 (𝑟) 𝜎sym

−2 (𝑟) 𝜎sym

(2.4.3)

, , ,

where 𝜎left , 𝜎right , and 𝜎sym are the standard deviations of the estimates. Thus, the inverse variances are used as the weights of the partial symmetric, left and right estimates for fusing in the final one. Similarly, the adaptive window size median [69] (Median-ICI) is introduced in the form of the symmetric and nonsymmetric left and right estimates as

𝑈̂ ℎsym (𝑟) = median{𝑈inp (𝑟 + 𝑛)}, 𝑛 = −(ℎ − 1)/2, . . ., (ℎ − 1)/2 , 𝑈̂ ℎleft (𝑟) = median{𝑈inp (𝑟 + 𝑛)}, 𝑛 = −(ℎ − 1)/2, . . ., 0 , 𝑈̂ ℎright (𝑟) = median{𝑈inp (𝑟 + 𝑛)}, 𝑛 = 0, . . ., (ℎ − 1)/2 ,

symmetrical median

nonsymmetrical left median nonsymmetrical right median

,

(2.4.4) where 𝑈̂ inp is the input process and ℎ ∈ 𝐻. The ICI rule makes these median filters data adaptive with the varying window size optimizing each of these symmetric and

96 | 2 Unknown noisy signal shape estimation by bispectrum-filtering techniques nonsymmetric estimates. The final estimate is found again in the weighted mean form (2.4.3). We apply these adaptive linear and median filters to the real and imaginary parts of the recovered Fourier spectra. The following parameters of the ICI rule are used in the experiments. The window sizes are 𝐻 = {1, 5, 7, 11, 21, 43, 61} for the linear LPA filters of order 2 and 𝐻 = {1, 3, 5, 11, 21, 43} for the median filters. In the experiments, the threshold parameter 𝛤 is fixed to be 1.5 for both the cases. The two-pulse test signal has been employed in computer experiments with pulse width of Δ𝑡 = 7. The values of SNRinp are varied from 0.35 to 3.5. The criterion 𝜒 is introduced additionally to criteria (1.5.7–1.5.11) for filtering performance estimation as 𝜀 𝜒= , (2.4.5)

𝜀0

where 𝜀0 is the parameter calculated according to (1.5.11) for the conventional BLW technique [10] (denoted as Technique 1, below) for a given SNRinp . The plots of 𝜒 (2.4.5) as the functions of SNRinp are presented in Figure 2.4.2. The results are given for the following techniques: (A) The conventional Technique 1; (B) The combined technique that employs smoothing the real and imaginary parts of signal Fourier spectrum recovered from bispectrum by 𝛼-trimmed filter with 𝑁𝑝 = 9, 𝑁𝛼 = 2; (C) The same filtering as in B) performed by 𝛼-trimmed filter with 𝑁𝑝 = 13, 𝑁𝛼 = 3; (D) The combined technique based on adaptive filtering ADAPT-NLIN; (E) The combined signal reconstruction technique using adaptive filtering ADAPTLIN; (F) Technique 4 (median filter) with 𝑁𝑝 = 5; (G) Technique 5 (mean filter) with 𝑁𝑝 = 5; (H) Technique 5 (mean filter) with 𝑁𝑝 = 9; (I) Technique 5 (mean filter) with 𝑁𝑝 = 13. The plots in Figure 2.4.2 (a) give us the possibility to compare the performance of the adaptive filter ADAPT-NLIN and the corresponding switch nonlinear filters. Similarly, Figure 2.4.2 (b) gives the same opportunity for comparative analysis in the case of ADAPT-LIN that is based on the mean filters with different scanning window sizes [49]. The analysis of these data allows us to conclude the following. – The two types of adaptive filters considered (see curves D and E) show good performance and they lose only a little to the best among the nonadaptive (switch) filters for a wide range of values of SNRinp (SNRinp < 1), the adaptive technique ADAPT-LIN (curve E) giving better 𝜒 values in comparison to the ADAPT-NLIN (curve D). The situation is opposite for large SNRinp values (SNRinp > 2). This can be explained by the worse dynamic properties of linear filters in comparison to the nonlinear ones.

2.4 Adaptive 1-D filtering applied for bispectrum-based signal reconstruction

Probability of detection

2.5

|

97

A F D B C

2

1.5

1 0

0.5

1

(a)

1.5

2

2.5

3

Probability of detection

3

A I E G H

2.5 2 1.5 1 0.5

(b)

3.5

SNR, dB

0

0.5

1

1.5 2 SNR, dB

2.5

3

3.5

Fig. 2.4.2. Criterion 𝜒 as the function of SNRinp for different techniques: The curves A-I correspond to the following techniques: A – to the conventional Technique 1; B – to the combined technique that employs smoothing by 𝛼-trimmed filter with 𝑁𝑝 = 9, 𝑁𝛼 = 2; C – to the combined technique that uses smoothing by 𝛼-trimmed filter with 𝑁𝑝 = 13, 𝑁𝛼 = 3; D – to the combined technique based on adaptive filtering ADAPT-NLIN; E – to the combined signal reconstruction with ADAPT-LIN; F – to Technique 4 (median filter) with 𝑁𝑝 = 5; G – to Technique 5 (mean ADAPT-NLIN technique (a) and ADAPT-LIN technique (b) filter) with 𝑁𝑝 = 5; H – to Technique 5 (mean filter) with 𝑁𝑝 = 9; I – to Technique 5 (mean filter) with 𝑁𝑝 = 13.

– –

The efficiency of both the adaptive and nonadaptive techniques is the largest for low SNRinp (which is important and useful for practical applications). Note that in some cases the use of the combined methods can lead to a worse performance with respect to the conventional technique. For example, the use of the mean filter with 𝑁𝑝 = 13 (see curve I in Figure 2.4.2 (b)) produces 𝜒 < 1 for SNRinp > 1.5, this is due to the large distortions introduced by the mean filter with a rather large 𝑁𝑝 . The goal of using adaptive filtering is to avoid such situations for the considered application of combined bispectrum-filtering data processing.

98 | 2 Unknown noisy signal shape estimation by bispectrum-filtering techniques The reconstructed test signal plots (example of pulse length of Δ𝑡 = 3) obtained for the conventional [10] and the proposed combined technique ADAPT-LIN [49] are shown in Figures 2.4.3 (a) and 2.4.3 (b), respectively. It can be seen that the noise in the reconstructed signal is sufficiently suppressed in the case of filtering the real and imaginary parts of the recovered Fourier spectrum separately. For comparison purpose we have also considered another method to reconstruct the signal waveform from a set of mutually shifted noisy realizations. For this method, the first realization 𝑥(1) (𝑖) in (1.4.1) was used as a reference one. Then, for all the other observed realizations 𝑥(𝑚) (𝑖) (𝑚 = 2, . . ., 𝑀), cross-correlation functions for the given 𝑚th realization and the reference realization have been calculated. The position of the global maximum of each cross-correlation function has been determined and used as ̂ . After this, cyclic shifting of the realizations an estimate of the signal random shift 𝜏𝑚 (𝑚) ̂ . Finally, averaging of all the 𝑀 𝑥 (𝑖) (𝑚 = 2, . . ., 𝑀) has been performed by 𝜏𝑚 aligned realizations has been carried out. The resulting output is presented in Figure 2.4.3 (c). As can be seen, due to the errors in the mutual shift estimation, waveform pulses that are originally rectangular become smeared. Their destroyed amplitudes (about 1 and 3) are considerably smaller than the original ones (2 and 6). The numerical simulation results obtained in [49] for three different test signals (all in the form of two rectangular pulses with Δ𝑡12 = 5 and the amplitudes 𝐴 1 = 2, 𝐴 2 = 6, but with different Δ𝑡) are given in Table 2.4.1 for the suggested combined bispectrum-filtering techniques. For the conventional Technique 1 [10], the notation NONE is used in Table 2.4.1. It should be noted, that there is no obvious favorite among these techniques. For the short time pulses (Δ𝑡 = 3), the filter ADAPT-LIN provides the best performance, that is, the largest values of 𝜀 have been reached. The results for ADAPT-NLIN are better than those for LPA-ICI if SNRinp is rather large and vice versa. For the test signal with Δ𝑡 = 7, ADAPT-LIN is the best for small SNRinp = 0.35 but it is the worst for SNRinp = 3.49 and 2.09. Note that for the test signals with Δ𝑡 = 3 and 7, all the three adaptive filters produce improvement for the entire range of SNRinp considered. However, for the test signal with the length of Δ𝑡 = 11 samples, the improvement is observed for only small SNRinp . Both ADAPT-LIN and ADAPT-NLIN produce a reduction in 𝜀 in comparison to the conventional technique (NONE) for SNRinp = 3.2. At the same time, LPA-ICI still performs well. It ensures improvement for the entire range of SNRinp considered. Since the signal and properties of the recovered spectrum are a priori unknown, the advantage of LPA-ICI is that it produces improvement for wider conditions (possible variations of the signal waveform and SNRinp ). Median-ICI always produces worse results than LPA-ICI. Thus, use of local adaptive filters instead of nonadaptive ones is one solution to improve the combined bispectrum-filtering performance. It is shown [49] that the

2.5 Conclusions

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(c) Fig. 2.4.3. Reconstructed signal obtained by: the conventional [10] (a), the proposed method [49] (b), and the method based on cross-correlation and realization alignment (Δ𝑡 = 3 and the minimal considered SNRinp = 0.15) (c).

adaptive filters based on Z-parameter and ICI rule provide stable and considerable benefits of the combined bispectrum filtering techniques in comparison to the conventional bispectrum based processing. The improvement provided is the largest for low (around and less than unity) input SNR.

2.5 Conclusions It has been demonstrated that there are quite a few different approaches to combining bispectrum-filtering processing of mutually shifted noisy realizations with the purpose of unknown signal shape reconstruction. Analysis of statistical properties of noise leaked to bispectrum estimates and signal Fourier spectrum estimates shows that these properties are rather complicated. This makes it problematic for a unique

100 | 2 Unknown noisy signal shape estimation by bispectrum-filtering techniques Table 2.4.1. Numerical simulation results obtained [49] for adaptive filters. Technique

SNRinp

𝜀 SNRinp

𝜀 SNRinp

𝜀 SNRinp

𝜀 SNRinp

𝜀

NONE ADAPT-NLIN ADAPT-LIN LPA-ICI Median-ICI

1.53

9.07 9.92 10.18 9.39 9.08

Test signal 1: Δ𝑡 = 3 0.92 12.80 0,46 16.80 14.50 20.75 15.09 22.40 13.77 20.06 12.94 18.06

0.23 17.07 27.94 34.63 28.74 22.01

0.15 14.17 30.99 42.52 34.95 22.21

NONE ADAPT-NLIN ADAPT-LIN LPA-ICI Median-ICI

3.49

9.42 10.19 9.88 10.02 8.85

Test signal 2: Δ𝑡 = 7 2.09 11.25 1.05 16.30 12.52 21.11 12.31 22.24 12.66 21.61 10.87 16.39

0.52 14.75 23.38 28.30 25.43 18.29

0.35 13.50 27.67 36.85 33.68 20.38

NONE ADAPT-NLIN ADAPT-LIN LPA-ICI Median-ICI

3.20 17.61 15.79 11.83 20.73 15.37

Test signal 3: Δ𝑡 = 11 1.60 16.12 0.80 15.84 0.53 13.57 18.23 24.95 27.55 16.29 26.38 33.43 21.96 32.29 38.95 16.14 19.38 20.08

0.32 10.35 26.55 37.25 44.41 20.54

selection of filtering methods to be able to perform well under conditions of complexvalued nonstationary and non-Gaussian noise when a priori information about signal parameters is limited. Several different approaches have been studied by us during recent years. The basic conclusions are the following. It is expedient to carry out separate filtering of the real and imaginary parts of complex-valued bispectrum or signal Fourier spectrum recovered from bispectrum. Adaptive nonlinear filters are, in general, able to provide a better performance than linear and nonadaptive filters. Recently, several novel adaptive 1-D and 2-D filters developed for the considered application have been proposed and studied. Within combined bispectrum-filtering framework, these filters are able to provide performance improvement for a wide range of input SNR values and typical signal shapes and parameters. The largest benefit due to combining the bispectrum approach with nonlinear and adaptive filtering is commonly provided for low input SNR and/or a limited number of processed realizations. This is very important for practice since performance improvement is a crucial task just for such conditions. Analysis carried out by computer simulations has allowed giving practical recommendations concerning filter parameters like the parameter 𝛽 for adaptive DCT-based filters.

3 Bispectrum-based digital image reconstruction using tapering pre-distortion 3.1 Additive predistortions in reconstruction of the images contaminated by noise and jitter Recently, the majority of the attention in the bispectrum-based signal processing community has been paid to the problems of 1-D unknown waveform signal reconstruction from bispectrum estimates. It is well known that the information about signal shape resides primarily in the signal phase Fourier spectrum and not in the magnitude Fourier spectrum or power spectrum. Since bispectrum estimation provides preservation of the signal Fourier phase, as well as low sensitivity to AWGN and signal translation invariance, it is natural to expect promising results in cases of bispectrum-based 2-D image processing. In fact, several bispectrum-based image reconstruction algorithms have been proposed recently [18, 73–75]. However, existing approaches usually deal with the following restrictions: (1) due to signal shift invariance property, image row Fourier spectrum recovered from bispectrum corresponds to a circularlyshifted row that might cause image distortions and result in problems of image row alignment; (2) signal phase Fourier spectrum recovery from a bispectrum argument provides correct results only when bispectrum and Fourier phase values are within the phase main value interval limited by (−𝜋, +𝜋). Otherwise, phase discontinuities observed as phase wrapping at -𝜋 and +𝜋 values, phase ambiguity and image reconstruction errors arise; (3) phase unwrapping procedure can lead to considerable distortions in the presence of AWGN. Note that phase wrapping is the crucial problem to be solved and phase unwrapping may provoke very large errors in the cases of considerable phase fluctuations and noise influence. Because of this, it will be preferable to search for the ways to avoid phase unwrapping procedure. No phase unwrapping is required and principal arguments of the phase bispectrum are necessary for image reconstruction technique [75] based on the derivation of log-magnitude estimates. However, the main requirement of the approach suggested in [75] is to have no zeros for signal z-transform on the unit circle nor in complex conjugate pairs. However, the last requirement is commonly right only for the limited class of minimum phase signals. Unfortunately, image rows usually belong to the class of nonminimum phase signals. In order to alleviate abovementioned drawbacks, a novel approach to bispectrumbased digital image reconstruction has been proposed in [76–80]. The main idea of the approach suggested in [76–80] is twofold. First, we offer the correction of random shifts of image rows by computing the differences between the adjacent row cross-correlation maximum coordinates that correspond to jittery image and center of gravity values of the neighbor row cross-correlation functions reconstructed from bispectrum

102 | 3 Bispectrum-based digital image reconstruction estimates. Second, in order to reduce phase errors caused by abovementioned rapid phase variations, to avoid phase unwrapping and, hence, to obtain nondistorted reconstructed Fourier phase values, we propose to introduce predistortions added to every processed image row after jitter removal. Let us consider the approach based on introducing additive predistortions placed in each digital image row in the form of large amplitude 𝛿-impulses [76, 78]. We suppose that an unknown object depicted in a digital image is corrupted by AWGN and suppose that relative positions of the image rows are randomly circular shifted due to jitter influence. We also assume that each 𝑘th (𝑘 = 1, 2, 3, . . ., 𝐼) image row is a (𝑚) real valued nonnegative sequence {𝑥𝑘 (𝑖)} (𝑖 = 1, 2, 3, . . ., 𝐼) that is observed at the digital reconstruction and object recognition system input as the following 𝑚th (𝑚 = 1, 2, 3, . . ., 𝑀) realization (𝑚) (𝑚) 𝑥(𝑚) 𝑘 (𝑖) = 𝑠𝑘 (𝑖 − 𝜏𝑘 ) + 𝑛𝑘 (𝑖) , (𝑚)

(3.1.1)

where 𝜏𝑘 is a random shift of the original real-valued deterministic nonminimum phase signal 𝑠𝑘 (𝑖) (original, i.e. noise- and jitter-free image of 𝑘th row), for which bis(𝑚) pectrum is supposed to be nonzero; 𝑛𝑘 (𝑖) is the 𝑚th realization of AWGN with spec2 ified variance 𝜎𝑚 . AWGN is assumed to be uncorrelated to a priori unknown signals {𝑠𝑘 (𝑖)}. We also assume that {𝜏𝑘(𝑚) } are uniformly distributed random variables. The problem is restoring the digital image in order to recognize visually an unknown object depicted in a digital image and contaminated by AWGN and jitter. In practice, jitter that appears in the form of relative random displacement between adjacent rows can be provoked by stochastic properties of telecommunication transmitting/receiving channels, mechanical raster scanning system errors, as well as by data digitizing from a noisy analog image. In the latter case, synchronization pulses are corrupted by AWGN affecting the loss of “lock” in the digitizing device (see, for example, [81]). It should be noted that large jitter value may be one of the main restrictions in high speed digital telecommunication systems. Notice that the considered model (3.1.1) is more complicated compared to the commonly used models described, for example, in [18] or [73]. In these papers, to simulate interference, the total sequence of 16 256 samples (size of original 2-D image was of 127 × 128 pixels) was randomly placed repeatedly in a 1-D noisy frame of 16 384 samples. However, an important aspect of the problem of adjacent rows de-jittering was not treated yet. Furthermore, images restored by the approaches proposed in [18] and [73] are circularly shifted and these images need manual realignment that is quite a time-consuming process. Moreover, this is practically inappropriate for automatic pattern recognition systems. Furthermore, in [18] and [73] only model of tone interference given in the form of five sinusoids with random phases added to each realization was considered and studied.

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To alleviate these shortcomings and restrictions, an original novel approach to image reconstruction in jitter and the AWGN environment is considered below. The proposed image reconstruction technique contains the following sequence of signal processing stages and steps [76, 78]. Stage 1. Jitter removal by correlation and bispectrum-based image row processing. Step 1.1. Evaluation of cross-correlation function estimates computed between two neighbor jittered and noisy image rows (3.1.1) belonging to 𝑚th realization in the form of 𝐼

(𝑚) (𝑚) 𝑅̂ (𝑚) 𝑘,𝑘+1 (𝑙) = ∑ 𝑥𝑘 (𝑖)𝑥𝑘+1 (𝑖 − 𝑙) ,

(3.1.2)

𝑖=1

(𝑚)

where 𝑙 is the delay index and the total number of the functions 𝑅̂ 𝑘,𝑘+1 (𝑙) is equal to

𝐼 − 1 for every 𝑚th realization in (3.1.1).

Step 1.2. Evaluation and storage of coordinates of the maximums of functions (3.1.2) as (𝑚) {𝑅̂ (𝑚) 𝑘,𝑘+1 (𝑙)}max ⇒ {𝑙max 𝑘 }jittered .

(3.1.3)

Step 1.3. Bispectrum reconstruction (BR) of the row cross-correlation function estimates {𝑅̂ 𝑘,𝑘+1 (𝑙)}BR using ensemble averaging of the 𝑀 bispectrum estimates (3.1.2) and the conventional BLW algorithm [1]. Step 1.4. Evaluation and storage of the centers of gravity (CG) {𝑙CG𝑘 }BR of the BR crosscorrelation estimates as follows

{𝑅̂ 𝑘,𝑘+1 (𝑙)}BRCG ⇒ {𝑙CG𝑘 }BR .

(3.1.4)

Steps 1.3 and 1.4 need a more detailed explanation. Since BR always provokes the suppression of the linear Fourier phase factor, bispectrum reconstructed signal will always be centered due to bispectrum properties. According to the abovementioned translation invariance property of bispectrum (1.2.12), the position of a signal reconstructed from bispectrum is determined by signal CG value. Hence, the centering of the functions {𝑅̂ 𝑘,𝑘+1 (𝑙)}BR using the values {𝑙CG𝑘 }BR (see formula (3.1.4)) always takes place. Therefore, after BR of the correlation estimates {𝑅̂ 𝑘,𝑘+1 (𝑙)}BR , they will be centered according to the image row CGs defined as CG𝑘 =

∑𝐼𝑙=1 𝑙𝑅̂ 𝑘,𝑘+1 (𝑙) . ∑𝐼 𝑅̂ 𝑘,𝑘+1 (𝑙)

(3.1.5)

𝑙=1

Due to the low sensitivity of the bispectrum-based signal recconstruction techniques to AWGN, the latter influences quite feebly on the CG values. Therefore, the CGs (3.1.5) can be chosen as reliable reference values for the proposed jitter removal in the AWGN environment.

104 | 3 Bispectrum-based digital image reconstruction Step 1.5. Jittered rows’ alignment procedure carried out on the basis of the following differences (𝑚) Δ(𝑚) (3.1.6) 𝑘 = {𝑙max 𝑘 }jittered − {𝑙CG𝑘 }BR . After jitter removal by using the differences (3.1.6) in (3.1.1), the latter expression can be rewritten as (𝑚) 𝑥(𝑚) (3.1.7) 𝑘corrected (𝑖) ≅ 𝑠𝑘 (𝑖) + 𝑛𝑘 (𝑖) . Stage 2. Bispectrum-based image reconstruction with insertion of the predistortions in the image rows. Step 2.1. Adding to the primary function (3.1.7) some secondary function for which its Fourier spectrum pronouncedly has no zeros and, hence, the total magnitude Fourier spectrum does not contain zeros. As a simple example, just two 𝛿-impulses (here 𝛿(. . . ) is the Kronecker delta function) can be chosen. These 𝛿-impulses are placed in the first and the last pixel contained in the image rows (3.1.7). After jitter removal and introduction of such predistortions, the modified, that is, pre-distorted image row can be written as

𝑓𝑘(𝑚) (𝑖) = 𝐴 0 𝛿(𝑖 − 1) + 𝑥(𝑚) 𝑘corrected (𝑖) + 𝐴 0 𝛿(𝑖 − 𝐼) ,

(3.1.8)

where 𝐴 0 is the pre-distortion function amplitude. In order to provide the absence of zeros in the Fourier spectrum, the amplitude 𝐴 0 must satisfy the following condition 𝐼

𝐴 0 ≫ ∑ 𝑥(𝑚) 𝑘 (𝑖) .

(3.1.9)

𝑖=1

Due to satisfaction of the last condition (3.1.9), the modified image rows (3.1.8) will belong to the minimum phase signal class. The proposed transformation to a minimum phase signal class lets us avoid the abovenoted phase wrapping problem. Therefore, recovery of the image row phase Fourier spectrum from the bispectrum estimate can be obtained uniquely, that is, avoiding ambiguity. At the same time, it should be stressed that CGs of the image rows (3.1.8) occur to be of approximately fixed values due to: – satisfaction of the condition (3.1.9); – quite high robustness of bispectrum-based reconstruction techniques to the AWGN influence. Therefore, in accordance to the proposed strategy, the CGs obtained for different image rows take approximately the same values. Step 2.2. Computation of the bispectrum estimates of the image rows (3.1.8) by the conventional recursive BLW algorithm [10].

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In contrast to the iterative reconstruction technique [75] that is quite complicated and seriously limited by unpredictability of its convergence, the BLW algorithm [10] seems more simple and reliable to us. It should be emphasized especially that the recursive algorithm operates perfectly if the processed signal belongs to the minimum phase signals. Step 2.3. Image row phase and magnitude Fourier spectra recovery from bispectrum estimates by the conventional BLW recursive algorithm [10]. Step 2.4. Row-by-row image reconstruction by inverse Fourier transform of the complex-valued image row Fourier spectra. In order to demonstrate the performance of the suggested approach, let us consider an example referring to the reconstruction of the 8-bit typical test image with dimensions of 𝐼 × 𝐼 = 256 × 256 pixels. The corruption has been simulated in the following way: zero mean AWGN with 2 fixed variance of 𝜎𝑚 = 100 units was added independently to each image row. Ran(𝑚) dom shift 𝜏𝑘 was supposed to be a uniformly distributed value with fixed maximum deviation of ±20 pixels. Every 𝑘th (𝑘 = 1, 2, 3, . . ., 256) image row of the original image was randomly circularly shifted to simulate the jitter influence. The predistortion 𝛿-impulse amplitudes were given as 𝐴 0 = 50 000 to avoid phase wrapping reliably, on one hand, and to provide CGs of different lines to be fixed values, on the other hand. Only a small number of the observed realizations, for example, 𝑀 = 5 frames, were used in our computer simulations for modeling a practically important case often arising in the real-life object recognition systems. Noise- and jitter-free original test image is shown in Figure 3.1.1.

Fig. 3.1.1. The original noise- and jitter-free test image.

106 | 3 Bispectrum-based digital image reconstruction

Fig. 3.1.2. Image contaminated by AWGN and jitter.

Fig. 3.1.3. The cross-correlation function estimate derived between neighbor jittery and noisy image rows.

One arbitrary 𝑚th (𝑚 = 1, 2, . . ., 5) realization of the original image corrupted by 2 = 100 and with row jitter with maximum deviation AWGN with sample variance of 𝜎𝑚 of ±20 pixels is represented in Figure 3.1.2. As it is evident from Figure 3.1.2, the image is completely contaminated by distortions and it is practically impossible to recognize visually an unknown object contained in this image. In accordance to the suggested approach, the cross-correlation function estimates 𝑅̂ (𝑚) 𝑘,𝑘+1 (𝑙) (see Equation (3.1.2)) between every two neighbor jittered and noisy image rows were computed. The cross-correlation estimates are demonstrated in Figure 3.1.3 as the number of 255 1-D functions (“rows”). Lighter color corresponds to larger intensity values.

