Advances in signal processing 9783030403119, 9783030403126

Table of contents :
Preface......Page 6
Contents......Page 7
About the Editors......Page 11
1 Signal Processing for Practical Applications......Page 13
1.2 Chapters Included in the Book......Page 14
References......Page 16
2.1 Introduction......Page 17
2.2 The Synthesis of the p-ary D-code and the Polyphase Coherent Additional Signal......Page 19
2.3 Correlation Characteristics of Polyphase Coherent Additional Signals......Page 22
2.4 The Compression Device of Polyphase Coherent Additional Signals......Page 25
References......Page 30
3.1 Introduction......Page 32
3.2.1 Memoryless Amplifier......Page 34
3.2.2 Amplifier with Memory......Page 37
3.3.1 Memoryless Amplifier......Page 43
3.3.2 Amplifier with Memory......Page 47
3.4 Conclusions......Page 51
References......Page 55
4.1 Introduction......Page 56
4.2 Types of Distorting Effects......Page 57
4.3 Signal Quality Assessment Metrics......Page 62
4.4 Distortion Identification at the Transmitter Output......Page 64
4.5 Conclusions......Page 70
References......Page 71
5.1 Introduction......Page 72
5.2 Parameter Estimation Quality Bounds and Threshold Effect in Classical Inference......Page 73
5.4 Threshold Indication for Microwave Imaging Problem......Page 76
References......Page 83
6 The Development of the Algorithm for Estimating the Spectral Correlation Function Based on Two-Dimensional Fast Fourier Transform......Page 85
6.1 Introduction......Page 86
6.2 Dyadic Cyclostationary Description......Page 87
6.3 Two-Dimensional FFT Algorithm......Page 89
6.4 Simulation Results......Page 91
6.5 Conclusions......Page 95
References......Page 96
7.1 Introduction......Page 97
7.2 Problems of Discrete Two-Dimensional Signal Processing in Fourier Bases......Page 98
7.2.1 System Analysis of the Nature of Origin, Sources, and Subject Areas of Applications of Discrete Two-Dimensional Signals......Page 99
7.2.2 The Main Directions of Scientific and Applied Research in the Field of Digital Spectral Processing of One-Dimensional and Two-Dimensional Signals......Page 100
7.3 Two-Dimensional Discrete Fourier Transform and Its Basis......Page 103
References......Page 105
8 Sliding Spatial Frequency Processing of Discrete Signals......Page 107
8.2 Direct Two-Dimensional Discrete Fourier Transform......Page 108
8.3.1 First Method and Fast Algorithm 1 of HS+ Sliding Processing......Page 112
8.3.2 Method of HS– Sliding Processing......Page 115
8.3.3 Second Method and Fast Algorithm 2 of HS+ Sliding Processing......Page 117
8.4 Conclusions......Page 118
References......Page 119
9.1 Introduction......Page 120
9.2 Direct and Inverse Two-Dimensional Discrete Fourier Transform......Page 122
9.2.1 Algebraic Form of Direct and Inverse Two-Dimensional Discrete Fourier Transforms......Page 123
9.2.2 Matrix Form of Direct and Inverse Two-Dimensional Discrete Fourier Transform......Page 124
9.2.3 Fast Algorithms of Direct and Inverse Two-Dimensional Discrete Fourier Transform......Page 125
9.3 Interpolation Method of Two-Dimensional Discrete Real and Complex Signals in Spatial Domain......Page 126
9.4 Conclusions......Page 130
References......Page 131
10 Topography of the z-Plane Discretized by Quantizing the Coefficients of the Canonical Form of Recursive Digital Filter......Page 132
10.1 Introduction......Page 133
10.2 The Number-Theoretic Nature of the Zeros and Poles of IIR Digital Filters with Finite Word