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Fig. 3.1.4. The cross-correlation function estimate recovered from the noisy bispectrum estimate.

As can be seen from Figure 3.1.3, the maximums of the cross-correlation functions are distributed randomly “from row to row” due to jitter influence. For jitter removal and correct 2-D image alignment we need to derive the differ(𝑚) ences Δ 𝑘 (see Equation (3.1.6)) that have been computed according to the CG values of the abovementioned cross-correlation estimates recovered from bispectrum estimates evaluated for 𝑀 = 5 realizations in our demonstrative example. The cross-correlation function estimates recovered by conventional recursive BLW algorithm [10] are represented in Figure 3.1.4. It is clearly seen from Figure 3.1.4 that the BR cross-correlation function maximums have “ranged” in a vertical white “column”. This “column” will serve us as quite a reliable reference to derive the correction differences (3.1.6). To illustrate typical behavior of an original, that is, noise- and jitter-free image row and its magnitude and phase Fourier spectra, the corresponding plots are shown in Figures 3.1.5–3.1.7, respectively. Because of the presence of several zeros in the magnitude Fourier spectrum (see Figure 3.1.6), the Fourier phase varies very rapidly and phase wrapping is clearly seen in Figure 3.1.7. Such Fourier phase behavior provokes phase ambiguity problems and causes the phase errors in image recovery by the abovementioned recursive bispectrum-based algorithm. That is why, in order to avoid a phase ambiguity problem and to decrease phase errors at the same time, the use of predistortions were proposed by us. In the simplest case, the predistortions just in the form of power 𝛿-impulses (see Equation (3.1.8)) have been employed. To illustrate this important peculiarity, a pre-distorted original image row, its magnitude and phase Fourier spectra are shown in Figures 3.1.8–3.1.10, respectively.

108 | 3 Bispectrum-based digital image reconstruction

Fig. 3.1.5. An arbitrary image row of the original image represented in Figure 3.1.1 (no predistortion).

Fig. 3.1.6. The magnitude Fourier spectrum of the image row in Figure 3.1.5.

Fig. 3.1.7. Image row phase Fourier spectrum corresponding to the original image row in Figure 3.1.5 (no pre-distortion).

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Fig. 3.1.8. Pre-distorted image row.

Fig. 3.1.9. Magnitude Fourier spectrum of the row in Figure 3.1.8.

Fig. 3.1.10. Phase Fourier spectrum of the row in Figure 3.1.8.

110 | 3 Bispectrum-based digital image reconstruction

17x1012 15x1012 13x1012 11x1012 9x1012 7x1012 5x1012 3x1012 1x1012 0

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Fig. 3.1.11. Magnitude bispectrum of a noisy and jittered image row (no pre-distortion).

Since the amplitudes 𝐴 0 of the pre-distortion 𝛿-impulses were chosen in accordance to the condition (3.1.9), each image row will have no zeros in its magnitude Fourier spectrum (see Figure 3.1.9) and, hence, no phase wrapping (see Figure 3.1.10). Therefore, due to the introduction of predistortions, it is possible to uniquely recover signal phase Fourier spectrum by recursive BLW algorithm [10]. In this case, the recursive algorithm operates accurately in the sense of unique phase recovery, decreasing phase errors and suppression of AWGN at the same time. In order to prove this important statement, the plots of the image row magnitude and phase bispectra without and with presence of predistortions are represented in Figures 3.1.11–3.1.14, respectively. From these plots, one can see that magnitude bispectrum corresponding to the image row including predistortions (see Figure 3.1.13) does not have zeros in opposite to the magnitude bispectrum shown in Figure 3.1.11 (predistortions are absent). The phase wrapping values that are clearly seen in Figure 3.1.12 in the total 2-D bispectrum domain are “pushed off” from the limits of the principal calculation triangle domain on the bispectrum plane in Figure 3.1.14. In other words, due to the proposed introduction of predistortions, each processed image row has been transformed to the class of minimum phase signals. Test image reconstructed row by row by the proposed technique is demonstrated in Figure 3.1.15.

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Fig. 3.1.12. Phase bispectrum of a noisy and jittered image row (no pre-distortion).

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Fig. 3.1.13. Magnitude bispectrum estimate of a noisy and jittered image row (pre-distortion is introduced).

112 | 3 Bispectrum-based digital image reconstruction

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Fig. 3.1.14. Phase bispectrum estimate of a noisy and jittered image row (pre-distortion is introduced).

Fig. 3.1.15. Image reconstructed by the proposed technique.

As can be seen, the reconstructed image is essentially cleaner than the distorted version shown in Figure 3.1.2. The object presented in Figure 3.1.15 can clearly be recognized visually. However, there are some artifacts in the reconstructed image: the ringing at the left and right image ends caused by the introduced predistortions. This ringing can be attributed to the Gibbs phenomenon.

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3.2 Bispectrum-based image reconstruction by using multiplicative predistortions Despite solving the problem of recognition of unknown object, image reconstruction technique [76, 78] considered in the previous Subsection results in arising the distortions that are mainly concentrated at the leftmost and rightmost pixels of each reconstructed image row. These errors appear due to the difficulties of additive predistortion compensation at the final stage of data reconstruction, as well as due to spectral leakage. In this Subsection, we consider the technique [77, 79] that produces three benefits: jitter removal, spectral leakage decreasing, and phase ambiguity avoidance. The technique suggested in [77, 79] contains the following data processing stages and steps. Stage 1. De-jittering of adjacent image rows. (𝑚) Step 1.1. Estimation of the sampled cross-correlation function 𝑅̂ 𝑘,𝑘+1 (𝑙) evaluated for each two adjacent jittery and noisy image rows (3.1.1) according to the procedure (3.1.2). (𝑚) Step 1.2. Evaluation and storage of the maximum coordinates {𝑙max 𝑘󸀠 }jit of the functions (3.1.2) according to (3.1.3). (𝑚) Step 1.3. Computations of the jitter correction values Δ 𝑘 by using (3.1.4). Step 1.4. Alignment of jittery rows by (3.1.4) and restoration of the de-jittered rows in the form of (3.1.5). Stage 2. Spectral leakage decreasing and avoidance of Fourier phase spectrum discontinuities. Step 2.1. Multiplication of the de-jittered functions (3.1.5) by some pre-distortion function whose Fourier spectrum pronouncedly has no zeros and, hence, the total function magnitude Fourier spectrum does not contain zeros. As for such pre-distortion, the tapering Gaussian shape function has been chosen in the simplest case. Pre-distorted image row then can be expressed as

𝑓𝑘(𝑚) (𝑖) = 𝑤pr (𝑖)𝑥(𝑚) 𝑘cor (𝑖), 𝑓𝑘(𝑚) (𝐼

− 𝑖 + 1) =

𝑤pr (𝑖)𝑥(𝑚) 𝑘cor (𝐼

𝑖 ∈ [1, 𝐿]

− 𝑖 + 1) ,

(3.2.1)

where 𝑤pr (𝑖) is the pre-distortion tapering function defined by 2

𝑤pr (𝑖) = 𝑒[𝜇(𝐿−𝑖)] ,

(3.2.2)

where variables 𝐿 < 𝐼/2 and 𝜇 determine spread and slope of the multiplicative prefunction (3.1.7), respectively. It should be noted, that signals (3.1.6) will be of maximum phase signals if maximum of the pre-distortion function (3.2.2) satisfies the following condition 𝐼

{𝑤pr (𝑖)}max ≫ ∑ 𝑥(𝑚) 𝑘 (𝑖), 𝑖=1

𝑘 = 1, 2, 3, . . ., 𝐼.

(3.2.3)

114 | 3 Bispectrum-based digital image reconstruction Step 2.2. Computation of the sampled 𝑚th bispectrum estimates according to (𝑚) (𝑚) (𝑚)∗ ∗ 𝐵̂ (𝑚) 𝑓𝑘 (𝑝, 𝑞) = [𝑋𝑘cor (𝑝) ⊗ 𝑊pr (𝑝)] [𝑋𝑘cor (𝑞) ⊗ 𝑊pr (𝑞)] [𝑋𝑘cor (𝑝 + 𝑞) ⊗ 𝑊pr (𝑝 + 𝑞)] ,

(3.2.4) (𝑚) where 𝑋𝑘cor (. . .) and 𝑊pr (. . .) are the direct discrete Fourier transforms of the functions (3.1.5) and (3.2.2), respectively; ⊗ and ∗ denote the convolution and complex conjugation, respectively; 𝑝 = 1, 2, 3, . . ., 𝐼 and 𝑞 = 1, 2, 3, . . ., 𝐼 are the independent spatial frequency indices. Notice that the role of the tapering pre-distortion (3.2.2) is threefold: – to obtain an improved bispectrum estimate (3.2.4) due to spectral leakage decrease in the sense of bias decrease; – to eliminate bispectrum phase wrapping due to the transformation of image rows to the maximum-phase signals; – to fix the coordinate of each 𝑘th image row center of gravity and, hence, to perform automatic alignment of image rows after bicpectrum-based image row reconstruction. Stage 3. Bispectrum-based image row reconstruction. Step 3.1. Image row phase and magnitude Fourier spectra recovery from bispectrum estimates (3.2.4) by the conventional recursive algorithm [10]. Step 3.2. Image row reconstruction by discrete inverse Fourier transform of the image row Fourier spectrum recovered from the bispectrum estimate. Step 3.3. Compensation of the predistortions (3.2.2) by multiplying the reconstructed image rows by the function inverse to (3.2.2). Test images participated in computer simulations are shown in Figures 3.2.1 and 3.2.2.

Fig. 3.2.1. The original noise- and jitter-free test image “Barbara”.

3.2 Using of the multiplicative tapering pre-distortions

|

115

Fig. 3.2.2. The original noise- and jitter-free test image “Letters”.

Fig. 3.2.3. Image “Barbara” corrupted by AWGN and jitter.

Two test objects (“Barbara”) and (“Letters”) corrupted by AWGN with fixed variance of 100 units and jitter are shown in Figures 3.2.3 and 3.2.4, respectively. As can be seen from Figures 3.2.3 and 3.2.4, the images are completely concealed by jitter and AWGN and it is impossible to recognize visually an a priori unknown object (to percept this image). Figures 3.2.5 and 3.2.6 illustrate the images reconstructed by the proposed technique for 𝐿 = 32 pixels and 𝜇 = 0.065 that corresponds to {𝑤pr (𝑖)}max = 14 787 (see the condition (3.2.3)). As can be seen, the reconstructed images are essentially cleaner than the distorted ones represented in Figures 3.2.3 and 3.2.4. The reconstructed objects demonstrated in Figures 3.2.5 and 3.2.6 can be confidently recognized despite the slightly jagged vertical image edges.

116 | 3 Bispectrum-based digital image reconstruction

Fig. 3.2.4. Image “Letters” corrupted by AWGN and jitter.

Fig. 3.2.5. Reconstructed object “Barbara”.

Fig. 3.2.6. Reconstructed object “Letters”.

3.3 Optimal parameters for additive and multiplicative pre-distortion functions

|

117

Unfortunately, there are distortions in the reconstructed images in the form of little circular shifts of some image rows. These distortions are caused by centering of the reconstructed 𝑘th image row with respect to the CG𝑘 coordinate. If the original image row was 𝑠𝑘 (𝑖), then the reconstructed image row 𝑠𝑘̂ (𝑖) is centered with respect to CG𝑘 coordinate and 𝑠𝑘̂ (𝑖) = 𝑠𝑘 (𝑖 − CG𝑘 ). Hence, despite satisfaction of the condition (3.2.3) as a whole, the CG𝑘 coordinates of some pre-distorted image rows may be slightly shifted (jagged) with respect to the central image row pixel. The residual jags of CG𝑘 coordinate can be explained by large gradient of intensities in different image rows.

3.3 Search of the optimal parameters used for additive and multiplicative pre-distortion functions The problem of searching the optimal parameters used both for additive and multiplicative pre-distortion functions has been considered in [80]. Note that in practice, it is really impossible to uniquely assess the performance for existing image reconstruction algorithms. In real-life situations it is possible to estimate approximately the enhancement of a reconstructed image only for uniform image segments by analyzing the interference variance 𝜎2 computed for corrupted image 2 estimated in the reconstructed image. However, and residual interference variance 𝜎res in practice, due to the problem of selecting some really uniform image test segments, 2 the estimates 𝜎2 and 𝜎res can considerably differ from the corresponding true values. Therefore, it is reasonable to use the standard test images owing a priori known structure for assessing correct performance of the image reconstruction algorithms. The interferences are usually inserted artificially in these segments for standard image testing. Usually, investigated images can contain both flat segments and different heterogeneities. The flat image segments contain the pixel intensities close to a local mean value corresponding to large-scale homogeneous image formations. The heterogeneities contain both small-sized or point-like objects and 1-D prolonged objects. Because of this, we have selected several standard different test images for statistical investigations. They are the following: “Baboon” is the high-textured test image containing number of small-sized objects (details) and abrupt intensity edge) (see Figure 3.3.1 (a)); “Barbara” is the test image containing a variety of objects with linear shape and several homogeneous segments (Figure 3.3.1 (b)), and “Lenna” is the test image containing homogeneous objects having sharp edges and a small number of textured segments (Figure 3.3.1 (c)). Original 8-bit test images shown in Figure 3.3.1 have been artificially corrupted by additive mixture of Gaussian noise with zero mean and different variances equal to 70, 100 and 130, as well by impulsive noise of the uniform probability density function.

118 | 3 Bispectrum-based digital image reconstruction

(a)

(b)

(c)

Fig. 3.3.1. Original test images: (a) – Baboon; (b) – Barbara; (c) – Lenna.

(a)

(b)

(c)

Fig. 3.3.2. Original test images distorted by jitter of deviation equal to ±30 pixels: (a) – Baboon; (b) – Barbara; (c) – Lenna.

The amplitudes of the impulses are distributed within the interval of [0, 255]. Different probabilities of pulse appearance equal to 1%, 3% and 5% have been studied. Random deviations of adjacent image rows (jitter) have been modeled by the values equal to ±10, ±30 and ±50 pixels. The test images contaminated by jitter of deviation equal to ±30 pixels are shown in Figure 3.3.2. Note that visual recognition of the original images of Baboon, Barbara and Lenna is impossible by visual inspection of the images represented in Figure 3.3.2. Our goal is to reconstruct these distorted images and to compare the effectiveness of the proposed image reconstruction techniques. The following parameters computed for the estimation of reconstructed image quality and comparative analysis of performances of two different image reconstruction techniques [76, 78] and [77, 79] considered in previous Subsections have been used in our computer simulations: 2 (a) the fluctuation variance 𝜎̄inp for the distorted image 2 𝜎̄ inp =⟨

𝐼 𝐽 2 1 ∑ ∑ [𝑥(𝑚) (𝑖, 𝑗) − 𝑠(𝑖, 𝑗)] ⟩ , 𝐼𝐽 − 1 𝑖=1 𝑗=1 𝑀 1

(3.3.1)

3.3 Optimal parameters for additive and multiplicative pre-distortion functions |

119

where 𝑀1 is the number of statistically independent image frames participating in ensemble averaging denoted in formula (3.3.1) by ⟨. . .⟩; 𝑥(𝑚) (𝑖, 𝑗) is the 𝑚th realization (𝑚 = 1, 2, . . ., 𝑀1 ) of an arbitrary (𝑖, 𝑗)th pixel intensity in the distorted image; (b) the input signal-noise-ratio SNRinp computed for the distorted image as SNRinp = 𝐼−1

𝑃𝑠 , 2 𝜎̄ inp

𝐽−1

(3.3.2) 𝐼−1

𝐽−1

where 𝑃𝑠 = (1/𝐼𝐽) ∑𝑖=0 ∑𝑗=0 [𝑠(𝑖, 𝑗) − 𝐸]2 , and 𝐸 = (1/𝐼𝐽) ∑𝑖=0 ∑𝑗=0 𝑠(𝑖, 𝑗);

2 computed for the reconstructed image as (c) the fluctuation variance 𝜎̄out 2 𝜎̄out = ⟨min {

𝐼 𝐽 2 1 ̂ (𝑖, 𝑗) − 𝑠(𝑖 − 𝑡, 𝑗)] }⟩ , ∑ ∑ [𝑠(𝑚) 𝐼𝐽 − 1 𝑖=1 𝑗=1 𝑀

(3.3.3)

1

̂ (𝑖, 𝑗) and 𝑠(𝑖, 𝑗) are the (𝑖, 𝑗)th pixel intensity in the reconstructed and where 𝑠(𝑚) original images, respectively; 𝑡 is the shift placed in formula (3.3.3) taking into account the bispectrum invariance property with respect to signal shift; (d) the output signal-to-noise ratio SNRout computed for the reconstructed image as SNRout =

𝑃𝑠 ; 2 𝜎̄ out

(3.3.4)

where 𝑃𝑠 is the image power. (e) the parameter 𝜀 which allows assessing the improvement of SNR in the reconstructed image SNRout 𝜀= . (3.3.5) SNRinp The set of parameters (3.3.1)–(3.3.5) serves as an assessment of the reconstructed image quality in computer simulations given below in this Subsection. The problem of estimation of optimal 𝛿-impulse amplitudes from the point of view of the best reconstructed image quality will be considered in this Subsection. Computer simulation results obtained with additive pre-distortion function in jitter and AWGN interference environment are shown in Figure 3.3.3. Comparison of the distorted test images shown in Figure 3.3.2 and the corresponding reconstructed images in Figure 3.3.3 permits concluding that, in opposite to the corrupted images, reliable object recognition is possible by visual inspection of the images shown in Figure 3.3.3. The results of computing the parameter 𝜀 (3.3.5) are demonstrated in Figures 3.3.4– 3.3.6. Analysis of the graphs plotted in Figures 3.3.4–3.3.6 permits noting the following peculiarities:

120 | 3 Bispectrum-based digital image reconstruction

(a)

(b)

(c)

Fig. 3.3.3. Examples of the reconstructed test images that have been artificially corrupted by jitter with deviation of ±30 pixels and Gaussian noise with variance of 130 units: (a) – the pre-distortion 𝛿-impulse amplitude is equal to 21 000; (b) – the pre-distortion 𝛿-impulse amplitude is equal to 49 000; (c) – the pre-distortion 𝛿-impulse amplitude is equal to 44 000.

1,7

1,6

1,5 ɛ 1,4

1,3

1,2 2·104

4·104

6·104

8·104

Pre-distortion amplitude ○○○ xxx ◊◊◊ □□□ +++

Only jitter Jitter + Gaussian noise with variance of 70 Jitter + Gaussian noise with variance of 100 Jitter + Gaussian noise with variance of 130 Jitter + Gaussian noise with variance of 100 + impulsive noise of 3%

Fig. 3.3.4. Parameter 𝜀 as a function of pre-distortion 𝛿-impulse amplitudes computed for the image Lenna.

3.3 Optimal parameters for additive and multiplicative pre-distortion functions |

121

1.4

1.3

ε 1.2

1.1

1 2.104

4.104

6.104

Pre-distortion amplitude ○ ○ ○ Only jitter x x x Jitter + Gaussian noise with variance of 70 ◊ ◊ ◊ Jitter + Gaussian noise with variance of 100 □ □ □ Jitter + Gaussian noise with variance of 130 + + + Jitter + Gaussian noise with variance of 100 + impulsive noise of 3% Fig. 3.3.5. Parameter 𝜀 as a function of pre-distortion 𝛿-impulse amplitudes computed for the image Barbara.

– – –

–

–

the dependence of the reconstructed image efficiency on the pre-distortion 𝛿-impulse amplitude is of nonlinear behavior; small fluctuations of 𝜀 value are observed nearby the global maximum; the curves in Figures 3.3.4 and 3.3.5 are bi-modal, that is, the parameter 𝜀 is small both for too small and too large pre-distortion amplitudes. The curves in Figure 3.3.6 have one global maximum; the optimum pre-distortion 𝛿-impulse amplitudes are of values equal to approximately 45 000 for smooth Lenna and Barbara images, however, the optimum amplitude is equal to 20 000 for high-textured Baboon image; the parameter 𝜀 reduces if additive noise variance increases, there are optimal pre-distortion 𝛿-impulse amplitudes, but the reconstructed image quality is approximately the same in cases of nonoptimal selection.

122 | 3 Bispectrum-based digital image reconstruction

2

1,8 ɛ

1,6

1,4

0

2·104

4·104

6·104

8·104

Pre-distortion amplitude ○ ○ ○ Only jitter x x x Jitter + Gaussian noise with variance of 70 ◊ ◊ ◊ Jitter + Gaussian noise with variance of 100 □ □ □ Jitter + Gaussian noise with variance of 130 + + + Jitter + Gaussian noise with variance of 100 + impulsive noise of 3% Fig. 3.3.6. Parameter 𝜀 as a function of pre-distortion 𝛿-impulse amplitudes computed for the image Baboon.

The parameter 𝜀 decreases, on the average, by 0.00025 due to the influence of small intensity impulsive noise for the probability of pulse appearance equal to 3%. However, the parameter 𝜀 decreases more considerably by about 0.042 if impulsive noise probability increases to 5%. Thus, computer simulation results demonstrated in Figures 3.3.4–3.3.6 indicate good efficiency of the approach suggested for solving the unknown object recognition problems under joint influence of heavy jitter, additive Gaussian and impulsive noise. In order to estimate an optimal multiplicative pre-distortion function shape (3.2.2) from the point of view of maximum value (3.3.5), statistical investigations have been performed for the test image reconstruction. The values of 𝐿 and μ in (3.3.2) have been varied within wide limits. The computer simulation results are presented in Figures 3.3.7–3.3.12. Analysis of the graphs plotted in Figures 3.3.7, 3.3.9, and 3.3.11 shows that 𝜀 value depends nonlinearly upon the varied 𝐿 and 𝜇 parameters. Clear extremum of 𝜀 is observed for the optimal 𝐿 and 𝜇 values.

3.3 Optimal parameters for additive and multiplicative pre-distortion functions

ε

|

123

1,5

1

0

0,5

2 4 6

L

8 10

0,8

4

2

0

8

6 μ

Fig. 3.3.7. Parameter 𝜀 as a function of 𝐿 and 𝜇 obtained for Lenna image corrupted by jitter with maximal deviation of ±30 pixels and mixture of additive Gaussian noise with variance of 100 units and impulsive noise with probability of 3%.

μ

0,4

0,1

0,6 L 0,2 20

40

60

80

100

120

ε

2

Fig. 3.3.8. The curve illustrating optimal selection of the parameters 𝐿 and 𝜇 obtained according to the data represented in Figure 3.3.7.

1,5

1 2 0,5

6 L 10

0

2

6

4 μ

8

Fig. 3.3.9. Parameter 𝜀 as a function of 𝐿 and 𝜇 obtained for Barbara image corrupted by jitter with maximal deviation of ±30 pixels and mixture of additive Gaussian noise with variance of 130 and impulsive noise with probability of 5%.

124 | 3 Bispectrum-based digital image reconstruction 0,8 μ 0,4

0,1

0,6 L 0,2 20

40

60

80

100

120

ε

Fig. 3.3.10. The curve illustrating optimal selection of the parameters 𝐿 and 𝜇 obtained according to the data represented in Figure 3.3.9.

2 1,5 1 0,5

2 L

6 10

0

2

8

6

4 μ

Fig. 3.3.11. Parameter 𝜀 as a function of 𝐿 and 𝜇 obtained for Baboon image corrupted by jitter with maximal deviation of ±30 pixels and mixture of additive Gaussian noise with variance of 130 and impulsive noise with probability of 5%.

0,8 μ 0,4

0,1

0,6

L

0,2 20

40

60

80

100

120

Fig. 3.3.12. The curve illustrating optimal selection of the parameters 𝐿 and 𝜇 obtained according to the data represented in Figure 3.3.11.

3.4 Conclusions

| 125

Behavior of 𝜀 value depending on the optimal 𝐿 and 𝜇 parameters (see Figures 3.3.8, 3.3.10, and 3.3.12) is close to exponential. Though the curves plotted in Figures 3.3.8, 3.3.10, and 3.3.12 look similarly, the function 𝜀(𝐿, 𝜇) differs for each particular image. It should be stressed that 𝜀 value decreases on the average by 0.5 and it decreases slightly under the influence of impulsive noise. Therefore, image reconstruction technique exploiting multiplicative pre-distortion is robust in regard to impulsive noise of small level. Analysis of the computer simulation results permits noting that the proposed technique removes jitter distortions quite effectively. However, different artifacts are observed in the reconstructed images in the form of some brightness distortions. Thus, computer simulation results demonstrate quite a good performance for reliable unknown object recognition under influence of heavy jitter. However, object recognition reliability decreases with presence of additive Gaussian and impulsive noise. Accuracy of bispectral density estimation improves with increasing the observed image frame number. Therefore, one may expect an improvement of the image reconstruction performance.

3.4 Conclusions A novel approach that is promising for reconstruction, recognition and identification of unknown objects in the noise and jitter environment has been proposed and investigated. The proposed approach is based on jitter removal by image alignment with correction coefficients evaluated as the difference between coordinates of maximums of the jittery and noisy line image cross-correlations and the centers of gravity of the cross-correlations recovered from bispectrum estimates. Due to the introduction of the predistortions to the processed image, only the principal arguments of the phase bispectrum are evaluated. Therefore, phase unwrapping is not necessary. Two digital image reconstruction techniques, by using additive and multiplicative predistortions, are studied. Computer simulation results demonstrate high robustness of the proposed techniques in the sense of suppression of quite intensive additive Gaussian noise and jitter removal with only five realizations. Further improvements can be expected if more realizations are available. Computer simulation results performed for three typical test images give an opportunity to select optimal parameters for additive and multiplicative pre-distortion functions from the point of view of maximum reconstructed image quality. The proposed approach can be recommended for automatic object recognition systems that operate under a priori unknown object and noise characteristics and with lack of processing frames in the jitter and noise environment.

4 Signal detection by using third-order test statistics in communications and radar applications 4.1 Detection of deterministic signals by using third-order test statistics and likelihood ratio criterion Matched filtering is a widespread procedure used in modern telecommunication [82] and radar [83] systems for known signal waveform detection in additive Gaussian noise environments. Under influence of AWGN or in cases of known interference spectral density which is necessary for performing de-noising operations, as well as by carrying out synchronization in a receiver, the matched filter (MF) is an optimal device in the sense of minimizing false alarm probability of signal detection and maximizing output SNR. Optimal signal processing is performed commonly by the second-order test statistic (TS) estimation. The latter value is estimated in the form of correlation integral evaluated for a noisy received signal and reference signal with further comparison to the correlation integral value to a threshold [82, 83]. A decision about the presence of a desirable signal is made if the threshold is exceeded. In this Chapter, we consider a novel approach dedicated to solving the problem of known signal waveform detection by using novel third-order TS formed at the MF output [84]. An approach suggested in [84] possesses some attractive and promising benefits compared to the traditional approach based on using second-order TSs. These benefits become apparent, first, in higher noise immunity respective to AWGN and, second, nonsensitivity to random processed signal delays. First, we consider a third-order TS referred to the AWGN and random signal delay environment. Assume that a discrete temporal process {𝑥(𝑚) (𝑖)}𝐼−1 𝑖=0 , 𝑚 = 1, 2, . . ., 𝑀 is observed at the MF input in the form of a sequence of 𝑀 realizations as

𝑥(𝑚) (𝑖) = 𝑠(𝑖 − 𝜏(𝑚) ) + 𝑛(𝑚) (𝑖) ,

(4.1.1)

where 𝑠(𝑖) is an a priori known deterministic noise-free signal; 𝑖 = 0, 1, . . ., 𝐼 − 1 is the temporal sample index; 𝜏(𝑚) is the signal delay that value randomly varies from one realization to another; 𝑛(𝑚) (𝑖) is the 𝑚th arbitrary realization of AWGN with zero mean. A signal waveform 𝑠(𝑖) is assumed to be unchangeable for all 𝑀 observed realizations. Signal and noise are supposed to be statistically independent from each other in (4.1.1). When the process (4.1.1) arrives at the MF input, the MF output 𝑦(𝑚) (𝑖) is equal to 𝐼−1

𝑦(𝑚) (𝑖) = ∑ 𝑥(𝑚) (𝑗)𝑠(𝐼 − 𝑖 + 𝑗) . 𝑗=0

(4.1.2)

4.1 Detection of deterministic signals

|

127

Second, we consider the dual-alternative signal detection problem by using the following two hypotheses H1 (signal is present) and H0 (no signal) as

H1 : 𝑥(𝑚) (𝑖) = 𝑠(𝑖 − 𝜏(𝑚) ) + 𝑛(𝑚) (𝑖) , (𝑚)

H0 : 𝑥

(𝑚)

(𝑖) = 𝑛

(4.1.3a)

(𝑖) .

(4.1.3b)

In order to solve this problem, we propose using a novel detection TS given by third-order autocorrelation function (TOAF) evaluated at the MF output in the point of TOAF origin of coordinates. According to the hypothesis H0 (no signal at the MF input), the response at the MF output is equal to 𝐼−1

𝑦𝑛(𝑚) (𝑖) = ∑ 𝑛(𝑚) (𝑗)𝑠(𝐼 − 𝑖 + 𝑗) ,

(4.1.4a)

𝑗=0

and according to the hypothesis H1 (desirable signal is present at the MF input), the MF output can be written as 𝐼−1

𝑦𝑠(𝑚) (𝑖) = ∑ {[𝑠(𝑗 − 𝜏(𝑚) ) + 𝑛(𝑚) (𝑗)] 𝑠(𝐼 − 𝑖 + 𝑗)} .

(4.1.4b)

𝑗=0

Sampled, that is, estimated for an arbitrary 𝑚th realization (4.1.1), detection TSs given in the form of TOAF and computed in the point of its origin of coordinates (0, ̂ (𝑚) 0), and corresponding to the hypotheses H0 − 𝑅̂ (𝑚) 𝑦𝑛 (0, 0) and H1 − 𝑅𝑦𝑠 (0, 0) can be defined, respectively, as 3

1 𝐼−1 (𝑚) 3 1 𝐼−1 𝐼−1 (𝑚) (𝑚) ̂ (0, 0) = 𝑅 = ∑ [𝑦 (𝑖)] = ∑ [ ∑ 𝑛 (𝑗)𝑠(𝐼 − 𝑖 + 𝑗)] , 𝑅̂ (𝑚) 𝑦𝑛 𝑦𝑛 𝐼 𝑖=0 𝑛 𝐼 𝑖=0 𝑗=0

(4.1.5a)

𝐼−1 3 ̂ (𝑚) = 1 ∑ [𝑦(𝑚) (𝑖))] 𝑅̂ (𝑚) (0, 0) = 𝑅 𝑦𝑠 𝑦𝑠 𝐼 𝑖=0 3

=

1 𝐼−1 𝐼−1 ∑ { ∑ [𝑠 (𝑗 − 𝜏(𝑚) ) + 𝑛(𝑚) (𝑗)] 𝑠(𝐼 − 𝑖 + 𝑗)} . 𝐼 𝑖=0 𝑗=0

(4.1.5b)

Assume that the proposed third-order TS is a random variable with statistical distribution that asymptotically obeys Gaussian law for 𝑀 → ∞. Hence, conditional probability densities corresponding to the signal presence 𝑝(𝑅̂ 𝑦𝑠 |𝐻1 ) and signal ab-

sence 𝑝(𝑅̂ 𝑦𝑛 |𝐻0 ) at the MF output can be written, respectively, as − 12 ( 1 𝑒 𝑝 (𝑅̂ 𝑦𝑠 |𝐻1 ) = √2𝜋𝜎(𝑅̂ 𝑦 )

2

(𝑚) 𝑅̂ 𝑦 −𝑅̄ 𝑦𝑠 𝑠 𝜎(𝑅̂ 𝑦𝑠 )

)

(𝑚) 𝑅̂ 𝑦 −𝑅̄ 𝑦𝑛 𝑛 𝜎(𝑅̂ 𝑦𝑛 )

)

,

(4.1.6)

,

(4.1.7)

𝑠

− 12 ( 1 𝑒 𝑝 (𝑅̂ 𝑦𝑛 |𝐻0 ) = √2𝜋𝜎(𝑅̂ 𝑦 ) 𝑛

2

128 | 4 Signal detection by using third-order test statistics 𝐼−1

𝐼−1

𝐼−1

where 𝑅̄ 𝑦𝑠 = 𝑅̄ 𝑦𝑠 (0, 0) = (1/𝐼) ∑𝑖=0 [𝑦𝑠 (𝑖))]3 = (1/𝐼) ∑𝑖=0 [∑𝑗=0 𝑠(𝑗)𝑠(𝐼 − 𝑖 + 𝑗)]3 is the TOAF at the MF output computed in the origin of coordinates (0, 0) in absence of noise at the MF input;

1 𝑀 1 𝐼−1 (𝑚) 3 1 𝑀 1 𝐼−1 𝐼−1 (𝑚) ∑ ∑ [𝑦𝑛 (𝑖)] = ∑ ∑ [ ∑ 𝑛 (𝑗)𝑠(𝐼 − 𝑖 + 𝑗)] 𝑅̄ 𝑦𝑛 = 𝑅̄ 𝑦𝑛 (0, 0) = 𝑀 𝑚=1 𝐼 𝑖=0 𝑀 𝑚=1 𝐼 𝑖=0 𝑗=0

3

is the third-order MF output averaged by 𝑀 realizations and computed when the signal is absent at the MF input;

𝜎2 (𝑅̂ 𝑦𝑠 ) =

2 1 𝑀 ̂ (𝑚) ∑ [𝑅𝑦𝑠 (0, 0) − 𝑅̄ 𝑦𝑠 (0, 0)] 𝑀 𝑚=1

1 𝑀 1 𝐼−1 𝐼−1 = ∑ { ∑ [ ∑ (𝑠 (𝑗 − 𝜏(𝑚) ) + 𝑛(𝑚) (𝑗)) 𝑠(𝐼 − 𝑖 + 𝑗)] 𝑀 𝑚=1 𝐼 𝑖=0 𝑗=0 3

3

2

1 𝐼−1 𝐼−1 − ∑ [ ∑ 𝑠(𝑗)𝑠(𝐼 − 𝑖 + 𝑗)] } 𝐼 𝑖=0 𝑗=0

is the variance of the third-order TS estimate computed under condition of receiving the additive mixture of desirable signal and noise at the MF input;

𝜎2 (𝑅̂ 𝑦𝑛 ) =

2 1 𝑀 ̂ (𝑚) ∑ [𝑅 (0, 0) − 𝑅̄ 𝑦𝑛 (0, 0)] 𝑀 𝑚=1 𝑦𝑛 3

=

3

1 𝑀 1 𝐼−1 𝐼−1 (𝑚) 1 𝑀 1 𝐼−1 𝐼−1 (𝑚) ∑ { ∑ [ ∑ 𝑛 (𝑗)𝑠(𝐼 − 𝑖 + 𝑗)] − ∑ ∑ [ ∑ 𝑛 (𝑗)𝑠(𝐼 − 𝑖 + 𝑗)] } 𝑀 𝑚=1 𝐼 𝑖=0 𝑗=0 𝑀 𝑚=1 𝐼 𝑖=0 𝑗=0

2

is the variance of the third-order TS estimate computed under condition of receiving only noise (signal is absent). Likelihood ratio for the considered dual-alternative detection problem can be written as 𝐻0

𝑝 (𝑅̂ 𝑦𝑠 |𝐻1 ) > 𝐿, 𝑝 (𝑅̂ 𝑦𝑛 |𝐻0 ) 𝐻
ln [𝐿 − ] . 𝜎2 (𝑅̂ 𝑦𝑠 ) 𝐻< 𝜎2 (𝑅̂ 𝑦𝑛 )

(4.1.9)

1

For comparison of signal detection performance obtained with the proposed thirdorder TSs and the conventional second-order TSs, we consider common autocorrelation functions calculated at the MF output, below.

4.1 Detection of deterministic signals

|

129

In contrast to the above considered third-order TS (see formulas (4.1.5a, b)), we will denote the second-order TSs by symbol 𝑟. Conditional probability densities corresponding to the signal presence 𝑝(𝑟𝑦̂ 𝑠 |𝐻1 ) and signal absence 𝑝(𝑟𝑦̂ 𝑛 |𝐻0 ) at the MF input can be represented as − 12 ( 1 𝑒 𝑝(𝑟𝑦̂ 𝑠 |𝐻1 ) = √2𝜋𝜎(𝑟𝑦̂ ) 𝑠 − 12 ( 1 𝑒 𝑝(𝑟𝑦̂ |𝐻0 ) = √2𝜋𝜎(𝑟𝑦̂ ) 𝑛 𝐼−1

2

(𝑚) ̂ ̄ 𝑠 𝑟𝑦 −𝑟𝑦 𝑠 ̂ 𝑠) 𝜎(𝑟𝑦

)

(𝑚) ̂ ̄ 𝑛 𝑟𝑦 −𝑟𝑦 𝑛 ̂ 𝑛) 𝜎(𝑟𝑦

)

,

(4.1.10)

,

(4.1.11)

2

𝐼−1

̂ 𝑠 = 𝑟𝑦(𝑚) ̂ 𝑠 (0) = (1/𝐼) ∑𝑖=0 {∑𝑗=0 [𝑠(𝑗 − 𝜏(𝑚) )+𝑛(𝑚) (𝑗)]𝑠(𝐼 − 𝑖 + 𝑗)}2 is the samwhere 𝑟𝑦(𝑚) pled autocorrelation function estimate evaluated in the origin of coordinates at the MF 𝐼−1 𝐼−1 input for an arbitrary realization (4.1.1); 𝑟𝑦̄ 𝑠 = 𝑟𝑦̄ 𝑠 (0) = (1/𝐼) ∑𝑖=0 {∑𝑗=0 𝑠(𝑗)𝑠(𝐼−𝑖+𝑗)}2 is the signal autocorrelation function evaluated in the origin of coordinates at the MF 𝐼−1

2

𝐼−1

̂ 𝑛 = 𝑟𝑦(𝑚) ̂ 𝑛 (0) = (1/𝐼) ∑𝑖=0 [∑𝑗=0 𝑛(𝑚) (𝑗)𝑠(𝐼 − 𝑖 + 𝑗)] input with the absence of AWGN; 𝑟𝑦(𝑚) is the noise autocorrelation function evaluated in the origin of coordinates in the case of the signal absence (only noise 𝑛(𝑚) (𝑖) arrives at the MF input) at the MF input; 𝑀 𝐼−1 𝑖−1 𝑟𝑦̄ 𝑛 (0) = (1/𝑀) ∑𝑚=1 (1/𝐼) ∑𝑖=0 [∑𝐽=0 𝑛(𝑚) (𝑗)𝑠(𝐼 − 𝑖 + 𝑗)]2 is the TS averaged by 𝑀 𝑀

𝐼−1

𝐼−1

realizations for signal absence at the MF input; 𝜎2 (𝑟𝑦̂ 𝑠 ) = (1/𝑀) ∑𝑚=1 {(1/𝐼) ∑𝑖=0{∑𝑗=0 𝐼−1

𝐼−1

[𝑠(𝑗 − 𝜏(𝑚) + 𝑛(𝑚) (𝑗)]𝑠(𝐼 − 𝑖 + 𝑗)}2 − (1/𝐼) ∑𝑖=0 [∑𝑗=0 𝑠(𝑗)𝑠(𝐼 − 𝑖 + 𝑗)]2 }2 is the variance of the second-order TS computed for the case of signal and noise mixture at the MF input; 2

𝜎2 (𝑟𝑦̂ 𝑛 ) =

1 𝑀 1 𝐼−1 𝐼−1 (𝑚) ∑ { ∑ [ ∑ 𝑛 (𝑗)𝑠(𝐼 − 𝑖 + 𝑗)] 𝑀 𝑚=1 𝐼 𝑖=0 𝑗=0

2

1 𝑀 1 𝐼−1 𝐼−1 (𝑚) − ∑ ∑ [ ∑ 𝑛 (𝑗)𝑠(𝐼 − 𝑖 + 𝑗)] } 𝑀 𝑚=1 𝐼 𝑖=0 𝑗=0

2

is the variance of second-order statistic estimate computed for the case of signal absence (only noise is observed). The likelihood ratio given for the second-order TS is defined as 𝐻0

𝑝 (𝑟𝑦̂ |𝐻1 ) > 𝐿, 𝑝 (𝑟𝑦̂ |𝐻0 ) 𝐻
0 (𝑟𝑦̂(𝑚) 𝜎2 (𝑟𝑦̂ 𝑠 ) 𝑠 ]. ln [𝐿 − 𝜎2 (𝑟𝑦̂ 𝑛 ) 𝜎2 (𝑟𝑦̂ 𝑠 ) 𝐻< 1

(4.1.13)

130 | 4 Signal detection by using third-order test statistics The condition (4.1.13) permits to evaluate signal detection probability by using secondorder TSs estimated at the MF output in the form of autocorrelation functions. Three typical signal models have been used in computer simulations (see Figure 4.1.1): – S1 – a video pulse of rectangular shape of the amplitude 𝐴 video and the width 𝑇 (see Figure 4.1.1 (a)); – S2 – a linear frequency modulated (LFM) pulse 𝑠(𝑖) = 𝐴 LFM cos[2𝜋(𝑓min + 𝛽𝑖)𝑖], 𝛽 = (𝑓max − 𝑓min )/𝐼(see Figure 4.1.1 (b)); – S3 – two pulses of triangular shape of different amplitudes 𝐴1triangle and 𝐴2triangle . It is the simplest model of aerial target range profile observed in high range resolution radars (see Figure 4.1.1 (c)). Signal observation interval of 𝐼 = 256 samples has been selected for examination. The probabilities of signal detection have been studied under influence of two typical kinds of interferences that are of paramount interest in wireless communication and radar applications. We pay attention to two kinds of interferences: AWGN and random signal delay. According to the equation (4.1.1), a received signal can be randomly shifted by value ±𝜏(𝑚) that varies from one arbitrary 𝑚th realization to another. In our simulations, we used uniform distribution of this random variable with maximal deviation of ±Δ 𝜏 . In radar applications, this kind of interference can be caused by random target or antenna platform displacement and, as a consequence, random delays can appear in received signal. Deviation of random signal delay might be larger than a signal pulse length. For studying the behavior of probability densities of third-order TSs 𝑝(𝑅̂ 𝑦𝑠 |𝐻1 )

given in (4.1.6) and 𝑝(𝑅̂ 𝑦𝑛 |𝐻0 ) in (4.1.7), as well as second-order statistics 𝑝(𝑟𝑦̂ 𝑠 |𝐻1 ) given in (4.1.10) and 𝑝(𝑟𝑦̂ 𝑛 |𝐻0 ) in (4.1.11), the corresponding histograms have been computed. The graphs of the histograms computed for LFM signal with 𝑓min = 10 Hz and 𝑓max = 1000 Hz and for 𝑀 = 10 000 realizations are plotted in Figures 4.1.2 and 4.1.3 for the second- and third-order TSs, respectively. Analysis of the histograms shown in Figures 4.1.2 and 4.1.3 confirms correctness of the abovementioned assumption about Gaussian shapes of conditional probability densities given by formulas (4.1.6) and (4.1.7) for the third-order statistics and formulas (4.1.10) and (4.1.11) given for the second-order statistic, respectively. One should also pay attention to a better separation of the maximums of Gaussian functions in Figure 4.1.3 compared to Figure 4.1.2. The latter peculiarity seems promising for the enhancement of signal detection performance by using third-order statistics. It should be noted that low input SNR values are of prime interest for computer simulations, because just under these conditions noise leaks to the MF output and it causes a decrease of the correct detection probability.

4.1 Detection of deterministic signals

|

131

3

S(i)

2

1

0 0

50

100

(a)

150

200

250

150

200

250

150

200

250

i 4

S(i)

2

0

–2

–4 0

50

100

(b)

i 3

S(i)

2

1

0 0 (c)

50

100 i

Fig. 4.1.1. Signals used in computer simulations: (a) single pulse of rectangular shape S1; (b) LFM signal S2; (c) two pulses of triangular shape S3.

132 | 4 Signal detection by using third-order test statistics

12 11 10 9 8 7 6 5 4 3 2 1 0 5000

10000

15000

20000

25000

30000

Fig. 4.1.2. The histograms of conditional probability distributions plotted for the third-order statistics 𝑝(𝑅̂ 𝑦𝑠 |𝐻1 ) (continuous line) and 𝑝(𝑟𝑦̂ 𝑛 |𝐻0 ) (dashed line).

8 7 6 5 4 3 2 1 0

1000

1600

2200

2800

3200

4000

Fig. 4.1.3. The histograms of conditional probability distributions plotted for the second-order statistics 𝑝(𝑟𝑦̂ 𝑠 |𝐻1 ) (continuous line) and 𝑝(𝑟𝑦̂ 𝑛 |𝐻0 ) (dashed line).

MF, output

4.1 Detection of deterministic signals

|

133

600 400 200 0 0

20

40

60

80

100

120

(a)

160

180

200

220

240

140

160

180

200

220

240

140

160

180

200

220

240

MF, output

1000 500 0 –500 0

20

40

60

80

100

120 i

(b)

MF, output

140 i

1000 500 0 –500 –1000 0

(c)

20

40

60

80

100

120 i

Fig. 4.1.4. Examples of MF outputs (4.1.2) for different input SNR values: (a) noise-free case; (b) input SNR = 0 dB; (c) input SNR = −3 dB.

Typical examples of the processes observed at the MF output are demonstrated in Figure 4.1.4 for the case of the test signal S1 given in a single video pulse embedded in AWGN: pulse length 𝑇 = 6 samples and amplitude 𝐴 video = 10. Analysis of the plots represented in Figure 4.1.4 demonstrates the following behavior of the process arrived at the MF output. The maximum video pulse energy equal to 𝐸 = (𝐴 video )2 𝑇 = 600 corresponds to the peak of MF output of triangle shape and its coordinate corresponds exactly to the video pulse end (see Figure 4.1.4 (a)). False energy peaks caused by AWGN are observed in Figures 4.1.4 (b) and 4.1.4 c. Note that amplitudes of false energy peaks increase with decreasing the input SNR. This results in decreasing the probability of signal detection. The plots of signal detection probabilities as functions of SNR at the MF input are represented in Figures 4.1.5–4.1.7 for three shapes of abovementioned test signals (see Figure 4.1.1). Probability of false alarm 𝑃𝐹𝐴 has been of fixed value and equal to 𝑃𝐹𝐴 = 10−5 . The number of realizations participating in statistical computer simulations 𝑀 = 1000. Continuous curves in Figures 4.1.5–4.1.7 correspond to the suggested third-order TSs and the dashed curves correspond to the common second-order TSs.

134 | 4 Signal detection by using third-order test statistics 1

Probability of detection

0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0 –2 –1 0

1

2

3

4

5

6

7 8 9 10 11 12 13 14 15 16 17 18 SNR, dB

Fig. 4.1.5. Probabilities of signal detection as a function of SNR at the MF input: signal S1, 𝐴 video = 10, 𝑇 = 6 samples, Δ 𝜏 = 11 samples.

1

Probability of detection

0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0 –16 –15 –14 –13 –12 –11 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 SNR, dB

1

2

3

4

Fig. 4.1.6. Probabilities of signal detection as a function of SNR at the MF input: signal S2, 𝐴 LFM = 4, 𝑇 = 𝐼 = 256 samples, 𝑓min = 10 Hz and 𝑓max = 4000 Hz, no random signal shift.

SNR at the MF input is computed as SNR = 10 lg ( 𝐼−1

2

𝑃𝑠 ) , 𝜎𝑛2 𝐼−1

(4.1.14)

where 𝑃𝑠 = (1/𝐼) ∑𝑖=0 [𝑠(𝑖) − 𝑚𝑠 ] ; 𝑚𝑠 = (1/𝐼) ∑𝑖=0 𝑠(𝑖); 𝜎𝑛2 is the variance of AWGN. Analysis of the computer simulation results demonstrated in Figures 4.1.5–4.1.7 permits noting the following conclusions. – Exploiting the proposed third-order TS provides a better performance for detection of signals embedded in AWGN compared to the common second-order TS;

4.1 Detection of deterministic signals

|

135

1

Probability of detection

0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 SNR, dB

3

4

5

6

7

8

9 10 11

Fig. 4.1.7. Probabilities of signal detection as a function of SNR at the MF input: signal S3, 𝐴1triangle = 5 and 𝐴2triangle = 10, the same pulse widths 𝑇1 = 𝑇2 = 9, and shift between the pulses 𝑑𝑡 = 7 samples, Δ 𝜏 = 30 samples.

– –

–

–

Probability of detection referred to the third-order TS is not sensitive to random signal shifts at the MF input; Probability of detection tends approximately to unity for input SNR = 11 dB for the third-order TS and for input SNR = 14 dB for the second-order TS for S1, respectively (see Figure 4.1.5). Therefore, the benefit achieved by using the thirdorder TS is of about 3 dB; Probability of detection tends to unity for input SNR = −2 dB for the thirdorder TS and for input SNR = 1 dB for the second-order TS for S2, respectively (see Fig.4.1.6). Therefore, the benefit achieved with using the third-order TS is about 3 dB; Probability of detection tends to unity for input SNR = 5 dB for the third-order TS and for input SNR = 7 dB for the second-order TS for S3 (see Figure 4.1.7), respectively. Therefore, the benefit obtained with the third-order TS is of approximately 2 dB.

It should be stressed that the produced benefits in signal detection can be explained by high noise immunity of third-order moment functions to AWGN and by invariance property to the random signal shifts. Note that the mentioned benefits are observed for low input SNR values and it is very important for practical applications in digital communication and radar systems. Thus, comparative analysis performed for signal detection performance evaluation with the second- and third-order statistics computed at the MF output permits to making up the conclusion about a promising perspective of the suggested approach in telecommunications and radar systems. Applications of this approach for radio communications and radar systems will be considered in the next Subsections.

136 | 4 Signal detection by using third-order test statistics

4.2 Bispectrum-based encoding technique developed for noisy, multipath and fading radio links Interference immunity is the paramount requirement presented in the modern information transport systems commonly operating under influence of multipath fading and intensive noise in radio communication links. Detection and discrimination of received signals contaminated by radio channel noise and fading is the primary aim for digital mobile, satellite, Wi-Fi and terrestrial radio relay wireless communication systems. Considerable signal attenuation often occurs in a real-life multipath environment makes it extremely difficult for the receiver to detect and discriminate the transmitted signals unless the receiver is provided with some form of spatial (MIMO) or frequency (OFDM) diversity. Considering the fact that receivers are typically required to be of small sizes and, at the same time, they must be robust against multipath fading, it motivates us to pay attention to frequency diversity. We propose a novel frequency diversity strategy whose distinctive property is in generating special multifrequency and bispectrum-organized modulating triplet-signals [85]. The suggested novel frequency encoding technique exploits frequency modulation of the carrier by multiharmonic oscillations generated in the synthesizer in the form of specifically bispectrum-organized pair of triplet-signals. Distinctive property of the suggested approach is in employing such two mutually orthogonal triplet-signals that contain the group of three phase coupled frequency tones. The certain phase coupling is inserted for further signal detection and discrimination of the triplet-signals by using bispectrum estimator built-in at the receiver. Discrimination of received triplet-signals is performed by estimation and comparison of the maximum bimagnitude peaks to each other evaluated in the bifrequency domain. Thus, the main idea behind the suggested bispectrum-based approach is in extraction of phase coupling contained in high-frequency carriers due to the frequency modulation performed by using bispectrum-organized triplet-signals. As opposed to common digital modulation strategy, distinctive information features required for discrimination of two triplet-signals at the receiver corrupted by AWGN and fading are not given by traditional signal parameters (magnitude, frequency or phase values) but by the phase relationships a priori specifically given in the triplet-signals. Assume that low-frequency triplet-signal 𝑠0 (𝑡) modulating the high-frequency carrier and related to the “0” symbol in transmitted binary data is given as 3

𝑠0 (𝑡) = 𝐴 0 ∑ cos(2𝜋𝑓0𝑘 𝑡 + 𝜙0𝑘 ) ,

(4.2.1a)

𝑘=1

and low-frequency triplet-signal 𝑠1 (𝑡) corresponding to “1” symbol in transmitting bit stream is 3

𝑠1 (𝑡) = 𝐴 0 ∑ cos(2𝜋𝑓1𝑘 𝑡 + 𝜙1𝑘 ) , 𝑘=1

(4.2.1b)

4.2 Bispectrum-based encoding technique for wireless communications

| 137

where both frequency and phase relationships are given in the triplet-signals 𝑠0 (𝑡) and 𝑠1 (𝑡) in the form of 𝑓03 = 𝑓01 + 𝑓02 ; 𝜙03 = 𝜙01 + 𝜙02 ; 𝑓13 = 𝑓11 + 𝑓12 ; 𝜙13 = 𝜙11 + 𝜙12 . A block-diagram depicting the outlines of the suggested frequency encoding and the bispectrum-based modulation is demonstrated in Figure 4.2.1. 3

A0Ʃcos(2πf0kt+φ0k) k=1 f03 = f01+f02; φ03 = φ01+φ02 x0(t)

x(t)

x1(t)

X

S0(t) S1(t)

3

A0Ʃcos(2πf1kt+φ1k) k=1 f13 = f11+f12; φ13 = φ11+φ12

Fig. 4.2.1. Bispectrum-organized modulation schema.

In Figure 4.2.1, initial digital data 𝑥(𝑡) arrives at the input of the electronic key. Depending on the symbol 0 or 1 contained in the initial bit stream, the key switches on upper or lower position, respectively. After that, analogue triplet-signal 𝑥0 (𝑡) corresponding to 0 or 𝑥1 (𝑡) corresponding to 1 arrives at the multiplier (denoted by “x” in the schema) in order to perform frequency bispectrum-organized modulation of highfrequency carrier 𝑠(𝑡) to be transmitted. In order to provide required discrimination of the triplet-signals (4.2.1a) and (4.2.1b), special attention should be paid to their orthogonality. It means, that the frequencies 𝑓03 = 𝑓01 + 𝑓02 and 𝑓13 = 𝑓11 + 𝑓12 contained in the triplet-signals (4.2.1a) and (4.2.1b) must be selected in such a manner to provide required frequency resolution of bimagnitude peaks referred to the triplet-signals. This important problem will be considered below. Schema of suggested bispectrum receiver providing detection and discrimination of triplet-signals corrupted by channel noise and fading is shown in Figure 4.2.2. Decision-making is performed on the basis of the ML rule by comparison between each other and the third-order test statistic values evaluated as the triplet-signal bimagnitude peaks at the “Decision making” block. Note that the suggested bispectrum-based approach can be referred to noncoherent detection which does not need received carrier to be phase locked with the transmitted carrier. Therefore, no assumption about carrier phase is given. The following additive mixture 𝑟(𝑡) of low-frequency demodulated triplet-signal 𝑠𝑖 (𝑡) and AWGN 𝑛(𝑡) is observed at the bispectral receiver input as

𝑟(𝑡) = 𝑠𝑖 (𝑡) + 𝑛(𝑡),

𝑖 = 0, 1 .

(4.2.2)

138 | 4 Signal detection by using third-order test statistics Ṡ(f02 ) Ṡ(f03 ) |B0|

x

r(t) = si(t)+n(t)

Decision making |B0| < |B1| xi = 1

FFT

|B0| > |B1| xi = 0

x |B1| Ṡ(f12 ) Ṡ(f13 )

Fig. 4.2.2. Bispectrum-based signal processing in receiver.

Note that the simplified model (4.2.2) describes received signal embedded in AWGN in a single-beam communication channel. The following signal processing procedures are performed at the bispectral receiver. (1) In order to obtain a smoothed triplet-signal bispectrum estimate, ensemble averaging must be executed. For this reason, it is necessary to split the triplet-signal (4.2.2) received during the entire bit length 𝑇𝑏 into 𝑀 short-time segments of the same length. Short-time Fourier transform referred to the 𝑚th arbitrary transient segmented signal (𝑚 = 1, 2, . . ., 𝑀) can be written as (𝑚+1)𝑇𝑏 /𝑀

𝑅̂̇ 𝑖𝑚 (𝑓) =

∫

[𝑠𝑖𝑚 (𝑡) + 𝑛(𝑡)]𝑒−𝑗2𝜋𝑓𝑡𝑑𝑡,

𝑖 = 0, 1

(4.2.3)

𝑚𝑇𝑏 /𝑀

where 𝑠𝑖𝑚 (𝑡) is the transient signal belonging to the 𝑚th arbitrary segment. (2) Two groups of the spectral replicas 𝑆̇0𝑚 (𝑓) and 𝑆̇1𝑚 (𝑓) evaluated in the form of noise-free segmented triplet-signal Fourier transforms are stored in the receiver memory as the following Fourier spectral references (𝑚+1)𝑇𝑏 /𝑀

𝑆̇ 0𝑚 (𝑓) =

∫

𝑠0𝑚 (𝑡)𝑒−𝑗2𝜋𝑓𝑡 𝑑𝑡 ,

(4.2.4a)

𝑠1𝑚 (𝑡)𝑒−𝑗2𝜋𝑓𝑡 𝑑𝑡 .

(4.2.4b)

𝑚𝑇𝑏 /𝑀 (𝑚+1)𝑇𝑏 /𝑀

𝑆̇ 1𝑚 (𝑓) =

∫ 𝑚𝑇𝑏 /𝑀

(3) Segmented triplet-signal bispectrum estimates 𝐵̂̇ 0𝑚 (𝑓1 , 𝑓2 ) and 𝐵̂̇ 1𝑚 (𝑓1 , 𝑓2 ) related to the binary data (0 s and 1 s), respectively, are computed as

𝐵̂̇ 0𝑚 (𝑓1 , 𝑓2 ) = 𝑅̂̇ 𝑖𝑚 (𝑓1 )𝑆̂̇0𝑚 (𝑓2 )𝑆̂̇∗0𝑚 (𝑓1 + 𝑓2 )𝑊(𝑓1 , 𝑓2 ), 𝐵̇̂ 1𝑚 (𝑓1 , 𝑓2 ) = 𝑅̂̇ 𝑖𝑚 (𝑓1 )𝑆̂̇1𝑚 (𝑓2 )𝑆̂̇∗ (𝑓1 + 𝑓2 )𝑊(𝑓1 , 𝑓2 ), 1𝑚

𝑖 = 0, 1

(4.2.5a)

𝑖 = 0, 1

(4.2.5b)

4.2 Bispectrum-based encoding technique for wireless communications

| 139

where 𝑊(𝑓1 , 𝑓2 ) is the window function commonly used for suppression of spectral leakage. For instance, it should be selected Rao–Gabr window [86] given in the form of

𝑊(𝑓1 , 𝑓2 ) =

√3 𝑓12 + 𝑓22 + 𝑓1 𝑓2 [1 − ] , 𝜋3 ( 𝑁2 )2

(4.2.6)

where 𝑓1 , 𝑓2 , ∈ [1, 𝑁] are the discrete frequency variables, 𝑁 is the number of samples contained in a separate arbitrary 𝑚th segment. (4) Computation of smoothed bimagnitude estimates |𝐵̇ 0 (𝑓1 , 𝑓2 )| and |𝐵̇ 1 (𝑓1 , 𝑓2 )| by using ensemble averaging of complex-valued bispectrum estimates (4.2.5a) and (4.2.5b) accomplished over 𝑀 short-time segments as

󵄨󵄨 󵄨󵄨 ̇ 󵄨 󵄨󵄨 󵄨󵄨𝐵0 (𝑓1 , 𝑓2 )󵄨󵄨󵄨 = 󵄨󵄨󵄨⟨𝐵̂̇ 0𝑚 (𝑓1 , 𝑓2 )⟩ 󵄨󵄨󵄨 , 󵄨 𝑀󵄨 󵄨 󵄨 󵄨󵄨󵄨𝐵̇ 1 (𝑓1 , 𝑓2 )󵄨󵄨󵄨 = 󵄨󵄨󵄨⟨𝐵̂̇ 1𝑚 (𝑓1 , 𝑓2 )⟩ 󵄨󵄨󵄨 . 󵄨 󵄨󵄨 󵄨 𝑀 󵄨󵄨

(4.2.7a) (4.2.7b)

(5) Comparing the maximum peak values of the averaged bimagnitude estimates (4.2.7a) and (4.2.7b) to each other. Maximum peak value should appear in the bifrequency coordinate point (𝑓01 , 𝑓02 ) for the function (4.2.7a) in the case of arrival of the triplet-signal (4.2.1a). Maximum peak value of the function (4.2.7b) should appear in the bifrequency coordinate point (𝑓11 , 𝑓12 ) in the case of receiving the triplet-signal (4.2.1b). One makes a decision about receiving the 0 symbol or 1 symbol by comparison of the latter bimagnitude peak values by using ML rule. The problem to be solved at the bispectral receiver can be formulated as follows. After receiving the mixture of signal and AWGN (4.2.2), it is necessary to detect what signal arrived at the receiver input from two possible transmitted binary triplet-signals (4.2.1a) or (4.2.1b). Therefore, the problem to be solved is referred to as discrimination of two known signals in the AWGN environment. Solution of this problem reduces to developing such a system that provides minimum average probability of erroneous decisions with reference to what signal 𝑠0 (𝑡) (4.2.1a) or 𝑠1 (𝑡) (4.2.1b) arrived at the receiver input. BER value usually serves as a conventional quantitative measure of digital communication system performance. Condition for decision-making can be written as 𝑠∗ =𝑠0

𝑃(R/𝑠0 ) > < 𝑃(R/𝑠1 )

(4.2.8)

𝑠∗ =𝑠1

where R is the multidimensional signal space considered in the general case for description of a priory probability density; 𝑃(R/𝑠0 ) and 𝑃(R/𝑠1 ) are the conditional probability densities; 𝑠∗ = 𝑠0 or 𝑠∗ = 𝑠1 is referred to as one of two possible hypotheses.

140 | 4 Signal detection by using third-order test statistics In order to define the optimal rules for signal detection in the AWGN environment, it is commonly accepted to consider conventional correlation functions and energy spectrum as the test statistics. These second-order moment functions are usually exploited in optimal receiving theory based on the optimal ML decision-making rule in energy-based detection. A novel bispectrum-based strategy to solving the problem of discrimination of two known binary triplet-signals in AWGN and fading environments is proposed in [85]. Similarity of the suggested approach to known energy-based approaches is in exploiting the conventional optimal statistical receiving theory by using the likelihood function. It is well known that BER performance of energy-based optimal signal detection using energy-based test statistics is dependent only on signal energy and independent of signal form [87]. Distinctive feature of suggested signal detection and discrimination philosophy is in exploiting the novel decision-making rule based on the estimation of higher-order statistics. The most important information used for suggested bispectrum-based signal detection and discrimination is contained in the phase relationships (phase coupling) a priori given in triplet-signals. The difference between bimagnitude estimates of received triplet-signals embedded in AWGN and bispectrum related to original triplet-signal is proposed as a novel discrimination measure. Proposed likelihood functions using novel third-order test statistics and the bispectrum-organized triplet-signals 𝑠0 (𝑡) and 𝑠1 (𝑡) can be expressed, respectively, as 2 󵄨 󵄨 󵄨 󵄨 [󵄨󵄨󵄨⟨𝐵̂̇ 𝑟 (𝑓1 , 𝑓2 )⟩𝑀 󵄨󵄨󵄨max − 󵄨󵄨󵄨𝐵̇ 0 (𝑓1 , 𝑓2 )󵄨󵄨󵄨max ] 𝑃(R/𝑠0 ) = 𝑐 exp {− }, 󵄨 󵄨2 2󵄨󵄨󵄨𝐵̇ 𝑛(𝑓1 , 𝑓2 )󵄨󵄨󵄨 2 󵄨 󵄨 󵄨 󵄨 [󵄨󵄨󵄨⟨𝐵̇̂ 𝑟 (𝑓1 , 𝑓2 )⟩𝑀 󵄨󵄨󵄨max − 󵄨󵄨󵄨𝐵̇ 1 (𝑓1 , 𝑓2 )󵄨󵄨󵄨max ] 𝑃(R/𝑠1 ) = 𝑐 exp { − }, 󵄨 󵄨2 2󵄨󵄨󵄨𝐵̇ 𝑛(𝑓1 , 𝑓2 )󵄨󵄨󵄨

(4.2.9a)

(4.2.9b)

where |⟨𝐵̂̇ 𝑟 (𝑓1 , 𝑓2 )⟩𝑀 |max is the maximum value of the ensemble averaged bimagnitude estimate, that is, maximum value of modulus of bispectrum estimate 𝐵̂̇ 𝑟 (𝑓1 , 𝑓2 )

of received signal (4.2.2) computed as ⟨𝐵̂̇ 𝑟 (𝑓1 , 𝑓2 )⟩𝑀 = ⟨𝑅̇ 𝑖 (𝑓1 )𝑆̇𝑖 (𝑓2 )𝑆̇∗𝑖 (𝑓1 + 𝑓2 )⟩𝑀 , 𝑖 = 0, 1; ⟨. . .⟩𝑀 denotes the ensemble averaging procedure performed according to (4.2.7a) or (4.2.7b) over 𝑀 segments; 𝑅̇ 𝑖 (𝑓) is the received signal Fourier transform; 𝑆̇ 𝑖 (𝑓) is the Fourier transform related to original and noise free binary tripletsignals (4.2.1a) or (4.2.1b); |𝐵̇ 0 (𝑓1 , 𝑓2 )|max is the maximum bimagnitude value of bispectrum 𝐵̇ 0 (𝑓1 , 𝑓2 ) = 𝑆̇ 0 (𝑓1 )𝑆̇0 (𝑓2 )𝑆̇∗0 (𝑓1 + 𝑓2 ) related to original triplet-signal 𝑠0 (𝑡); |𝐵̇ 1 (𝑓1 , 𝑓2 )|max is the bimagnitude of bispectrum 𝐵̇ 1 (𝑓1 , 𝑓2 ) = 𝑆̇ 1 (𝑓1 )𝑆̇1 (𝑓2 )𝑆̇∗1 (𝑓1 + 𝑓2 ) related to original triplet-signal 𝑠1 (𝑡); |𝐵̂̇ 𝑛 (𝑓1 , 𝑓2 )| is the bimagnitude estimate, that

̇ 1 )𝑁(𝑓 ̇ 2 )𝑁(𝑓 ̇ 1 + 𝑓2 )⟩𝑇 related is, modulus of bispectrum estimate 𝐵̂̇ 𝑛 (𝑓1 , 𝑓2 ) = ⟨𝑁(𝑓 𝑏 to the contribution of AWGN; 𝑐 is the constant. Detailed description of bispectrumbased signal detection and discrimination, as well as obtaining the likelihood functions (4.2.9a) and (4.2.9b) are represented in the previous Sections.

4.2 Bispectrum-based encoding technique for wireless communications

| 141

Taking into account the monotonicity of the exponential function, one can consider not the likelihood functions (4.2.9a) and (4.2.9b) but their arguments. Hence, decision-making rule (4.2.8) related to suggested bispectrum-based test statistics can be expressed as 𝑠∗ =𝑠∗0

2 2 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 [󵄨󵄨󵄨⟨𝐵̂̇ 𝑟 (𝑓1 , 𝑓2 )⟩ 󵄨󵄨󵄨 − 󵄨󵄨󵄨𝐵̇ 0 (𝑓1 , 𝑓2 )󵄨󵄨󵄨max ] > [󵄨󵄨󵄨⟨𝐵̂̇ 𝑟 (𝑓1 , 𝑓2 )⟩ 󵄨󵄨󵄨 − 󵄨󵄨󵄨𝐵̇ 1 (𝑓1 , 𝑓2 )󵄨󵄨󵄨max ] . < 󵄨 󵄨 𝑀 󵄨max 𝑀 󵄨max 𝑠∗ =𝑠∗1

(4.2.10) Discrimination rule (4.2.10) evaluated for two known bispectrum-organized tripletsignals by using the ML rule means that the decision is made for the benefit of the triplet-signal 𝑠0 (𝑡) in the case when bimagnitude estimate |𝐵̂̇ 𝑟 (𝑓1 , 𝑓2 )| of the received signal is less discriminated from bimagnitude |𝐵̇ 0 (𝑓1 , 𝑓2 )| computed for this noise-free

triplet-signal, that is, |𝐵̂̇ 𝑟 (𝑓1 , 𝑓2 )| function is more approximated to |𝐵̇ 0 (𝑓1 , 𝑓2 )| function. Decision is made for the benefit of the triplet-signal 𝑠1 (𝑡) in the case when bimagnitude estimate |𝐵̂̇ 𝑟 (𝑓1 , 𝑓2 )| of received signal is less discriminated from bimagnitude

|𝐵̇ 1 (𝑓1 , 𝑓2 )| computed for this noise-free triplet-signal, that is, |𝐵̂̇ 𝑟 (𝑓1 , 𝑓2 )| function is more approximated to |𝐵̇ 1 (𝑓1 , 𝑓2 )| function.

In order to examine the reliability of the proposed decision-making rule (4.2.10), a computer test for study of “Gaussianity” has been performed for suggested bispectral test statistics |⟨𝐵̂̇ 𝑟 (𝑓1 , 𝑓2 )⟩𝑀 |max by using the following parameters of triplet-signals (4.2.1a) and (4.2.1b): 𝑓01 = 400 Hz, 𝑓02 = 800 Hz, 𝑓11 = 700 Hz, 𝑓12 = 1100 Hz, 𝜙01 = −𝜋/4, 𝜙02 = −𝜋/4, 𝜙11 = 0, 𝜙12 = 𝜋. Optimal bit length required for mutual orthogonality of signal triplets has been given as 𝑇𝑏 = 0.01 s. Test binary message contained 1024 bits that has been transmitted through the noisy channel and signalto-noise ratio (SNR) has been given as the following ratio 𝐸𝑏 /𝑁𝑜 = 10 dB of bit energy 𝐸𝑏 to noise spectral density 𝑁𝑜 . Histograms of the bimagnitude estimates corresponding to transmitting 0 and 1 symbols are shown in Figures 4.2.3 (a) and 4.2.3 (b), respectively. It can be seen from Figure 4.2.3 that bimagnitude estimates are distributed close to standard Gaussian law with mathematical expectation values equal to 𝑚(𝐵0 ) = 1.8 and 𝑚(𝐵1 ) = 1.85. According to the ML rule (4.2.8), signal processing procedures performed at the receiver can be described as follows. First, complex-valued segmented bispectrum estimates are computed, accumulated and ensemble averaged over 𝑀 segmented triplet-signals according to (4.2.7a) and (4.2.7b). Second, the ensemble averaged bimagnitudes are computed and their maximums are evaluated. Third, the latter maximum values are compared with corresponding maximums of bimagnitudes related to the noise-free triplet-signals 𝑠0 (𝑡) and 𝑠1 (𝑡). Finally, a decision is made for benefit of triplet-signal 𝑠0 (𝑡) or 𝑠1 (𝑡) on the basis of the ML rule (4.2.10).

142 | 4 Signal detection by using third-order test statistics N

300 250 200 150 100 50 0

0

1,4

1,6

1,8

2

2,2

2,4

2,6

B0

(a) N

200

150

100

50

0 0 (b)

1,4

1,5

1,6

1,7

1,8 B1

1,9

2

2,1

2,2

Fig. 4.2.3. (a) Histogram of the bimagnitude estimate corresponding to transmitting the symbol 0. (b) Histogram of the bimagnitude estimate corresponding to transmitting the symbol 1.

It should be noted that suggested bispectrum-organized modulation exploits the frequency modulation of the carrier for further discrimination of triplet-signals at the receiver. Because of this, common MFSK scheme [88], that is, strategy exploiting frequency modulation has been selected for comparative analysis performed against suggested bispectrum-organized modulation/demodulation technique. Allocated bandwidth, bit length and bit rate values were selected to be approximately of the same values for the suggested bispectrum-based technique and MFSK. For this reason, FSK4 communication system has been selected as the most similar format for comparison to the suggested bispectrum-organized modulation. First, optimal bispectrum-organized bit length must be defined from the point of view of mutual orthogonality of two triplet-signals.

4.2 Bispectrum-based encoding technique for wireless communications

|

143

Minimum frequency spacing between frequency tones 𝑓1 and 𝑓2 commonly required for providing the orthogonality of the FSK signals is defined by known relationship [89] as 𝑇𝑏FSK = 1/(𝑓1 − 𝑓2 ) (4.2.11) where 𝑇𝑏FSK is the minimum bit length providing maximum bit rate; 𝑓1 and 𝑓2 are the frequencies corresponding to the symbols “0” and “1”, respectively. Maximum period of triplet-signal depends on the contribution of minimum frequency value contained in the triple-frequency signal. According to (4.2.11), frequency spacing between two minimum frequencies belonging to symbols “0” and “1” can be derived as

𝑇𝑏 =

1 , min {𝑓01 , 𝑓02 , 𝑓03 } − min {𝑓11 , 𝑓12 , 𝑓13 }

(4.2.12)

where 𝑇𝑏 is the minimum length of bispectrum-organized bit; min{𝑓01 , 𝑓02 , 𝑓03 } and min{𝑓11 , 𝑓12 , 𝑓13 } are the minimum frequency values corresponding to the symbols “0” and “1”, respectively. Computer simulations dedicated to the search of optimal triplet-signal length were performed. It was defined that the best frequency resolution between the test statistics (4.2.9a) and (4.2.9b) can be achieved according to the following inequality

𝑇𝑏 >

1 . min {𝑓01 , 𝑓02 , 𝑓03 } − min {𝑓11 , 𝑓12 , 𝑓13 }

(4.2.13)

Optimal bispectrum-organized bit length providing mutual orthogonality between two triplet-signals has been derived as

𝑇𝑏opt =

𝑛 GCD {𝑓01 , 𝑓02 , 𝑓03 , 𝑓11 , 𝑓12 , 𝑓13 }

,

(4.2.14)

where 𝑛 is the positive integer; GCD denotes the greater common divisor. Note that maximum bit rate can be achieved for 𝑛 = 1 in (4.2.14). Suggested bispectrum-based approach has been compared with the common M-ary FSK (FSK-4) as modulation format [88] contained four frequency tones. Mutually orthogonal four frequencies related to the symbols 00, 01, 10 and 11 were given in our computer simulations as 𝑓00 = 400 Hz, 𝑓01 = 900 Hz, 𝑓10 = 1400 Hz, and 𝑓11 = 1900 Hz, respectively. Frequency values used in bispectrum-organized binary triplet-signals (4.2.1a) and (4.2.1b) were given according to (4.2.12)–(4.2.14) for computer simulations as: 𝑓03 = 𝑓01 + 𝑓02 (𝑓01 = 400 Hz, 𝑓02 = 800 Hz) and 𝑓13 = 𝑓11 + 𝑓12 (𝑓11 = 700 Hz, 𝑓12 = 1100 Hz). Phase coupling was given by the following values of initial phases in the triplet-signals (4.2.1a) and (4.2.1b) as 𝜙01 = −𝜋/4, 𝜙02 = −𝜋/4, 𝜙03 = −𝜋/2, 𝜙11 = 0, 𝜙12 = 𝜋, 𝜙13 = 𝜋. Bit length 𝑇𝑏 and bit rate 𝑅 values were fixed in computer simulations for the suggested bispectrum-based technique and FSK-4 and equal to 𝑇𝑏 = 100 ms and 𝑅 = 100 bit/s, respectively.

144 | 4 Signal detection by using third-order test statistics Periodogram power spectral density estimate

Power / frequency (dB/Hz)

0 –20 –40 –60 –80 –100

0

1

2

(a)

3 4 Frequency (kHz)

5

6

Periodogram power spectral density estimate

Power / frequency (dB/Hz)

0 –20 –40 –60 –80 –100 (b)

0

0.5

1

1.5 2 Frequency (kHz)

2.5

3

Fig. 4.2.4. Periodograms computed for test bit stream: (a) triplet-signals; (b) FSK-4 signals.

In order to estimate and compare allocated bandwidths occupied by FSK-4 and triplet-signals, the corresponding periodograms have been computed for test bit stream containing 100 transmitted bits. The graphs of the periodograms are shown in Figure 4.2.4. It can be seen from Figure 4.2.4 that allocated bandwidths occupied by binary signal-triplets and FSK-4 signals are equal to Δ𝑓triplet = 1400 Hz and Δ𝑓FSK-4 = 1500 Hz, respectively. Therefore, the bandwidth Δ𝑓triplet referred to triplet-signal is approximately of the same value as compared with bandwidth Δ𝑓FSK-4 occupied by FSK-signal. The spectral efficiencies estimated for bispectrum-based technique and FSK-4 are equal to 𝑅/Δ𝑓triplet = 100 bit/s/1400 Hz = 0.071 bit/s/Hz and 𝑅/Δ𝑓FSK-4 = 100 bit/s/1500 Hz = 0.067 bit/s/Hz, respectively. Therefore, the latter values are selected to be approximately of the same values for correct comparison of their performances.

4.2 Bispectrum-based encoding technique for wireless communications

B0 x 10–3

B1 x 10–4

X: 78 Y:39 Z:0,002334

3 2

145

X: 79 Y:37 Z:0,005525

6 4

1

2

0 100 q

50 0 0

(a)

|

50

100 p

150

0 100 q

50 0 0

50

100 p

150

(b)

Fig. 4.2.5. Bimagnitudes computed by using averaging performed over eight segments (𝑀 = 8) and 50% segment overlapping, the Rao–Gabr window is used: (a) |𝐵̇ 0 (𝑝, 𝑞)|, (b) |𝐵̇ 1 (𝑝, 𝑞)|.

The graphs of the noise-free digital-valued bimagnitudes |𝐵̇ 0 (𝑝, 𝑞)| and |𝐵̇ 1 (𝑝, 𝑞)| (𝑝 and 𝑞 are the frequency indexes) corresponding to test transmitting the triplet-mark and using the Rao–Gabr smoothing window (4.2.6) are shown in Figure 4.2.5. Number of segments participated in computer simulations was equal to 𝑀 = 8 segments and 50% overlapping of the segments was exploited to obtain considerably smoothed bimagnitude estimates. Comparing the graphs in Figure 4.2.5 to each other demonstrates the following peculiarities. First, rather good resolution and discrimination of the bimagnitude peaks are observed as follows: |𝐵̇ 0 (𝑝, 𝑞)|max = 0.002334 and |𝐵̇ 1 (𝑝, 𝑞)|max = 0.0005525. Second, considerable reducing of spectral leakage in bimagnitude functions is clearly seen provided by using the Rao–Gabr window (4.2.6). Performance has been evaluated by computation of BER values depending upon SNR = 𝐸𝑏 /𝑁0 . Two scenarios have been considered and studied. According to the first scenario, simplified AWGN channel model, that is, nonfading environment has been examined. According to the second scenario, a fading communication channel has been investigated. In order to study fading immunity, only simple flat fading mode has been considered. In this case, the coherence bandwidth of the communication link is supposed to be larger than the bandwidth of the signal. Therefore, all frequency components of the signal will have the same influence of fading. It is well known that fading provokes signal loss after signal transmitting over realistic wireless communication links. Because of this, in our computer simulations fading influence has been studied as follows. We pay attention to the most difficult, that is, destructive interference and fast fading case. It means that deep attenuation of signal amplitude arriving at the receiver input occurs and received signal magnitude can fluctuate randomly within rather a short time interval, that is, within the bit length. Under this assumption, simplified received signal model (4) has been transformed to the following form

𝑟fading (𝑡) = 𝜀(𝑡)𝑠𝑖 (𝑡) + 𝑛(𝑡),

𝑖 = 0, 1 ,

(4.2.15)

146 | 4 Signal detection by using third-order test statistics where

𝜀(𝑡) = {

0, 𝑡 < 𝜏fad 1, 𝑡 ≥ 𝜏fad

𝑡 ∈ [0, 𝑇𝑏 ]

is the fading coefficient randomly attenuating triplet-signal amplitude within the given short-time interval 𝜏fad . We suppose that the location of short-time interval 𝜏fad given in (4.2.15) can randomly vary within the bit length 𝑇𝑏 from one bit to another in the received test bit stream. The graphs of the averaged BER as a function of SNR are plotted in Figure 4.2.6 for the test binary message contained 210 randomly generated bits both for common FSK4 modulation and suggested the bispectrum-organized encoding technique. In order to estimate the averaged BER value, thirty independent Monte Carlo runs have been generated, that is, thirty separate transmitting attempts have been produced. Limited BER value equal to 10−3 was selected for demonstrating the gain provided by the suggested bispectrum-based technique as compared to the common FSK-4 technique. As can be seen from Figure 4.2.6, BER value equal to 10−3 is reached by using the common FSK-4 technique for SNR = 𝐸𝑏 /𝑁0 = 11 dB. The suggested bispectrumbased technique provides averaged BER equal to 10−3 for SNR ≈ 2 dB using the Rao– Gabr window and SNR = 3 dB without windowing. Therefore, the suggested technique outperforms the common FSK-4 technique by 9 dB and 8 dB with windowing and without it, respectively. Thus, the suggested bispectrum-based approach provides considerable gain of noise immunity in the considered simplified AWGN wireless communication channel model. BER 100 3 10–1

2 10–2 1 10–3 –2

0

2

4

6

8

10

12

Eb/No Fig. 4.2.6. Nonfading case: simplified AWGN channel model. Averaged BER as a function of SNR: (1) bispectrum-organized modulation by using the Rao–Gabr window; (2) bispectrum-organized modulation without using windowing; (3) common FSK-4.

4.2 Bispectrum-based encoding technique for wireless communications

100

| 147

BER

4

10–1 1

3

2

–2

10

10–3

0

2

4

6

8 10 Eb/No

12

14

16

18

Fig. 4.2.7. Case of fading: averaged BER as a function of SNR computed for FSK-4: (1) no fading; (2) case of fading with 𝜏fad = 𝑇𝑏 /16; (3) case of fading with 𝜏fad = 𝑇𝑏 /4; (4) case of fading with 𝜏fad = 𝑇𝑏 /2.

10–1

BER

3 10–2 1 2 10–3 –2

–1

0

1

2 3 Eb/No

4

5

6

7

Fig. 4.2.8. Case of fast fading. Averaged BER as a function of SNR computed for bispectral technique: (1) no fading; (2) case of fading with 𝜏fad = 𝑇𝑏 /4; (3) case of fading with 𝜏fad = 𝑇𝑏 /2.

Computer simulation results obtained for mixture of fast fading and AWGN are represented in Figures 4.2.7 and 4.2.8 for common FSK-4 and suggested bispectrumbased techniques, respectively. From comparing the graphs represented in Figures 4.2.7 and 4.2.8 it shows that the bispectrum-based technique considerably outperforms the common FSK-4 format from the point of view related to fading resistance. Even though the amplitude of the triplet-signal hops to zero value during the fading time interval equal to 𝜏fad = 𝑇𝑏 /2, the suggested bispectrum-based technique provides BER = 10−3 corresponding to SNR = 6.5 dB (see the curve #3 in Figure 4.2.8). At the same time, common FSK-4 provides the BER value equal to 10−3 for 𝜏fad = 𝑇𝑏 /2 only for SNR = 17 dB (see the curve #4 in Figure 4.2.7). Hence, the novel bispectrum-based encoding technique outperforms the common FSK-4 technique by more than 10 dB.

148 | 4 Signal detection by using third-order test statistics Achieved fading resistance gain can be explained as follows. BER performance in common energy-based receivers largely depends on the transmitted signal energy, that is, decision-making is performed at the matched filter output by estimation of signal energy. Because of this, deep fading attenuating received signal amplitude provokes a considerable decrease of signal energy and, as a consequence, BER performance provided by the common energy-based decision-making strategy is worse. BER performance provided by the suggested bispectrum-based receiver depends, first of all, on phase coupling contained in the triplet-signal. Because of this, despite considerable attenuating triplet-signal amplitude, phase coupling information remains to be preserved for true triplet-signal discrimination and correct decision-making is provided for data transmission over the fading communication channel. The suggested strategy could be promising in digital communications from the aspect of transmitted power-saving in mobile communication base stations and modern Wi-Fi systems including the indoor wireless channels, or increasing the area of coverage for fixed transmitted power in these communication systems, or decreasing the number of base stations in cellular communication systems with preservation of coverage area. Therefore, the proposed technique may be a rather simple and costeffective one for modern telecommunications market demands. The proposed bispectrum-based technique does not require considerable frequency bandwidth expansion as, for example, orthogonal frequency division multiplexing (OFDM) multicarrier communication systems. The essential requirement in the approach suggested is in phase coupling given between the frequencies in tripletsignals. These low-frequency values given in modulating signals can be generated rather close to each other. The suggested approach could also be useful for decreasing fading influence by using one of the frequencies contained in triplet for channel transfer function blind diagnostics, that is, to determine the quality of radio links.

4.3 Naval surface target detection and recognition by estimation of radar range profiles Recently, automatic target recognition (ATR) techniques based on the estimation of radar signatures are of particular interest [90–94]. Radar range profile (RP) is a kind of radar signature that is often used in ATR systems. There is a variety of approaches to solving ATR problems such as optimal statistical pattern recognition using wideband and multifrequency signals [90], the use of neural network-based classifiers [94], the target shape reconstruction by bispectrum-based techniques [91–93], and so on. Such variability of approaches deals with the following. First, there are many different factors affecting the quality of the obtained RPs and the performance of ATR methods. These factors are target, as well as antenna platform rotation and translation

4.3 Naval surface target detection and recognition by estimation of radar range profiles | 149

during data measurements; noise and/or clutter influence; heterogeneity of propagation channel, and so on. Because of this, the ATR is a complicated problem. Second, the performance of ATR systems depends upon the radar characteristics and operation principles, the effectiveness of signal processing techniques, the robustness of RP characteristics (information features) and the effectiveness of a classifier. Considerable attention has recently been paid to improving the ATR system performance in different manners. For example, wideband radar signals are used to obtain information about target length and shape by RPs [90]. Bispectrum-based methods possess good immunity to target translation and rotation and, hence they permit to quasi-coherently accumulate the number of RPs [91–93]. Although naval surface targets have relatively low velocities in comparison to aerial targets, the naval object RPs obtained by conventional radar signal processing methods still depend on target aspect angle and they are translation variant. Another peculiar feature that restricts naval object identification in the maritime environment is the presence of radar echo interference caused by backscattering of electromagnetic waves from the surface of the sea. There are many papers dealing with sea clutter study (see, e.g., [95]) that demonstrate the complicated and nonstationary nature of the statistical characteristics of the backscattered signal which depends on random dynamic of sea clutter, grazing angle of observations, radar signal frequency and polarization. Because of this, it is usually rather difficult to suppress sea clutter to extract naval target RP from sea clutter. In this Subsection, we consider the bispectrum-based approach to naval object RP reconstruction using experimentally measured naval object signatures obtained by Xband polarimetric radar for the cases of non-Gaussian sea clutter interference [28]. First, we consider conventional approaches to RP estimation by using averaging of the received echo signal envelopes. Let us describe a spatial-temporal model of received signals that correspond to backscattering from a naval surface object and interference that is caused by sea surface backscattering. We assume that the dimensions of the object are considerably larger than radar wavelength. The relative location of transmitting-receiving on-land radar antenna, the backscattering area 𝛺 that is limited by antenna beamwidth, the naval object and backscattering sea surface are shown in Figure 4.3.1. The plane of Cartesian coordinate system XOY is consistent with antenna aperture and its origin is positioned onto the aperture center. Total area 𝛺 can be divided into a set of independent elementary surfaces Δ𝜃 with reflection coefficients that are different for the metallic object and the sea surface. Assume that the antenna is fixed and its pattern maximum is oriented to the object effective reflection center 𝜃0 . Define the coordinates of elements Δ𝜃 of the area 𝛺 by vector 𝜃 = (𝜃𝑥 , 𝜃𝑦 ) with directive cosines 𝜃𝑥 , and 𝜃𝑦 . Suppose that a polarimetric radar antenna irradiates narrow-band radar signal in the form of 𝑀 RF impulse chain with rectangular shape and without any carrier

150 | 4 Signal detection by using third-order test statistics y On-land radar θx

x θ0

θy θ y0 Δθ R Ω Sea

Target

Fig. 4.3.1. Locations of coastal radar and naval surface object.

̇ (𝑡) as modulation. Let us present the received signal 𝑠𝑖𝑘 ̇ (𝑡) = 𝑆̇𝑖𝑘 (𝑡) exp(𝑗2𝜋𝑓0 𝑡) , 𝑠𝑖𝑘

(4.3.1)

where 𝑡 ∈ [−𝑇/2, 𝑇/2] denotes time interval of observation; 𝑖, 𝑘 = {𝐻, 𝑉} are the indices that correspond to horizontal and vertical polarization, respectively;

1, |𝑡 − 𝑚𝑇𝑟 | ≤ 𝜏𝑝/2 𝑆̇𝑖𝑘 (𝑡 − 𝑚𝜏𝑝 ) = { 0, |𝑡 − 𝑚𝑇𝑟 | > 𝜏𝑝/2 is the complex envelope; 𝜏𝑝 denotes the pulse length; 𝑇 = 𝑇𝑟 𝑀 is the total radar transmitting/processing time, 𝑇𝑟 is the pulse repetition period; 𝑚 = 1, 2, 3, . . ., 𝑀 defines the irradiated pulse number; 𝑓0 is the central frequency, 𝑓 ∈ [𝑓0 − Δ𝐹/2, 𝑓0 + Δ𝐹/2] where the bandwidth Δ𝐹≪𝑓0 ; 𝑗 = √−1. During the total time interval 𝑇 the object position on the sea surface can considerably vary, its orientation and visibility can also vary. In order to obtain a smooth estimate of the RP, taking into account the dynamic behavior of the maritime environment, the common signal processing techniques [94] presume dividing the time interval 𝑇 into 𝑁 segments and carrying out the averaging performed over 𝑁 realizations. Assuming that additive noise (internal and background noise) is negligible, the 𝑛th (𝑛 = 1, 2, . . ., 𝑁) echo signal realization or the 𝑛th temporal scan observed at the amplitude detector output can be written as (𝑛) ̇ ̇ ̇ (𝜃, 𝑙Δ𝑡)𝑆[𝑙(Δ𝑡 𝑆(𝑛) + 𝑇𝑟 ) − 𝜏(𝜃)]𝑑𝜃} , 𝑖𝑘 (𝑙Δ𝑡) = Re{ ∫ 𝐹(𝜃)𝜀𝑖𝑘

(4.3.2)

𝛺 (𝑛)

̇ is the complex-valued antenna pattern; 𝜀̇ (𝜃) is the 𝑛th relative (normalwhere 𝐹(𝜃) 𝑖𝑘 ized for squared surface unit equal to 1 m2 ) complex-valued backscattering coefficient; 𝜏(𝜃) = 2𝑅(𝜃)/𝑐 denotes the time delay; 𝑅(𝜃) and 𝑐 are the slant range and

4.3 Naval surface target detection and recognition by estimation of radar range profiles

| 151

speed of light, respectively; 𝑙 = 1, 2, . . ., 𝐿 is the time pulse index; 𝐿 = 𝑀/𝑁 is the total sample number contained within one scan; Δ𝑡 is the pulse repetition period corresponding to the range bin Δ𝑅 = 𝑐Δ𝑡. Equation (4.3.2) is valid under spatial-temporal band-limitedness, that is, 2Δ𝐹/𝑓0 < 𝜆 0 /𝐷, 𝜆 0 = 𝑐/𝑓0 , where 𝐷 denotes the antenna aperture size. (𝑛) ̇ (𝜃, 𝑙Δ𝑡) in (4.3.2) can be evaluated by the sum of contribuThe coefficients 𝜀𝑖𝑘 (𝑛)

tions referred to random reflection components 𝜀𝑇̇ 𝑖𝑘 (𝜃, Δ𝑡) corresponding to the radar echoes from the elementary surfaces Δ𝜃 of the randomly moving target and the ran(𝑛) dom contribution 𝜀𝑆̇ 𝑖𝑘 (𝜃, 𝑙Δ𝑡) provoked by sea clutter. These two components are statistically independent because they result from a principally different physical origin. (𝑛) ̇ (𝜃, 𝑙Δ𝑡) can be expressed as Hence, 𝜀𝑖𝑘 (𝑛) (𝑛) (𝑛) ̇ (𝜃, 𝑙Δ𝑡) = 𝜀𝑇𝑖𝑘 ̇ (𝜃, 𝑙Δ𝑡) + 𝜀𝑆𝑖𝑘 ̇ (𝜃, 𝑙Δ𝑡) , 𝜀𝑖𝑘

(4.3.3) (𝑛)

where the first term related to the signal polarization matrix coefficients 𝜀𝑇̇ 𝑖𝑘 (𝜃, Δ𝑡) =

̇ 𝑖𝑘 (𝜃, Δ𝑡)| exp[𝑗𝜉𝑇(𝑛)𝑖𝑘 (𝜃, Δ𝑡)] and the second term is referred to the interference |𝜀𝑇(𝑛) (𝑛) (𝑛) (sea clutter) polarization matrix random coefficients 𝜀𝑆̇ 𝑖𝑘 (𝜃, Δ𝑡) = |𝜀𝑆̇ 𝑖𝑘 (𝜃, Δ𝑡)| ⋅ exp[𝑗𝜉𝑆(𝑛)𝑖𝑘 (𝜃, Δ𝑡)]. (𝑛) ̇ (𝜃, Δ𝑡) contains the contributions caused by the following Note that the term 𝜀𝑆𝑖𝑘 four electromagnetic wave propagation paths: – antenna-sea surface-antenna; – antenna-sea surface-target-antenna; – antenna-target-sea surface-antenna; – antenna-sea surface-target-sea surface-antenna. Conventional RP estimate is evaluated by using the averaging performed over 𝑁 realizations. Due to statistical independence of signal and interference contributions, we obtain (𝑛) ̃ (𝑛) ̄(𝑛) ]⟩ + ⟨𝑆̃(𝑛) 𝑆̂𝑖𝑘 (𝑙Δ𝑡) = ⟨𝑆(𝑛) 𝑆 [𝑙Δ𝑡 − 𝜏𝑆̄ ]⟩ , 𝑖𝑘 (𝑙Δ𝑡)⟩ = ⟨𝑆𝑇 [𝑙Δ𝑡 − 𝜏𝑇 (𝑛)

(𝑛)

(4.3.4)

where 𝑆̃𝑇 (. . .) and 𝑆̃𝑆 (. . .)are the 𝑛th radar echo envelopes smoothed by antenna pat(𝑛) tern and corresponding to the target and sea backscattering, respectively; 𝜏𝑇̄ denotes the temporal lag integrated relatively to contributions caused by all target backscat(𝑛) tering centers during the 𝑛th scan; 𝜏𝑆̄ is the time lag integrated relatively to all sea backscattering elements during the 𝑛th scan; ⟨. . . ⟩ denotes the averaging performed (𝑛) (𝑛) over 𝑁 observed scans. Both 𝜏𝑇̄ and 𝜏𝑆̄ are random values that vary from one realization to another. The analysis of the relationships (4.3.1)–(4.3.4) reveals the following. (1) The antenna integrates both the multiple reflections and sea clutter contributions ̇ cut-off property. As the reinto common temporal signals due to influence of 𝐹(𝜃) sult of random target displacements on the sea surface, the object response is a

152 | 4 Signal detection by using third-order test statistics fluctuation process whose parameters vary from one scan to another and it possesses specific PDF. (2) The presence of the abovementioned four propagation paths leads to different (𝑛) (𝑛) time lags 𝜏𝑇̄ and 𝜏𝑆̄ contained in received signals (4.3.4), and these lags vary randomly from one scan to another. The target and sea responses are independent processes for which the corresponding PDFs and correlation intervals differ from each other. The correlation intervals for the processes that correspond to reflections from sea surface elements are comparable to one scan processing time. Therefore, the response from the sea is a rapidly fluctuating nonstationary process that varies from one scan to another. Thus, the time-varying sea echo responses that usually overlap with the object response can destroy the estimate (4.3.4). Recall that the bispectrum-based RP estimation techniques have the following two benefits: 1) possibility to preserve Fourier spectrum phase information; 2) insensi-

̂̇ 𝑞) of observations tivity to processed signal temporal lags. Bispectrum estimate 𝐵(𝑝, (4.3.2) can be derived by direct technique and written as ̂ ̂̇ 𝑞)󵄨󵄨󵄨󵄨 𝑒𝑗𝛾(𝑝,𝑞) ̂̇ 𝑞) = 󵄨󵄨󵄨󵄨𝐵(𝑝, = ⟨𝑋̇ (𝑛) (𝑝)𝑋̇ (𝑛) (𝑞)𝑋̇ (𝑛)∗ (𝑝 + 𝑞)⟩ , 𝐵(𝑝, 󵄨󵄨 󵄨󵄨

(4.3.5)

̂̇ 𝑞)| and 𝛾(𝑝, ̂ 𝑞) are the magnitude and phase bispectrum estimates, rewhere |𝐵(𝑝, spectively; 𝑝 = 1, 2, . . ., 𝐿 and 𝑞 = 1, 2, . . ., 𝐿 are the independent frequency in(𝑛)

(𝑛) dices; 𝑋̇ (𝑛) (. . .) is the direct Fourier transform of (4.3.2): 𝑋̇ (𝑛) (𝑝) = 𝑆̇𝑇 (𝑝)𝑒𝑗2𝜋𝜏𝑇

𝑝

+ (𝑛) (𝑛) (𝑛) ̇ ̇ ̃ ̇𝑆(𝑛) (𝑝)𝑒 ; 𝑆𝑇 (𝑝) and 𝑆𝑆 (𝑝) are the Fourier transforms of the object 𝑆𝑇 (. . .) and 𝑆 (𝑛) sea 𝑆̃𝑆 (. . .) radar responses, respectively; * denotes complex conjugation. 𝑗2𝜋𝜏𝑆(𝑛) 𝑝

The expression (4.3.5) can be rewritten on basis of (4.3.4) with regard to statistical independence of the object and sea radar responses. Then, due to the signal shift invariance property of bispectrum, the following formula can be written

̂̇ 𝑞) = 𝐵̂̇ (𝑝, 𝑞) + ⟨𝑆̇(𝑛) (𝑝)𝑆̇(𝑛) (𝑞)𝑒−𝑗2𝜋𝜏𝑇(𝑛) (𝑝+𝑞) ⟩ ⟨𝑆̇ (𝑛)∗ (𝑝 + 𝑞)𝑒𝑗2𝜋𝜏𝑆(𝑛) (𝑝+𝑞) ⟩ 𝐵(𝑝, 𝑇 𝑇 𝑇 𝑆 (𝑛)

(𝑛)

(𝑛)

(𝑛)

𝑗2𝜋𝜏𝑇 𝑞 −𝑗2𝜋𝜏𝑆 𝑞 ̇(𝑛)∗ ⟩ ⟨𝑆̇(𝑛) ⟩ + ⟨𝑆̇(𝑛) 𝑇 (𝑝)𝑆𝑇 (𝑝 + 𝑞)𝑒 𝑆 (𝑞)𝑒 𝑗2𝜋𝜏𝑇 𝑞 −𝑗2𝜋𝜏𝑆 𝑝 ̇(𝑛)∗ ⟩ ⟨𝑆̇ (𝑛) ⟩ + ⟨𝑆̇(𝑛) 𝑇 (𝑞)𝑆𝑇 (𝑝 + 𝑞)𝑒 𝑆 (𝑝)𝑒 (𝑛)

(𝑛)

(𝑛)

(𝑛)

−𝑗2𝜋𝜏𝑇 𝑝 𝑗2𝜋𝜏𝑆 𝑝 ̇(𝑛)∗ ⟩ ⟨𝑆̇ (𝑛) ⟩ + ⟨𝑆̇(𝑛) 𝑇 (𝑝)𝑒 𝑆 (𝑞)𝑆𝑆 (𝑝 + 𝑞)𝑒 −𝑗2𝜋𝜏𝑇 𝑞 𝑗2𝜋𝜏𝑆 𝑞 ̇(𝑛)∗ ⟩ ⟨𝑆̇(𝑛) ⟩ + ⟨𝑆̇(𝑛) 𝑇 (𝑞)𝑒 𝑆 (𝑝)𝑆𝑆 (𝑝 + 𝑞)𝑒 (𝑛)

𝑗2𝜋𝜏𝑇 + ⟨𝑆̇(𝑛)∗ 𝑇 (𝑝 + 𝑞)𝑒

(𝑝+𝑞)

(𝑛)

−𝑗2𝜋𝜏𝑆 ̇(𝑛) ⟩ ⟨𝑆̇ (𝑛) 𝑆 (𝑝)𝑆𝑆 (𝑞)𝑒

(𝑝+𝑞)

⟩ + 𝐵̂̇ 𝑆 (𝑝, 𝑞) . (4.3.6)

The first term in formula (4.3.6) 𝐵̂̇ 𝑇 (𝑝, 𝑞) = ⟨𝑆̇ 𝑇 (𝑝)𝑆̇𝑇 (𝑞)𝑆̇𝑇 (𝑝 + 𝑞)⟩ is the original RP target bispectrum. The other terms in (4.3.6) are the interference terms that (𝑛)

(𝑛)

(𝑛)∗

4.3 Naval surface target detection and recognition by estimation of radar range profiles | 153

contaminate the original estimate 𝐵̂̇ 𝑇 (𝑝, 𝑞). Theoretically, sufficiently good accuracy of 𝐵̂̇ 𝑇 (𝑝, 𝑞) can be obtained under traditional assumptions that interference is zeromean, and its PDF is close to symmetric [43]. The case of sea clutter with nonzero mean and with long tail PDFs needs to be studied and this problem is one of the subjects of our experimental investigation. Bispectrum-based target RP estimate can be represented as the following inverse Fourier transform (IFT)

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 𝑆̂ range (𝑙) = 󵄨󵄨󵄨𝐼𝐹𝑇 {󵄨󵄨󵄨𝑆̂̇ bisp (𝑟)󵄨󵄨󵄨 𝑒𝑗𝜑̂ bisp (𝑟) }󵄨󵄨󵄨 , 󵄨 󵄨 󵄨 󵄨

𝑟 = 1, 2, . . ., 𝐿 ,

(4.3.7)

where the magnitude |𝑆̂̇bisp (𝑟)| and phase 𝜑̂ bisp (𝑟) object RP Fourier spectrum estimations can be recovered from (4.3.6) by recursive algorithms [10]. Experimental measurements were carried out in summer period using on-land Xband polarimetric radar operating with carrier of 𝑓0 = 9.370 MHz. The fixed antenna was located at the height of 𝑦0 = 8 m over sea level (see Figure 4.3.1) on the Black Sea shore in Crimea. The radar basic characteristics are listed below: – the antenna beam width is of 3.0∘ in both azimuth and elevation planes; – the transmitted peak power is of 10 kW; – the pulse width is of 3 μs; – the pulse repetition frequency is of 400 Hz for the total possible polarization combinations, that is, for HH, HV, VH and VV polarizations (the first letter corresponds to the transmitted wave polarization while the second one denotes the received signal polarization), thus, it is 100 Hz for each polarization component combination; – two synchronous receiving channels for H and V polarization were employed; the separation of the channels is over 30 dB; – the dynamic range of the receiver is not less than 120 dB; – the ADC capacity is 10 bits; – the access time is 50 ns; – the sample pulse step is equal to 250 ns that corresponds to the range bin of 75 m. After passing through analog intermediate frequency amplifier and amplitude detector, the received signals were digitized, recorded and accumulated in the memory block. The intermediate frequency amplifier and detector characteristics are close to linear. The sampled data were recorded in the form of scans (realizations) contained 𝐿 = 32 samples for each HH, HV, VH and VV polarization. The scan duration for each polarization was equal to 320 ms. Number of recorded and accumulated scans for each polarization was 𝑁 = 256. The antenna was pointed to the object with the aid of a video camera fixed to the antenna. The anchored metallic buoy served as the naval target. Its size was consid-

154 | 4 Signal detection by using third-order test statistics 0,0011

0,0002

0,001

0,00018

0,0009

0,00016

0,0008

0,00014

NRP

NRP

0,0007 0,0006 0,0005

0,00012 0,0001 0,00008

0,0004 0,0003

0,00006

0,0002

0,00004

0,0001

0,00002 0

0

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

(a)

I

(b)

0,0002 0,00018 0,00016

0,00012

NRP

NRP

0,00014

0,0001 0,00008 0,00006 0,00004 0,00002 0

I 0,00028 0,00026 0,00024 0,00022 0,0002 0,00018 0,00016 0,00014 0,00012 0,0001 0,00008 0,00006 0,00004 0,00002 0

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

(c)

I

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

(d)

0,0003

I 0,0004 0,00035

0,00025

0,0003

NRP

NRP

0,0002 0,00015

0,00025 0,0002 0,00015

0,0001

0,0001 0,00005

0,00005 0

0

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

(e)

I

(f)

I

Fig. 4.3.2. Consecutive chain of the scans: (a) scan#1; (b) scan#2; (c) scan#3; (d) scan#4; (e) scan#5; (f) scan#6.

erably smaller than the range bin. The object radar echoes were recorded under small grazing angles of about 0.4∘ . Figure 4.3.2 illustrates consecutive scans for HH polarization, that is, normalized range profiles (NRPs) as the function of the range sample index 𝑙. The slant range to buoy was 𝑅 = 1500 m and the buoy range position approximately corresponds to

4.3 Naval surface target detection and recognition by estimation of radar range profiles | 155

the range sample index of number 26. The total duration of 6 observed scans equals 6 × 320 ms = 1920 ms. Note that random buoy motion on the sea waves as well as random variations of the electromagnetic wave reflection angles can be expected to appear during this sufficiently long observation time interval. Hence, the behavior of (𝑛) (𝑛) ̇ (𝜃, Δ𝑡) and sea 𝜀𝑆𝑖𝑘 ̇ (𝜃, 𝑙Δ𝑡) reflection components in (4.3.3) can be exthe object 𝜀𝑇𝑖𝑘 pected as random values observed during each observed scan. As seen from Figure 4.3.2, the scan fragments corresponding to buoy location and to sea clutter have a random nature and their appearance considerably varies from one scan to another. For some scans, the buoy is not practically visible at all, for example, for scan #2 (see Figure 4.3.2 (b)). Though the target RP should appear itself as a single peak because its size is much smaller than the range bin, the object response shape is considerably contaminated by sea clutter. Due to the contribution of sea clutter, several peaks are observed in the neighborhood of the index 𝑙 = 26 (see the scans in Figure 4.3.2 (c)–(f)). Such RP distortions prevent target detection and recognition. In order to improve the ATR performance, one can simply average the scans over 𝑁 realizations (see (4.3.4)). The averaged NRPs evaluated for different polarizations are shown in Figures 4.3.3–4.3.8. Figures 4.3.3–4.3.5 correspond to irregular sea waves and absence of wind. Figures 4.3.6–4.3.8 correspond to sea state of 2. . . 2.5 and the wind speed of 7. . . 10 m/s. As seen from Figures 4.3.3–4.3.5, when the sea backscattering is low, the NRPs for different polarizations are considerably distorted. The averaged object NRPs are spread and their width at the half-amplitude level varies from approximately 300 m (see Figures 4.3.4 and 4.3.5) to 450 m (see Figure 4.3.3). These values are considerably larger than the range bin of 75 m that can be expected as the target response width. This effect appears due to sea clutter and the abovementioned influence of interference from several electromagnetic wave propagation paths. The effect can be also partly induced by the random motion of the naval object and averaging of the randomly shifted object responses according to (4.3.4). The sea clutter level observed in Figures 4.3.3–4.3.5 depends on polarization and its maximum varies from approximately −6 dB (see Figures 4.3.4 and 4.3.5) to −13 dB (see Figure 4.3.3). With wind presence and the sea state of 2. . . 2.5 (see Figures 4.3.6–4.3.8), the object NRP width at the half-amplitude level varies approximately between 900 m and 300 m. In this case, sea clutter level varies approximately from −13 dB (see Figures 4.3.6 and 4.3.8) to −8 dB (see Figure 4.3.7). Thus, analysis of the NRPs shown in Figures 4.3.3–4.3.8 permits noting that: - each observed object response shape is distorted due to the averaging of the set of randomly shifted received object signal envelopes (see Figure 4.3.2); - sea clutter level is sufficiently high that can mask the object response in some cases (see, for example, Figure 4.3.6).

156 | 4 Signal detection by using third-order test statistics

0,0006 0,00055 0,0005 0,00045

NRP

0,0004 0,00035 0,0003 0,00025 0,0002 0,00015 0,0001 0,00005 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Fig. 4.3.3. Averaged NRP (HH, no wind).

I

0,000002 0,000002 0,000002 0,000001

NRP

0,000001 0,000001 0,000001 0,000001 0,000000 0,000000 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

I

Fig. 4.3.4. Averaged RP (HV, no wind).

0,000026 0,000024 0,000022 0,00002 0,000018

NRP

0,000016 0,000014 0,000012 0,00001 0,000008 0,000006 0,000004 0,000002 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

I

Fig. 4.3.5. Averaged RP (VV, no wind).

4.3 Naval surface target detection and recognition by estimation of radar range profiles

| 157

0,0018 0,0016 0,0014

NRP

0,0012 0,001 0,0008 0,0006 0,0004 0,0002

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

I

Fig. 4.3.6. Averaged RP (HH, wind speed 7. . .10 m/s; sea state 2. . .2.5).

0,00003 0,000025

NRP

0,00002 0,000015 0,00001 0,000005

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

I

Fig. 4.3.7. Averaged RP (HV, wind speed 7. . .10 m/s; sea state 2. . .2.5).

0,0012 0,0011 0,001 0,0009

NRP

0,0008 0,0007 0,0006 0,0005 0,0004 0,0003 0,0002 0,0001 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

I

Fig. 4.3.8. Averaged RP (VV, wind speed 7. . .10 m/s; sea state 2. . .2.5).

158 | 4 Signal detection by using third-order test statistics

0,0016 0,0014

NRP

0,0012 0,001 0,0008 0,0006 0,0004 0,0002

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Fig. 4.3.9. Bispectrum-based NRP (HH, no wind).

I 0,000005 0,000004 0,000004

NRP

0,000003 0,000002 0,000002 0,000001 0,000001 0,000000

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

I

Fig. 4.3.10. Bispectrum-based NRP (HV, no wind).

Consequently, the conventional RP estimation technique based on the averaging of received signal envelopes has a low range resolution and low robustness to sea clutter. The bispectrum-based NRP estimates are shown in Figures 4.3.9–4.3.14. Figures 4.3.9–4.3.11 correspond to irregular sea waves and absence of wind. Figures 4.3.12– 4.3.14 correspond to sea state of 2. . . 2.5 and the wind speed of 7. . . 10 m/s. Note, that due to bispectrum shift invariance property, these NRPs are centered with respect to the NRP center of gravity (index 𝑙 = 16 corresponds to the object location in Figures 4.3.9–4.3.14). With wind absence, the NRP width at the half-amplitude level equals approximately to 150 m and the sea clutter levels have the values −20 dB (see Figure 4.3.10), −24 dB (see Figure 4.3.9) and −26 dB (Figure 4.3.11). The bispectrum-based technique performance slightly worsens in the case of the sea state of 2. . . 2.5 and wind speed of 7. . . 10 m/s (see Figures 4.3.12–4.3.14). The NRP width is between 150 m (see Figures 4.3.13 and 4.3.14) and 450 m (see Figure 4.3.12) and the sea clutter level becomes

4.3 Naval surface target detection and recognition by estimation of radar range profiles | 159

0,00009 0,00008 0,00007

NRP

0,00006 0,00005 0,00004 0,00003 0,00002 0,00001

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

NRP

I

Fig. 4.3.11. Bispectrum-based NRP (VV, no wind).

0,0028 0,0026 0,0024 0,0022 0,002 0,0018 0,0016 0,0014 0,0012 0,001 0,0008 0,0006 0,0004 0,0002 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

I

Fig. 4.3.12. Bispectrum-based NRP (HH, wind speed 7. . .10 m/s; sea state 2. . .2.5).

0,00012 0,00011 0,0001 0,00009

NRP

0,00008 0,00007 0,00006 0,00005 0,00004 0,00003 0,00002 0,00001 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

I

Fig. 4.3.13. Bispectrum-based NRP (HV, wind speed 7. . .10 m/s; sea state 2. . .2.5).

160 | 4 Signal detection by using third-order test statistics 0,005 0,0045 0,004 0,0035

NRP

0,003 0,0025 0,002 0,0015 0,001 0,0005

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

I

Fig. 4.3.14. Bispectrum-based NRP (VV, wind speed 7. . .10 m/s; sea state 2. . .2.5).

Table 4.3.1. Results obtained for simple common averaging of received signal envelopes according to the formula (4.3.4). Absence of wind HH HV VV 450

−13

Wind speed of 7. . . 10 m/s HH HV VV

NRP width at the half-amplitude level, m 300 300 900 300 300

−6

Sea clutter level, dB −6 −13 −8

−13

Table 4.3.2. Results obtained for bispectrum-based reconstruction of the received signal envelopes according to the formulas (4.3.5–4.3.7). Absence of wind HH HV VV 150

−24

Wind speed of 7. . . 10 m/s HH HV VV

NRP width at the half-amplitude level, m 150 150 450 150 150

−20

Sea clutter level, dB −17 −28

−26

−26

equal to −17 dB (see Figure 4.3.12), −28 dB (see Figure 4.3.13), and −26 dB (see Figure 4.3.14). In Tables 4.3.1 and 4.3.2, the key parameters of the RP estimates for two considered techniques are summarized as the range resolution and sea clutter contribution. As seen from Table 4.3.1, the range resolution is worse by from 2 to 12 times in comparison to the theoretical radar range bin of 75 m. The results represented in Table 4.3.2 demonstrate the stability of the range resolution that is worse only by 2 times in comparison to the abovementioned theoretical radar range bin except the case of wind speed of 7. . . 10 m/s and HH polarization.

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The values of the sea clutter contributions demonstrated in Tables 4.3.1 and 4.3.2 depend upon polarization type and sea state. As clearly seen from comparing the data in Tables 4.3.1 and 4.3.2, bispectrum-based technique provides suppression of sea clutter contribution from 11 dB (HH polarization) to 20 dB (VV polarization) in the case of absence of wind and from 4 dB (HH polarization) to 20 dB (HV polarization) in the case of wind speed of 7. . . 10 m/s. Thus, experimental results demonstrate the promising possibility of improving the range resolution of a priori unknown naval objects on the background of nonstationary sea clutter by bispectrum-based reconstruction of radar RPs. This bispectrumbased approach does not require a priori knowledge about object and sea clatter characteristics.

4.4 Using bicoherence-based features for aerial target classification The opportunity of aerial target classification by extraction of high resolution range profiles (HRRP) and micro-Doppler contributions contained in radar echoes has been of paramount interest to the modern radar community during the last decades [96, 97]. Jet engine modulation (JEM) is referred to the micro-Doppler characteristic of the rotating turbine or propeller blades contained in radar backscattering [97]. HRRP can be considered as a projection of a target spatial intensity distribution of the backscattered electromagnetic field on the radar line of sight. HRRP contains certain information about aircraft geometry. Both HRRPs and JEMs allow extraction classification features conforming to the aircraft geometry and engine characteristics. As a result of aerial target motion over a large azimuth angular sector, its HRRPs measured within this route section considerably suffers due to the target translations. Due to the speckle effect, HRRP can fluctuate essentially even for slight aspect angle variations approximately equal to tenths of one angle degree. In other words, only a slight translation of aerial target in elevation or aspect azimuth is enough for considerable variations in measured HRRP. It is well-known that efficiency of the learning procedure usually performed in ATR systems largely depends on the robustness and dimensionality of the classification feature vector. Therefore, extreme variability of the HRRPs is the most difficulty for ATR. In addition, when the input SNR is low, classification probability rate degrades considerably due to the contamination of the HRRP by the noise. The objective of this Subsection is twofold. First, novel algorithms of extraction cepstrum-, bispectrum- and bicoherence-based classification features from radar returns are suggested and studied. Second, performances of the state-of-the-art classifiers using different decision-making strategies, as well as both common classification features contained in the HRRPs and novel bispectrum- and bicoherence-based information features are examined and compared between each others.

162 | 4 Signal detection by using third-order test statistics Based on our survey, the approach most frequently cited for comparison of the performances of aircraft classification was proposed by A. Zyweck and R. E. Bogner in [98]. Below, we will refer to this approach as the Zyweck and Bogner technique (ZBT). ZBT contains the following signal processing procedures. (1) Evaluation of the number of N HRRPs in the form of

𝑌𝑛 (𝑚) = |𝐾𝑛(𝑚)|2 ,

𝑛 = 1, 2, . . ., 𝑁; 𝑚 = 1, 2, . . ., 𝑀

(4.4.1)

where 𝐾𝑛 (𝑚) = 𝑆(:, 𝑛) ∈ C𝑀 is the matrix representation corresponding to the sequence of 𝑁 radar returns accumulated in the form of 𝑀 quadrature samples and given in the matrix form as S ∈ C𝑀×𝑁 . In order to provide input signal energy invariance property, the HRRPs must be normalized. The normalized HRRP is denoted by 𝑌̂𝑛 . (2) Aligning consecutive HRRPs (4.4.1) by using correlation procedure and further averaging of normalized HRRPs 𝑌̂𝑛 (𝑚) as 𝑃

𝑅(𝑚) = ∑ 𝑌̂𝑛(𝑚) ,

(4.4.2)

𝑛=1

where 𝑃 is the number of the HRRPs participated in extraction of classification features. (3) Computation of direct discrete Fourier transform for the averaged HRRP (4.4.2). In order to provide target translation invariance property, magnitude Fourier spectrum is used for further extraction of classification features. In real-life situations, HRRPs consecutively accumulate according to the sequence of radar returns having different spatial shifts within fixed radar range strobes. These shifts are caused by the translation of the aircraft and, hence, the migration of intensity contributions from one range cell to another is observed. Because of this, ZBT exploits the aligning procedure based on correlation analysis to provide translation invariance property. One more classification feature extraction approach based on the evaluation of the HRRPs is proposed in [99] by K.-T. Kim et al. (further referred to as KT technique). The main idea behind KT is to use the first twenty central moments as the features for aircraft classification. KT contains the following signal processing steps. (1) Evaluation of number of the HRRPs according to (4.4.1). (2) Aligning the consecutive HRRPs by using correlation procedure and averaging of aligned range profiles according to (4.4.2). Computation of the first twenty central moments related to averaged range profile and further exploiting them for extraction of classification features. Time-frequency distribution (TFD) of multicomponent and multifrequency radar echo signals can serve as effective tools for solving the problem of aerial target classifica-

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tion [97]. The TFD can be derived from micro-Doppler content. The micro-Doppler averaged data can be extracted from the radar echo signal in the form proposed in [97] as: 𝑀

𝐷(𝑛) = ∑ 𝐾𝑛(𝑚) .

(4.4.3)

𝑚=1

Based on (4.4.3), joint TFD of radar returns can be formed by the following procedure of short-time Fourier transform: 𝐿

TFD(𝑓, 𝑛) = ∑ 𝑤(𝑙)𝐷(𝑙 + (𝑡 − 1)(𝐿 − 𝑄)) exp(−𝑗2𝜋𝑓𝑡/𝐿) ,

(4.4.4)

𝑙=1

100 150 200 250

50 100 150 200 250

100

200

300

400

50 100 150 200 250

100

Time, ms/2

(a) AH-64

Frequency, Hz x 8

50

Frequency, Hz x 8

Frequency, Hz x 8

where 𝐿 is the length of smoothing window 𝑤(𝑙); 𝑄 is the number of overlapping segments; 𝑓 and 𝑡 are the instantaneous frequency and time indices, respectively. The dimensionality of TFD (4.4.4) can be defined as TFD ∈ C𝐿×𝑇 and 𝑇 = (𝑃 − 𝑄)/(𝐿 − 𝑄). Examples of the spectrograms computed for three different aerial target models are demonstrated in Figure 4.4.1. Radar signal models related to the backscattering provoked by these three types of aircrafts were developed and described in [96]. It is seen from Figure 4.4.1 that each of the spectrograms contains the patterns inherent to the particular aerial target. These patterns are of periodical structure and period depends on the radial velocity of the rotating parts belonging to the type of aircraft. The contributions caused by front and back AH-64 helicopter blades can be discriminated in the spectrogram shown in Figure 4.4.1 (a). The front helicopter blade provokes larger radar cross-section (RCS) value, and in the position perpendicular to the radar line of sight excites harmonics in the whole spectrum. The back helicopter blade provokes smaller RCS value and excites harmonics within a smaller frequency bandwidth. The patterns related to the aircrafts An-26 and B-52 shown in Figures 4.4.1 (b) and 4.4.1 (c) demonstrate higher periodicity and more intricate frequency content. An interesting approach [100] based on the cepstrum transform provides translation invariance property for moving radar target classification. Rather low-dimensional and shift invariant classification features can be extracted from the cepstrum

200

300

400

100

Time, ms/2

(b) An-26

200

300

400

Time, ms/2

(c) B-52

Fig. 4.4.1. Results of computer simulations of the spectrograms for different aerial target models that correspond to aspect angle of 185∘ : (a) helicopter AH-64 Apache; (b) military transport aircraft An-26; (c) B-52 stratofortress.

164 | 4 Signal detection by using third-order test statistics coefficients C(.) computed as

󵄨󵄨 󵄨󵄨2 1 𝑇 𝐶(.) = 󵄨󵄨󵄨󵄨𝐹−1 ( log10 ( ∑ |TFD(:, 𝑡)|2 ))󵄨󵄨󵄨󵄨 , 𝑇 𝑡=1 󵄨 󵄨

(4.4.5)

where 𝐹−1 denotes the indirect discrete Fourier transform. The dimensionality of the cepstrum-based classification features evaluated by using (4.4.5) depends, first of all, on the required frequency resolution. Therefore, it is equal to 𝐿 according to (4.4.4). The next two considered classification feature extraction techniques are based on bispectrum- and bicoherence-based estimations. Short-time bispectrum estimates can be defined and computed as:

𝐵𝑡 (𝑓1 , 𝑓2 ) = TFD(𝑓1 , 𝑡) TFD(𝑓2 , 𝑡) TFD∗ (𝑓1 + 𝑓2 , 𝑡) .

(4.4.6)

Distinctive feature of bispectrum-based estimate (4.4.6) is that magnitude bispectrum (bimagnitude) contains information about phase coupling in a micro-Doppler content. Bicoherence [2] serves as the quantitative measure of phase coupling. Squared bicoherence 𝑏̂2 (𝑓1 , 𝑓2 ) can be interpreted as the proportion of signal energy at the frequency (𝑓1 + 𝑓2 ) which is coupled with the contribution of spectral components at the frequencies 𝑓1 and 𝑓2 as:

󵄨2 󵄨󵄨 1 𝑇 󵄨󵄨 ∑ 𝐵 (𝑓 , 𝑓 )󵄨󵄨󵄨 󵄨󵄨 𝑇 𝑡=1 𝑡 1 2 󵄨󵄨 󵄨 󵄨 , 𝑏̂2 (𝑓1 , 𝑓2 ) = ̂ ̂ 𝑋(𝑓1 , 𝑓2 )𝑃(𝑓1 + 𝑓2 )

(4.4.7)

𝑇

̂ 1 + 𝑓2 ) = (1/𝑇) ∑ |TFD(𝑓1 + 𝑓2 , 𝑡)|2 is the power spectrum estimate where 𝑃(𝑓 𝑡=1 ̂ 1 , 𝑓2 ) = (1/𝑇) ∑𝑇 |TFD(𝑓1 , 𝑡) TFD(𝑓2 , 𝑡)|2 . averaged over 𝑇 finite data samples; 𝑋(𝑓 𝑡=1 Squared bicoherence (4.4.7) takes the values within the limits of [0, 1]. If it tends to 𝑏̂2 (𝑓1 , 𝑓2 ) ≠ 0, there exists phase coupling at some pair of frequencies, and if 𝑏̂ 2 (𝑓1 , 𝑓2 ) = 0 there is no phase coupling. Bicoherence value (4.4.7) depends on variations of target velocity. In order to provide target velocity invariant property, the following bicoherence-based feature (referred below as BIC) is given as BIC = |𝐹2𝐷 [𝑏̂ 2 (𝑓1 , 𝑓2 )]| ,

(4.4.8)

where 𝐹2𝐷 denotes the 2-D Fourier transform. Examples of bicoherence estimates (4.4.7) computed for the considered three types of aircrafts are demonstrated in Figure 4.4.2 [101]. The bicoherence estimate computed for helicopter AH-64 (see Figure 4.4.2 (a)) indicates that 𝑏̂2 (𝑓1 , 𝑓2 ) ≈ 1 in bifrequency plane for number points. It means that numerous phase coupled harmonics are contained in this radar signature.

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01

500 450 400 350 300 250 200 150 100 50

Frequency, Hz/2

0.9

500 450 400 350 300 250 200 150 100 50

Frequency, Hz/2

Frequency, Hz/2

4.4 Using bicoherence-based features for aerial target classification

165

|

500 450 400 350 300 250 200 150 100 50

0.6 0.5 0.4 0.3 0.2 0.1 900

1000

800

500

700

600

300

400

100

Frequency, Hz/2

(b) An-26

200

1000

700

900

800

500

600

400

100

Frequency, Hz/2

(a) AH-64

300

200

0

1000

700

900

800

500

600

300

400

100

200

0

Frequency, Hz/2

(c) B-52

Fig. 4.4.2. Bicoherence estimates.

In contrast to helicopters, when phase coupling is caused by rotating blades, phase coupling related to the airplanes An-26 and B-52 is provoked by JEM. Therefore, the corresponding difference must be demonstrated by the evaluation of bicoherence. The difference between bicoherence frequency distributions is clearly seen from comparing the radar signature shown in Figure 4.4.2 (a) and the radar signatures in Figures 4.4.2 (b) and 4.4.2 (c). The bicoherence estimate computed for An-26 (see Figure 4.4.2 (b)) has the lowest amount: only two points in the bifrequency plane can be found with contribution larger than 0.1. It means that phase coupling in microDoppler content appears weakly for this type of aircraft. The bicoherence estimate computed for B-52 (see Figure 4.4.2 (c)) has a number of contributions caused by phase coupling with a bicoherence coefficient larger than 0.6. It can be explained by pronounced JEM contribution contained in the backscattering of B-52. We consider classification system performance by study of two types of classifiers. The first classifier is the Support Vector Machine (SVM) belonging to nonprobabilistic linear classifiers, and the second one is the Naive Bayes Classifier (NBC) belonging to probabilistic linear classifiers. The dimensionality of feature vector conforming to ZBT is of 𝑀 order, cepstrum-based technique is of 𝐿, bicoherence-based technique is of 𝐿×𝐿, and CDFB is of 𝐿. Linear Discriminant Analysis (LDA) is performed for reduction of dimensionality. Modeled radar returns are computed in the form of complex-valued radar backscattering caused by an aircraft. Bandwidth of chirp radar signal equal to Δ𝑓 = 150 MHz provides range resolution of Δ𝑅 = 𝑐/2Δ𝑓 = 1 m. Pulse repetition frequency and central wavelength are equal to 2 KHz and 3 cm, respectively. The coherent chain contained 𝑁 = 2000 chirp radar pulses is emitted within the time interval of 1 s. Three aerial targets are considered. Helicopter AH-64: speed is of 160 km/h, height is of 200 m. Aircrafts An-26: speed is of 400 km/h, height is of 2000 m. Aircraft B-52: speed is of 800 km/h, height is of 2000 m. For each aerial target a set of data is available. Each set contains 𝑁 = 2000 realizations of radar returns. Each radar return contains 𝑀 = 1200 quadrature digital samples. The aspect angle is considered to be one from a set of 𝐴 = {185∘ , 190∘ , 195∘ }. The aspect angle 𝛼 = 180∘ corresponds to the situation when the aircraft is flying towards the radar.

166 | 4 Signal detection by using third-order test statistics Classification probability rate has been examined depending upon the following different three scenarios. Under Scenario #1, a half of radar backscattering data related to each of the three abovementioned aspect-angles are used for training, and the remainder half is used as testing data. Note that for the TFD-based approach, the parameters of 𝐿 = 56 and 𝑄 = 46 are used in (4.4.4), and the Hamming window was exploited. The common destabilizing factor in aerial target classifications is that measurement radar echo data highly depends on an aspect angle. Even a slight change, for example, a few degrees in aspect angle causes transformations of observed data which leads to classification errors. However, feature extraction techniques could provide invariant properties to these transforms. According to Scenario #2, data conforming to one aspect angle are used for training, and the radar records related to other two aspect angles are used for testing. Stepby-step, each record from three available aspect angles will be used one time for training and two other times for testing. Under Scenario #3, data were accumulated for two aspect angles and they will be used for training. The remainder is used for testing. In the latter case, three sets of radar data corresponding to different aspect angles are examined and three combinations of testing and training sets are possible. For each of them, classification performance is evaluated and the final result is averaged. Table 4.4.1 shows the results obtained for three aerial targets under three considered scenarios. It can be seen from Scenario #1 that both SVM and NBC classifiers provide similar classification probability rates. The best performance is achieved by ZBT. HRRP-based schemes (ZBT and KT) outperform the performances of TFD-based techniques (Cepstrum, BIC and CDFB) approximately by 1. . . 4%. It can be explained by the fact that the optimal values of parameters 𝐿, 𝐺 and 𝑄 given in (4.4.4) are not estimated. In order to obtain better performances, additional investigations must be performed. The best performance within the features related to the TF distributions is achieved by bicoherence-based (BIC) features. The CDFB and Cepstrum-based techniques demonstrate the worst results. Table 4.4.1. Classification probability rates computed in percents.

Scenario #

Classifier

Scenario #1

SVM NBC SVM NBC SVM NBC

Scenario #2 Scenario #3

ZBT 99.99 99.97 63.65 71.8 78.52 75.92

Feature extraction technique KT Cepstrum BIC 99.98 93.06 76.5 79.32 85.63 79.46

95.51 98.67 76.89 77.38 90.97 88.15

98.98 100 83.9 – 87.14 82.78

CDFB 95.51 96.48 72.58 72.41 75.33 76.42

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Classification probability rates obtained under Scenarios ##2 and 3 demonstrate the similar behavior referred to Scenario #1. The results obtained by using NBC for BIC features are not included because of limitations of the classifier. The best classification performance referred to Scenario #2 is achieved by BIC features. It allows concluding that phase coupled information is a more robust feature regarding the aspect angle variations than other considered features. The BIC features outperform other considered features approximately by 4. . . 8%. The worst performance is achieved with the CDFB technique. At the same time, the highest probability of correct classification is achieved by cepstrum-based features and the SVM classifier for Scenario #3. The BIC features take the second place with lag of 4%. Summarizing the results represented in Table 4.4.1, the best robustness to the variations of aspect angle is demonstrated by TFD-based schemes providing extraction of cepstrum-based and bicoherence-based classification features. Taking into account the results obtained for Scenario #2, the information about phase coupling provides the best immunity to aspect angle variations. In real-life situations, testing radar records are contaminated by AWGN. Therefore, the study of classification performance under influence of AWGN is of great practical interest. Classification performance estimated for various SNRs is demonstrated in Figure 4.4.3. As can be seen from Figure 4.4.3 (a), the best performance is achieved by the KT approach. The BIC technique becomes comparable to the KT starting from SNR = 1 dB. The cepstrum-based approach outperforms the BIC method only for SNRs smaller than 2 dB. The worst performance is achieved by ZBT, and the loss is rather high compared to other techniques. Concluding analysis of the results is represented in Figure 4.4.3 (a), the best performance of 72% is achieved by KT even for SNR = −10 dB, where other techniques provide the probability of random guess of 33%. The Bayes classifier demonstrates quite different results shown in Figure 4.4.3 (b). The KT, cepstrum-based and CDFB methods provide the same result as the results obtained 100

90 80 70 60 Zyweck Kim Cepstrum Bic CDFB

50 40 30 –10

–5

0

5

10

15

20

25

30

Probability of correct classification

Probability of correct classification

100

90 80 70 60 50 40 30

Zyweck Kim Cepstrum Bic CDFB

20 10 0 –10

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0

(a) SVM classifier

5

10 SNR, dB

SNR, dB

(b) Bayes classifier

Fig. 4.4.3. Classification performance evaluated for various SNRs.

15

20

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168 | 4 Signal detection by using third-order test statistics with the SVM classifier. The main difference is affected by the BIC method. For SNR ≥ 1 dB, the BIC method outperforms all other methods. Moreover, errorless classification can be provided starting from SNR = 2 dB. Thus, novel bispectrum-, cepstrum- and bicoherence-based classification features are proposed to be extracted from radar aerial backscattering. The comparison between suggested classification schemes and two state-of-the-art classifiers using information features extracted from HRRP is performed. Analysis of the influence of aspect angle variation demonstrates the benefits achieved with proposed classification features. The best robustness to the aspect angle variability is provided by bicoherence-based classification features. Therefore, bicoherence-based classification features can be recommended as effective tools in modern ATR systems.

4.5 Time-frequency analysis of backscattering in ground surveillance Doppler radar One of the most important problems existing in radar ATR systems is the search for robust information features for target detection, recognition and classification. Recently, information contained in Doppler and micro-Doppler spectra has been widely used for a moving ground target detection, recognition and classification. Common energy density spectra or squared magnitude Fourier transform estimated during a rather long time interval are inadequate for analyzing the spectral content of nonstationary signals observed in ground surveillance radars because the notion of frequency becomes meaningless. An alternative assumption is that a signal is stationary during a short time interval, and instantaneous frequency (IF) should be used to understand how the spectral content varies in time [102]. The IF serves as an important timevarying signal parameter which allows one to analyze simultaneously the behavior of the spectral content in time and in frequency. In recent years, joint time-frequency (TF) distributions have extensively been employed for study of time-varying spectra and estimation of parameters of nonstationary radar backscattered signals [102–107]. One approach frequently used in nonstationary signal spectrum analysis is the estimation of energy per unit time per unit frequency. A spectrogram is a typical representative of these energy-based estimators for detection and recognition of ground moving targets [108, 109]. Since the possibly-existing phase relationships between harmonics are lost in energy and power spectra, it is impossible to extract phase coupled frequencies by means of time-varying energy or power spectra. In other words, energy and power spectra are unable to extract phase coupled instantaneous frequencies (PCIFs) contained in the signal processed. This peculiarity is the most serious shortcoming of energetic versions of evolutionary spectral estimators. Starting from the well known Wigner–Ville distribution [110] and its modifications with different kernels [111], the optimal estimates have been obtained for analysis of

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linear frequency modulated (LFM) signals. However, one of the most important drawbacks of these joint TF distributions is the appearance of a large number of spurious cross-terms when processing nonlinear FM signals or multicomponent FM signals. These cross-terms often mask the true features of the signal under study. The third-order Wigner distribution or Wigner bispectrum was originally proposed by N. L. Gerr in [112] as an extension of the Wigner–Ville distribution for higherorder spectral analysis. This distribution permits analysing the evolutionary behavior of the third-order moment functions of a non-Gaussian process in the time-bifrequency hyperplane. Other authors have extended this definition [113] and studied the properties of third-order Wigner distributions theoretically [114]. The attractive benefits of the Wigner bispectrum follow from the conventional properties of the bispectrum analysis described in detail in Chapter 1. They are as follows: 1) low sensitivity to any additive symmetrically distributed noise; this property allows reducing the noise influence; 2) insensitivity to target translation and rotation; this property permits removing the artifacts caused by target translations and rotations; 3) robust detection of phase coupling and discriminating phase coupled Fourier spectrum components from those that are not; this can be a useful tool for extracting relevant features of an unknown object. However, there are serious shortcomings of higher-order Wigner distributions. These are: huge computational complexity and low processing rate caused by the necessity of using 2-D FFT, as well as the impossibility of visual representation of evolution of higher-order spectrum estimates in the time-bifrequency hyperplane for analysis. Due to that, higher-order Wigner distributions are not widely employed in real-life radar signal processing. An approach based on extraction of phase coupled Doppler frequency pairs from time-varying bispectrum estimates measured by ground surveillance radar has been developed in [115–117]. First, we consider the Doppler ground surveillance radar echo signal model for a walking human. A walking person can be considered as a complex physical phenomenon of simultaneous motion of different body parts: the torso, the legs, and the arms. The parameters of the radar echo depend on several features of scattering by moving and spatially distributed surfaces. The size of the scattering surface compared with the wavelength; the electrical characteristics of the human body: the admittance and radiation efficiency of the human body parts; the scattering surface radial velocity; the grazing angle and reflecting power may contribute significantly to the backscattered signal. Therefore, for a human body the backscattered electromagnetic field is a sum of a number of contributions. The continuous-wave radar can provide the recording of the radar signatures of a walking human with high-resolution Doppler frequency shifts at the microwave band. The echo signals include the combined reflections from the three main body parts involved in human walking. They are the swings of torso, legs and arms having different

170 | 4 Signal detection by using third-order test statistics areas of scattering surfaces and moving with different and, possibly, time-varying velocities. The time-varying phase of the backscattered signal is of paramount importance for our study. Since the electromagnetic field backscattered by a walking human is a sum of a number of contributions provoked by different human body parts, the phase of the total backscattered signal does not vary with the same rate as that of the separate components. Therefore, under the assumption of a linear approximation, these swinging separate human body parts provoke a sum of different time-varying Doppler frequency shifts in the backscattered signal and these frequencies may be phase coupled due to kinematic relationships between separate human body parts. It should be especially noted, that backscattered radar signals measured by using vertical or horizontal polarizations differ one from another just by different spatial phase distributions of the reflection coefficients for vertical and horizontal polarizations. In other words, a human body can be considered as a “radiating antenna” having different and time-varying spatial amplitudes and phase distributions related to vertical or horizontal polarizations. This important peculiarity can serve for retrieving a novel class of information features for moving object classifications. Taking into consideration these peculiarities, assume that a received radar return is a nonstationary and multicomponent signal. For a multipoint scattering surface, that is, the surface containing a great number of scattering centers, the received discrete-time multicomponent and nonlinearly FM radar returns corresponding to vertical 𝑦𝑉 (𝑖) and horizontal 𝑦𝐻 (𝑖) polarizations can be written, respectively, as 𝑀

𝑦𝑉 (𝑖) = ∑ 𝐴𝑉𝑚 (𝑖) cos [𝛷𝑚𝑉 (𝑖)] 𝑚=1 𝑀

𝑉 (𝑖)]𝐹2 (𝜃𝑚 ) cos { = ∑ 𝑎𝑚𝑉 (𝑖) cos [𝜑𝑚 𝑚=1

4𝜋 [𝑟 (𝑖) − 𝑟0 (𝑖)]} , 𝜆0 𝑚

(4.5.1a)

4𝜋 [𝑟 (𝑖) − 𝑟0 (𝑖)]} , 𝜆0 𝑚

(4.5.1b)

𝑀

𝑦𝐻 (𝑖) = ∑ 𝐴𝐻𝑚 (𝑖) cos [𝛷𝑚𝐻 (𝑖)] 𝑚=1 𝑀

𝐻 (𝑖)]𝐹2 (𝜃𝑚 ) cos { = ∑ 𝑎𝑚𝐻 (𝑖) cos [𝜑𝑚 𝑚=1

𝑉 (𝑖) and 𝑎𝑚𝐻 (𝑖) are the time-varying magnitudes of the local reflection coeffiwhere 𝑎𝑚 cients corresponding to the 𝑚th object scattering center for vertical and horizontal po𝑉 𝐻 larizations, respectively; 𝜑𝑚 (𝑖) and 𝜑𝑚 (𝑖) are the local time-varying phases for vertical and horizontal polarizations, respectively; 𝑖 = 1, 2, . . .𝐼 is the temporal sample index; 𝐹(𝜃) is the radar antenna amplitude directional pattern (it is supposed that the same antenna is used for transmission and reception, the pattern shape is the same in the 𝐻 and 𝐸 planes); 𝜃𝑚 is the angle location of the 𝑚th object scattering center; 𝑟𝑚 (𝑖) and 𝑟0 (𝑖) are the time-varying distances between the antenna phase center and the arbitrary 𝑚th moving object scattering center and the object phase center, respectively; 𝜆 0

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is the radiated wavelength; 𝐴 𝑚 (𝑖) and 𝛷𝑚 (𝑖) are, respectively, the total time-varying radar echo signal magnitudes and phases corresponding to the 𝑚th object scattering center. According to (4.5.1a) and (4.5.1b), the backscattered radar signal is a result of collecting 𝑀 contributions caused by a large number of scattering centers distributed on the moving surfaces. Two motion components are involved in the model (4.5.1a) and (4.5.1b): the translation motion of the human torso (this contribution corresponds to an approximately constant low Doppler frequency shift due to the relatively constant and low velocity of the human body translation) and the swinging motion of legs and arms (this contribution corresponds to the sum of different time-varying Doppler frequency shifts caused by different nonuniform time-varying velocities of swinging body parts). The time-varying phase 𝛷𝑚 (𝑖) in (4.5.1a) and (4.5.1b) can be represented in the form of the following polynomial value of a priori unknown order 𝑅 depending on the characteristic features of a moving human 𝑅

𝛷𝑚 (𝑖) = ∑ 𝑏𝑚𝑟 𝑖𝑟 ,

(4.5.2)

𝑟=0

where 𝑏𝑚𝑟 are the unknown constant coefficients controlling the weighting of polynomial phase values. The discrete-time IF 𝑓𝑚 (𝑖) can be derived from (4.5.2) as

𝑓𝑚 (𝑖) =

1 𝑅 1 𝑑𝛷𝑚 (𝑖) = ∑ 𝑟𝑏 𝑖𝑟−1 . 2𝜋 𝑑𝑖 2𝜋 𝑟=0 𝑚𝑟

(4.5.3)

Depending on the physical properties of the moving object, the polynomial (4.5.2) can describe, for example, the quadratic FM law (for 𝑟 = 3). In general, the radar echo signal model (4.5.1)–(4.5.3) describes a nonstationary signal of multicomponent structure as a weighted sum of mono component signals, each one with its own IF 𝑓𝑚 (𝑖). This frequency is proportional to the radial velocity related to the 𝑚th scattering center located in the moving object. In other words, an unknown number of IFs might be contained in the backscattered signal. It should be supposed that the phases of some pairs of IFs may be related, that is, some of the IFs might be phase coupled. The latter assumption is the key hypothesis in the approach suggested in [115–117]. It is capable of providing novel features in object identification, recognition and classification in such circumstances. It should especially be stressed that the authors commonly only pay attention to the quadratic phase coupling phenomenon caused by passing a linear non-Gaussian process through a nonlinear device [2]. Existence of phase coupled spectral components in multicomponent radar signals backscattered by a moving human has been found experimentally and described for the first time in [115]. Phase coupling can be explained physically by the fact that the swinging torso, legs, and arms are not independent sources of Doppler frequency shifts, but they are

172 | 4 Signal detection by using third-order test statistics related mechanically via the “carrier”: human torso serves as a “carrier” for the swinging legs and arms. Furthermore, from the radio physical point of view, the microwave conductivity currents flowing in the human skin also may cause certain phase relationships in multicomponent radar echo signals. Therefore, one can expect the presence of phase coupled harmonics in radar backscattering (4.5.1a, b). According to this hypothesis, our objective is to detect, extract and verify the presence of phase coupling contribution, as well as to estimate evolutionary behavior of the PCIFs in the nonstationary and possibly multicomponent radar echoes collected from a moving object. The main idea of our approach utilizes one of the properties of the bispectrum [2], notably the bispectrum allows measuring the magnitude and phase of the autocorrelation of a signal at different Fourier frequencies. When a phase relationships exist, the phase coupled components contribute to the bispectrum estimate. In other words, the bimagnitude tends to nonzero value only if the radar responses at certain frequencies are correlated. Therefore, the quantitative measure of coupling between the harmonics may be evaluated by the bispectral estimation. On the other hand, for a stationary zero-mean Gaussian process, the bispectrum tends to zero [2]. It means that there are no phase coupled frequencies in a linear Gaussian uncorrelated process. Therefore, unlike the energy spectrum, the bimagnitude distribution contains peaks caused by the coherence between the bifrequency components. Let us consider the short-time bispectrum estimate as a triple product of the Fourier transforms of a transient signal 𝑦(𝑖, 𝑛) whose time duration is significantly shorter than the total observation interval of 𝐼 samples given in (4.5.1a, b). Note that the transient signal 𝑦(𝑖, 𝑛) can be separated from the process (4.5.1a, b) by a sliding window that step-by-step moves along the process recorded and takes 𝑛 = 1, 2, . . ., 𝑁 nonoverlapping positions. Then, the short-time bispectrum estimate can be written as the following triple product

𝐵(𝑝, 𝑞; 𝑛) = 𝑌(𝑝, 𝑛)𝑌(𝑞, 𝑛)𝑌∗ (𝑝 + 𝑞, 𝑛) = |𝐵(𝑝, 𝑞; 𝑛)| exp[𝑗𝛽(𝑝, 𝑞; 𝑛)] ,

(4.5.4)

where 𝑌(𝑝, 𝑛) = |𝑌(𝑝, 𝑛)| exp[𝑗𝜑(𝑝, 𝑛)] is the discrete short-time Fourier transform related to the transient signal 𝑦(𝑖, 𝑛); |𝐵(𝑝, 𝑞; 𝑛)| and 𝛽(𝑝, 𝑞; 𝑛) are the time-varying bimagnitude and biphase, respectively; 𝑝 = 1, . . ., 𝑃 and 𝑞 = 1, . . ., 𝑃(𝑃 ≪ 𝐼) are the independent frequency indices. Note that prior to the computation of time-varying bispectrum estimates (4.5.4), the mean values or DC component must be removed in the time series (4.5.1a, b). It is required in order to eliminate a huge spurious peak in the bifrequency plane at its origin of coordinates (𝑝 = 𝑞 = 0) that might mask the information peaks provoked by phase coupling to be detected and extracted. If there exists phase coupling between two harmonics corresponding to the frequency samples 𝑝, 𝑞 for an arbitrary 𝑛th location of the sliding window, a peak in the bimagnitude distribution |𝐵(𝑝, 𝑞; 𝑛)| appears at the intersection of these two sam-

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ples 𝑝 and 𝑞 in the bifrequency plane. Otherwise, in the discrete power spectrum, three peaks could appear in the samples with indices 𝑝, 𝑞, and 𝑝 + 𝑞 regardless of whether phase coupling exists or not. As a typical example for demonstrating the phase coupling phenomenon, an arbitrary 𝑛th time-varying bimagnitude estimate has been computed according to the formula (4.5.4). For this purpose, a radar signal of the total length of 200 000 samples was recorded as a radar signal backscattered by a metallic sphere (a pendulum) swinging in coherent continuous-wave microwave radar radiation with the operating wavelength of 𝜆 0 = 2 cm. A typical echo signal segment of the length of approximately 4 000 samples is shown in Figure 4.5.1.

S(i)

–10000 –12000 –14000 6.5

7

7.5

8

8.5

t, s. Fig. 4.5.1. The chirp-like radar backscattering recorded from a swinging sphere: the signal magnitude as a function of a sample number [117].

Because the radar microwave radiation was backscattered by a spatially distributed object (the diameter of the sphere significantly exceeded the radar wavelength 𝜆 0 ) the radar echo is not a pure LFM signal in the strict sense as in the case of a point scatterer. As clearly seen from Figure 4.5.1, the process is of chirp-like nature and demonstrates nonstationary behavior. The plot of the bimagnitude estimate of the transient signal truncated by the window of a simple rectangular shape with width of 𝑃 = 256 samples and an arbitrary location (𝑖 ∈ [514, 770]) is shown in Figure 4.5.2. The horizontal plane in Figure 4.5.2 is the bifrequency plane of the dimensionality 𝑝 × 𝑞 = 256 × 256 frequency samples and the bimagnitude values are represented on the vertical axis. Several pronounced sharp bimagnitude peaks in the bifrequency plane are clearly seen in Figure 4.5.2. This indicates and confirms the presence of phase coupled frequencies in the radar echo returns obtained from the swinging metallic sphere. The analysis of the bimagnitude function shown in Figure 4.5.2 has suggested the idea to project the 3-D PCIF bimagnitude distribution to the 2-D time-frequency (TF) plane convenient for analysis and further extraction of classification features [115– 117]. Experimental verification of the suggested approach has been performed by two coherent, homodyne, and continuous-wave radars. First radar operates at three wave-

174 | 4 Signal detection by using third-order test statistics

450 400 350 300 250 200 150 100 50 00

20

40

60

80 100 120 140 160 180 200 220 240

Fig. 4.5.2. Backscattering of swinging sphere: bimagnitude as a function of frequency samples 𝑝 and 𝑞 computed for an arbitrary signal segment.

lengths of 𝜆 0 = 2 cm, 3 cm and 8 mm with the vertical polarization. Second polarimetric radar operates at the wavelength of 𝜆 0 = 8.8 mm with the vertical and horizontal polarizations. The radars were designed at the Institute of Radiophysics and Electronics of the National Academy of Sciences, Kharkov, Ukraine [118]. The output of the 10 bits ADC allows collecting low frequency radar returns containing Doppler frequency shifts. Fixed transmitting and receiving horn antennas were mounted at the height of 150 cm above the ground. Experimental investigations were carried out during the spring and summer periods. A block diagram and photograph of the radar system are shown in Figures 4.5.3 and 4.5.4. First, we analyze the radar echo signals collected by the radar operating with vertical polarization. The radar returns from a swinging metallic sphere (a calibration test object) and a walking person was recorded during the radar data collection time of 63 s that corresponds to 200 000 digitized signal samples. This record of the mentioned total length was divided into 𝑁 segments by a rectangular-shaped window with the width of 256 samples which was sliding along the signal. The dividing procedure was nonoverlapping. As a result, maximum number of 𝑁 = 780 short segments, that is, 𝑁 transient partial records can be registered.

4.5 Time-frequency analysis of backscattering in ground surveillance Doppler radar

Channel 2

Channel 1

PC

|

175

LF amplifier

Mixer Hor

Impatt FC–1 Magic–T FC–2

Ferrite circulator Power supply

Twist

Ver Mixer

LF amplifier

Fig. 4.5.3. Block diagram of the radar.

Fig. 4.5.4. The radar system: power radiated is of 85 mW; antenna beam width is of 6∘ in both 𝐸 and 𝐻 planes; sensitivity of receiver is equal to 10−15 W/Hz).

176 | 4 Signal detection by using third-order test statistics The window width of 256 samples provides a frequency resolution of approximately 12.4 Hz. The sequence of time-varying bispectrum estimates of 256 × 256 samples has been calculated in the form of triple products of time-varying Fourier transforms (4.5.4) for each 𝑛th separate signal segment. The peaks of bimagnitude |𝐵(𝑝, 𝑞; 𝑛)| have been analyzed and their values exceeding some threshold level were selected for analysis. The bimagnitude values are less than one tenth of the maximum bimagnitude value and have not been taken into consideration because these values correspond to such PCIF coherence coefficients that can be neglected. Note that self-phase coupling is an artifact of the instantaneous bimagnitude estimates. It occurs when 2𝑝 = 𝑞 or 2𝑞 = 𝑝. Therefore, the corresponding bimagnitude slices located at (𝑝, 𝑝) were eliminated from further consideration. Finally, the values of the selected bimagnitude peaks were accumulated and stored. It should be stressed that in the analysis of evolutional behavior of the PCIF, the 3-D time-varying bispectrum estimates are difficult to visualize. Thus, we propose to map the phase coupling frequency values (which correspond to the peak of bimagnitude in the bifrequency plane) onto the TF plane as two separate values on the frequency axis in the frequency coordinates 𝑝 and 𝑞. The graphs of TF distributions were plotted on the basis of stored time-varying bimagnitude data [117]. We first verified our approach with radar echoes generated by the multipoint scattering surface of the oscillating metallic sphere. The metallic sphere was hung up with a thin line and oscillated as a pendulum in the far electromagnetic field of the radar. Received radar echo signals were analyzed during the time interval of 1 200 ms (𝑁 = 15) that approximately corresponded to one total pendulum oscillation period. The behavior of PCIF transformed onto the TF plane (PCIF TF distribution) is illustrated in Figure 4.5.5. 1200

Time (ms)

960 720 480 240 0

0

48

96 144 Frequency (Hz)

192

Fig. 4.5.5. PCIF TF distribution for an oscillating sphere-pendulum.

The time-frequency analysis of the graph in Figure 4.5.5 reveals the following interesting properties of the swinging sphere. First, there is the horizontal (parallel to the time axis) Doppler frequency shift response. This component of the phase coupling IF distribution in the TF domain corresponds to the approximately constant Doppler

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frequency shift due to the presence of the zero-order term in the polynomial expansion (4.5.3). Second, the quadratic law can clearly be seen for the PCIF behavior in the TF domain of the graph. It should be stressed that the described phase coupling phenomenon differs from phase coupling origin caused by a walking human. The phase coupling in the backscattered radar return measured from an oscillating sphere can be explained by the spatial phase distribution of microwave conductivity currents flowing on a spherical metallic surface. These microwave currents are phase coupled, that is, the spatial phase values of the microwave currents in different local backscattering centers distributed in the spherical surface are coupled by a certain law. In the course of time, the pendulum-like oscillations of the 3-D spherical shape cause simultaneous displacements of the all local backscattering centers. It is experimentally confirmed that the 3-D microwave current spatial distribution turns into the PCIF distribution in Figure 4.5.5 owing to behavior of the oscillations of the sphere. The same results were obtained for the wavelengths of 𝜆 0 = 3 cm and 8 mm. Therefore, the PCIF distribution projected onto the TF domain and obtained on basis of the time-varying bispectrum estimates of the radar return signal can be considered as promising information feature for object classification. The other radar target in our experimental investigations was a walking human. Unlike a swinging metallic sphere, a human body is a much more complicated backscattering radar object. An example of the received intricate signal (duration of recording is of 960 ms) backscattered by a walking human is displayed in Figure 4.5.6.

–40

y(i)

–60 –80 i

–100 58500

59500

60500

Fig. 4.5.6. The radar echo sample recorded from a walking human: the signal magnitude as a function of a sample number.

As compared to the signal in Figure 4.5.1, more magnitude and chirp-like fluctuations are observed in the arbitrary signal segment demonstrated in Figure 4.5.6. It is reasonable to suppose that the signal in Figure 4.5.6 is a nonstationary multicomponent process. This assumption has been examined by the histogram of the PCIF distribution computed for a walking human and plotted below in Figure 4.5.7. The pres-

178 | 4 Signal detection by using third-order test statistics

20

10 800

0 390

480 Time

260 Phase coupled frequencies (Hz)

130

160 0

Fig. 4.5.7. The histogram of the PCIF distribution computed for a walking human.

840 Time (ms)

700 560 420 280 140 0

60

120 180 240 360 420 Frequences (Hz)

Fig. 4.5.8. PCIF distribution for a walking human: vertical polarization, 𝜆 0 =2 cm.

ence of a large number of PCIFs contained in the signal represented in Figure 4.5.6 is confirmed by the histogram in Figure 4.5.7. The results of extraction of the PCIFs from the time-varying bispectrum estimates of the signal in Figure 4.5.6 are illustrated by the TF distribution plotted in Figure 4.5.8. A sliding window of the width of 256 samples covered 𝑁 = 12 positions without overlapping and the threshold corresponding to one tenth of maximum bimagnitude value were used. The analysis of the TF distribution in Figure 4.5.8 demonstrates the presence of several frequency peaks in the joint TF domain which confirms the above pronounced assumption regarding the multicomponent structure of the radar echo signal backscattered by a walking human. We would like to stress the presence of the main- (the swinging human torso) and micro-Doppler (swinging arms and legs) contributions.

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The swinging torso provokes the appearance of peaks in the low-frequency range. These peaks can be seen for the frequencies of about 30 Hz. The peaks corresponding to the swinging arms and legs are observed in the high-frequency area from approximately 120 Hz to 240 Hz. Polarimetric measurements allow additionally obtaining new promising information features like PCIFs extracted from short-time cross-bispectrum estimates by using radar returns recorded for the vertical (V) and horizontal (H) polarization for an object moving in a growing vegetation like trees and bushes. For this reason, in this Subsection, we pay attention to the analysis of the PCIF distributions measured by the polarimetric radar [116, 117]. The question is to know whether or not there is a difference between the radar Doppler TF signatures obtained with the vertical and horizontal polarizations and how much they differ one from another? Most radar specialists answer without a doubt that for metallic radar objects there is no difference (see, e.g., [119]). Even if this answer is usually correct for metallic objects, this is not always correct for the electromagnetic microwave fields backscattered by a moving human body. First, we illustrate the difference between the PCIF distributions obtained for V and H polarizations. In fact, according to the radar echo signal model (4.1.5a, b), the last terms (electrical distances) in the quadruple products in the sums are the same for both vertical and horizontal polarizations. Therefore, the distinction must arise 𝑉 𝐻 due to different behavior of the local time varying phases 𝜑𝑚 (𝑖) and 𝜑𝑚 (𝑖). The latter peculiarity can be explained by the difference in phase spatial distributions in human skin for vertical and horizontal polarizations. For experimental verification of our hypothesis, let us analyze the difference between the bimagnitude estimates obtained for the transient radar echo signals for V and H polarizations for a human walking towards the radar. These signals were truncated by a window of simple rectangular shape with a duration of 𝑃 = 256 samples and an arbitrary location. The frequency resolution for the polarimetric radar was equal to 31 Hz. The plots of the bimagnitude estimates calculated for vertical and horizontal polarizations are shown in Figures 4.5.9 and 4.5.10, respectively. The difference between these bimagnitude estimates is clearly seen because the number of bimagnitude peaks, their levels and PCIF values differ considerably one from another for V (Figure 4.5.9) and H (Figure 4.5.10) polarizations. It confirms our assumption about the difference between the spatial phase distributions in the human body for the vertical and horizontal polarizations. Now, let us consider an experimental example that demonstrates the difference between the PCIF distributions obtained with V and H polarizations. The PCIF distributions computed for a human standing still but with swinging arms are represented in Figures 4.5.11 and 4.5.12 for V and H polarizations, respectively. As can be seen from these Figures, a human swinging torso causes the appearance of the lowfrequency peaks in the TF distribution which are concentrated approximately near

180 | 4 Signal detection by using third-order test statistics

20 000 000 19 000 000 18 000 000 17 000 000 16 000 000 15 000 000 14 000 000 13 000 000 12 000 000 11 000 000 10 000 000 9 000 000 8 000000 7 000 000 6 000 000 5 000 000 4 000 000 3 000 000 2 000 000 1 000 000

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230

Fig. 4.5.9. Bimagnitude as a function of frequencies 𝑝 and 𝑞 obtained for a human walking towards the radar (vertical polarization).

the frequency of 60 Hz both for V and H polarizations. The swinging arms cause highfrequency micro-Doppler peaks evolutionary behaviors of which are different for vertical (Figure 4.5.11) and horizontal (Figure 4.5.12) polarizations. The TF distribution in Figure 4.5.11 contains more high-frequency peaks in comparison to the horizontal polarization in Figure 4.5.12. Thus, it has been demonstrated in [116, 117] that in opposite to the conventional energy-based estimators which do not allow detecting the difference between short time-varying Doppler spectra obtained for the vertical and horizontal polarizations, the time-varying bispectrum estimates produce an opportunity for recognizing the difference between PCIF distributions computed for the V and H polarizations. The approach suggested can serve as a new classification feature for improving the detection and recognition capability in radar ATR systems operating in vegetation clutter like the trees and bushes. TF analysis of ground surveillance polarimetric radar signals by using timevarying cross-bispectrum estimates containing radar echoes experimentally measured for V and H polarizations has been proposed in [116].

4.5 Time-frequency analysis of backscattering in ground surveillance Doppler radar |

181

16 000 000 15 000 000 14 000 000 13 000 000 12 000 000 11 000 000 10 000 000 9 000 000 8 000 000 7 000 000 6 000 000 5 000 000 4 000 000 3 000 000 2 000 000 1 000 000

0 10 20 30 40

50 60

70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230

Fig. 4.5.10. Bimagnitude as a function of frequencies 𝑝 and 𝑞 obtained for a human walking towards the radar (horizontal polarization).

1050 900

Time (ms)

750 600 450 300 150 0

0

60

120

240 300 180 Frequency (Hz)

360

420

Fig. 4.5.11. PCIF distribution for a human standing still but with swinging arms (vertical polarization).

182 | 4 Signal detection by using third-order test statistics 900 750

Time (ms)

600 450 300 150 0

0

60

120

180 240 300 Frequency (Hz)

360

420

480

Fig. 4.5.12. PCIF distribution for a human standing still but with swinging arms (horizontal polarization).

Let us define the short-time cross-bispectrum estimate as a triple product of the short-time Fourier transforms of the transient signals 𝑦V (𝑖, 𝑛) and 𝑦H (𝑖, 𝑛) related to the V polarization (4.5.1a) and H polarization (4.5.1b) responses and separated by a sliding window that shifts step-by-step along the process studied and takes 𝑛 = 1, 2, . . ., 𝑁 nonoverlapping locations. The short-time cross-bispectrum estimates can be written as

𝐵VH (𝑝, 𝑞; 𝑛) = 𝑌V (𝑝, 𝑛)𝑌V (𝑞, 𝑛)𝑌H∗ (𝑝 + 𝑞, 𝑛) = |𝐵VH (𝑝, 𝑞; 𝑛)| exp[𝑗𝛽VH (𝑝, 𝑞; 𝑛)] , (4.5.5) where |𝐵VH (𝑝, 𝑞; 𝑛)| and 𝛽VH (𝑝, 𝑞; 𝑛) are the time-varying cross-bimagnitude and cross-biphase, respectively. It should be stressed, that if there exists phase coupling between some two harmonics contained in backscattered signal (4.5.1a) for V and (4.5.1b) for H polarization, a peak in the cross-bimagnitude |𝐵𝑉𝐻 (𝑝, 𝑞; 𝑛)| must appear at the intersection of two corresponding samples 𝑝 and 𝑞 in the bifrequency plane. Otherwise, for a Gaussian uncorrelated process provoked by growing vegetations like trees and bushes, the cross-bimagnitude tends to zero due to the fact that the corresponding backscattered electromagnetic field tends to have Gaussian distribution. Therefore, we have supposed that the cross-bispectrum estimates (4.5.5) would contain the peaks caused by phase coupling of certain IFs in non-Gaussian signal backscattered by a moving human. This important peculiarity can serve for retrieving new detection and recognition features for a radar object moving in growing vegetation clutter.

4.5 Time-frequency analysis of backscattering in ground surveillance Doppler radar

|

183

A coherent, homodyne, continuous-wave and polarimetric ground surveillance radar operating at the millimeter wavelength of 𝜆 0 = 8.8 mm has been designed at the Institute of Radiophysics and Electronics of the National Academy of Sciences, Kharkov, Ukraine [118]. The radar parameters are: the power radiated is equal to 15 mW; the antenna beam width is 6∘ in both 𝐸 and 𝐻 planes; the value of receiver noise figure is of 20.2 dB; the level of side lobes is of −24 dB; the cross polarization level is less than −30 dB; the 16 bit two-channel ADC has been used. Transmittingreceiving horn antennas were mounted at the height of 150 cm above the ground. Experimental investigations were carried out during the summer period. The radar returns measured at V and H polarization from a walking human and vegetation like trees and bushes in a light breeze were recorded in PC memory during the data collection time of approximately one minute. The sequence of the backscattered signal samples has been truncated by a window of rectangular shape having the width of 𝑃 = 256 samples and sliding along the signal. As a result of truncation, 𝑁 short signal segments have been separated. The set of 𝑁 cross-bispectrum estimates (4.5.5) of 256 × 256 samples each has been computed. The peaks of cross-bimagnitude |𝐵𝑉𝐻 (𝑝, 𝑞; 𝑛)| have been analyzed and their values were projected in time-frequency domain. The plots of time-PCIF distributions measured for a human walking in vegetation clutter and only vegetation clutter contribution are shown in Figures 4.5.13 and 4.5.14, respectively. Several pronounced cross-bimagnitude peaks are observed in Figure 4.5.13: the swinging torso provokes the peaks in the low-frequency range and the swinging arms and legs micro-Doppler contributions are observed in a highfrequency range. Time-PCIF distribution measured for vegetation clutter represented in Figure 4.5.14 essentially differs from a walking human: the cross-bimagnitude peaks are observed only in low-frequency range and their values are sufficiently smaller in comparison to a walking human. It permits to discriminate and recognize a human walking in vegetation clutter by the proposed time-PCIF radar signature. Experimental results demonstrated for the cross-bimagnitude estimates can find application in ground radar surveillance automatic target recognition systems for security purposes. One of the most important problems usually arising in ground surveillance Doppler radars before object recognition is detection of a moving target observed for low input SNR. Because of this, paramount interest consists in studying the detection performance and robustness to AWGN for the bispectrum-based approach considered in the papers [115–118]. This important feature has been investigated experimentally in [120]. Ground surveillance radar having the characteristics described above served for experimental investigations of noisy backscattered signals recorded with a human walking and running towards or away the radar in vegetation and precipitation. The amplitudes of phase coupled Doppler harmonics extracted by using bispectrum estimates calculated in the form of (4.5.4) have been measured.

184 | 4 Signal detection by using third-order test statistics 900 800 700

Time, ms

600 500 400 300 200 100 0

100

200

300

400

500

Frequency (Hz) Fig. 4.5.13. PCIF distribution for a human walking in vegetation clutter.

900 800 700

Time, ms

600 500 400 300 200 100 0

20

40

60

80

100

120

140

Frequency (Hz) Fig. 4.5.14. PCIF distributions measured for only vegetation clutter contribution.

Relative information peak phase coupled amplitudes mapped onto the plane relative amplitude-frequency are computed as Relative Amplitude (dB) = 20 log

|𝐵(𝑝, 𝑞)| , |𝐵(𝑝, 𝑞)|max

where |𝐵(𝑝, 𝑞)|max is the maximum bimagnitude value.

(4.5.6)

4.5 Time-frequency analysis of backscattering in ground surveillance Doppler radar

|

185

0 Relative amplitude (dB)

–20 –40 –60 –80 –100 –120 –140 –160 500

1000

1500 2000 2500 3000 3500 Frequency (Hz)

Fig. 4.5.15. Bispectrum estimation: human walking away from the radar.

Relative amplitude (dB)

–35.0 –45.0 –55.0 –65.0 –75.0 –85.0 –95.0 –105.0 –115.0

40

70 100

200

400 700 1.0k

2.0k

Frequency (Hz) Fig. 4.5.16. SPECLAB estimation: human walking away from the radar.

Sliding window width equal to 1 024 samples has taken 𝑁 = 24 locations with 50 percent overlapping. The results of these bispectrum-based measurements are illustrated by graphs plotted in Figures 4.5.15, 4.5.17, and 4.5.19. The corresponding conventional amplitude Doppler spectra computed by using SPECLAB software with FFT size of 32 768 samples are demonstrated in Figures 4.5.16, 4.5.18, and 4.5.20 for comparative study and analyzing the benefits of the bispectrumbased approach with respect to Doppler Fourier spectrum analysis. The total measurement time interval is equal to approximately 3 s for analysis of Doppler spectrum content. Comparison of the graphs shown in Figures 4.5.15 and 4.5.16 permits noting that the differences between information Doppler peak at 400 Hz (this Doppler frequency value has been a priori evaluated) and interference levels estimated at the frequency of

186 | 4 Signal detection by using third-order test statistics 0 Relative amplitude (dB)

–20 –40 –60 –80 –100 –120 –140 –160 500

1000

1500 2000 2500 Frequency (Hz)

3000

3500

Fig. 4.5.17. Bispectrum estimation: human walking towards the radar.

Relative amplitude (dB)

–35.0 –45.0 –55.0 –65.0 –75.0 –85.0 –95.0 –105.0 –115.0

40

70 100

200

400 700 1.0k

2.0k

Frequency (Hz) Fig. 4.5.18. SPECLAB estimation: human walking towards the radar.

0 Relative amplitude (dB)

–20 –40 –60 –80 –100 –120 –140 –160 500

1000

1500 2000 2500 Frequency (Hz)

3000

3500

Fig. 4.5.19. Bispectrum estimation: human running away from the radar.

4.5 Time-frequency analysis of backscattering in ground surveillance Doppler radar |

187

Relative amplitude (dB)

–35.0 –45.0 –55.0 –65.0 –75.0 –85.0 –95.0 –105.0 –115.0 40

70 100

200 400 700 1.0k Frequency (Hz)

2.0k

Fig. 4.5.20. SPECLAB estimation: human running away from the radar.

2 KHz are equal to 68dB in Figure 4.5.15 and 45 dB in Figure 4.5.16, respectively. Therefore, the benefit obtained by the bispectrum-based approach in signal-to-interference ratio is equal to 23 dB. Comparative analysis of the results represented in Figures 4.5.17 and 4.5.18 as well as in Figures 4.5.19 and 4.5.20 allows concluding that the benefits provided by the bispectrum-based approach in SNR terms are approximately equal to 15 dB. One can conclude that the improvement of signal-to-interference ratio is achieved compared to the conventional Doppler Fourier spectrum estimation due to the coherent accumulation of the bimagnitude peaks caused by phase coupling of Doppler frequencies and smoothing the noncoherent interference frequency contributions.

5 Conclusions Properties and benefits of higher-order correlation functions and higher-order spectra were considered from the particular point of view of real-life digital signal processing in AWGN and non-Gaussian noise environment. In the book, the main attention has been paid to nonparametric techniques for signal recovery by using triple correlation function and bispectrum estimates. Statistical features of bispectrum estimates contaminated by noise have been analyzed and studied by computer simulations performed for different types of signals. It was demonstrated that the noise leaked in bispectrum estimates possesses non-Gaussian behavior both in real and imaginary parts in bispectrum estimates. On the basis of these statistical investigations, novel bispectrum-filtering techniques have been developed and studied with computer simulations. Suggested bispectrum-filtering techniques are based on linear and nonlinear, as well as nonadaptive and adaptive filtering of the bimagnitude and biphase, as well as filtering real and imaginary parts of bispectrum estimates. Novel modifications of adaptive DCT-based filters have been designed and shown to perform well for a wide range of input SNR and signal waveforms. The benefits of the combined bispectrumfiltering approach are demonstrated by numerous examples of computer simulations. Recommendations on selecting the appropriate filter type and parameters have been provided. Instead of common evaluation of biphase function usually provoking phase wrapping, a new approach without addressing direct standard phase calculations is proposed. The suggested recursive signal waveform reconstruction procedure is represented in the form of computation of the quadrature components of the complexvalued normalized bispectrum and signal Fourier spectrum. It provides an essential reduction of distortions caused by phase wrapping in reconstructed signal waveforms. Novel bispectrum-based techniques for reconstructing jittered and noisy digital images are proposed by using additive and multiplicative predistortion functions inserted in image rows. The optimal predistortion function parameters have been defined and studied for several standard digital test images. The problem of the detection of deterministic signals embedded in AWGN by using the novel proposed test statistic given in the form of third-order autocorrelation function evaluated at the matched filter output, has been investigated. By computer simulations, the merits of the proposed test statistic compared to the second-order statistic formed as conventional correlation integral have been demonstrated. In particular, it resulted in higher signal detection probability. A novel encoding strategy using frequency diversity and bispectrum-based signal processing is suggested for digital communications. According to the proposed approach, binary data are transmitted by using a pair of mutually orthogonal tripletsignals contained phase coupled frequency tones. Phase coupling in each triplet-signal is given by group of three tones such that phase related to one frequency value

5 Conclusions

| 189

is equal to the sum of phases related to other two frequencies. Novel third-order test detection statistics evaluated in the form of triplet-signal bimagnitude peaks are suggested for detection and discrimination of received triplet-signals in noisy and fading communication radio links. BER performance evaluated for additive Gaussian link noise and fast fading demonstrates considerable gain of noise and fading protection as compared with common MFSK scheme using conventional energy-based strategy for signal detection and discrimination. The benefits of the proposed bispectrum-based approach were demonstrated by experimental study of radar target detection, classification and identification performed for naval, aerial and ground moving objects. Experimental results represented in this book demonstrate sea clutter suppression and good naval object range resolution provided by polarimetric X-band radar. Reduction of aspect angle dependent speckle distortions in aerial high resolution range profiles were illustrated by computer simulation results performed for several aerial targets of different types, sizes and backscattering surface irregularities. Essential reduction of speckle distortions in high resolution range profiles obtained by the technique proposed was shown. Experimental results of time-frequency analysis of backscattered signals recorded by ground surveillance Doppler radar were represented and discussed. Novel approach permits extracting the phase coupled instantaneous frequency contributions contained nonstationary and multicomponent chirp-like radar backscattered signals. This makes it possible to obtain new information features for better radar target recognition and classification. The approach suggested can serve for improving the detection and recognition performances in radar ATR systems operating in vegetation clutter.

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Subject index A adaptive window size median filtering (Median-ICI) 95 aerial target classification features evaluated with bispectrum- and bicoherence-based estimations 164 aircraft classification by using high resolution range profiles 162 B bicoherence estimates 165 bimagnitude and biphase estimates contaminated by noise 50–52 bimagnitude evaluated for a human walking towards the radar 180–181 bispectrum-based image restoration 101–103 – additive predistortions 103–112 – multiplicative predistortions 113–117 bispectrum-based naval target range profile estimation 152–153 bispectrum-based signal processing in receiver 138–141 – Rao–Gabr window 139 bispectrum-based signal waveform reconstruction, recursive algorithm 12–14 – cumulant and moment functions 1–4 – energy spectrum, bispectrum, and trispectrum 3 bispectrum-organized modulation by using triplet-signals 136–137 bit error rate 146 bit error rate: case of additive white Gaussian noise (AWGN) 146 bit error rate: case of fading in wireless communication link 147 C cepstrum coefficients 164 classification performance evaluated for aerial targets 167 D direct and indirect techniques of bispectral density estimation 9–11 discrete cosine transform (DCT)-based filters 76–80, 83–92

E evaluation of the signal Fourier spectrum from the bispectrum estimate 12–14 F fast fading link 145–146 frequency shift keying: FSK-4 143 G ground surveillance Doppler radar 168 H high resolution range profiles (HRRPs) of aerial targets 161 Hodges–Lehmann estimate 81–82 I instantaneous frequency contained in radar backscattering 168 J jitter 106 K K-nearest neighbor (KNN) and the FIR-median hybrid (FMH) filters 54–62 L local adaptive filtering using Z-parameter 92–99 M matched filtering 126–127 micro-Doppler content in radar backscattering 161 model of signal backscattered by naval object 150–151 N Naive Bayes Classifier (NBC) 165 naval object range profile 148–150 normalized bispectrum 35–36 – performance of bispectral estimators 15–29, 42–46 – Kravchenko weight functions (windows) 28 normalized range profiles of naval target evaluated for different polarizations and sea states 156–160

Subject index | 199

O optimal bispectrum-organized bit length 143 optimal parameters for additive and multiplicative predistortions 117–124 P phase coupled instantaneous frequencies (PCIFs) contained in radar backscattering 168 phase wrapping 33–35 polarization matrix coefficients 153 probabilities of signal detection 134–135 properties of noisy bispectral estimate 63–67 properties of triple correlation functions and bispectrum 4–9 – invariance property of bispectrum to a signal time delay 8 – phase coupling 9 R radar time-frequency distribution (TFD) derived from micro-Doppler content 163 S short-time bispectrum estimate 172 short-time cross-bispectrum estimates 182 signal shape reconstruction by using normalized bispectrum 36–42 smoothed bimagnitude and biphase estimates evaluated by linear mean or nonlinear median filters 52–53

spectral efficiency 144 spectrograms computed for different aerial target models 163 squared bicoherence 164 Support Vector Machine (SVM) classifier 165 surveillance radar system 175 T third-order test statistic used for signal detection in noise 127–129 time-frequency distribution of phase coupled contributions – evaluated for a human walking in vegetation clutter 184 – evaluated for an oscillating sphere-pendulum 176 – evaluated for a walking human 178 total (TOSD) and truncated (TRSD) standard deviations of bispectrum estimator 19–27 V variance of bispectral estimate 46 vector filters used for smoothing noisy bispectrum estimates 67–70 W W-test 73–75 X X-band polarimetric radar 153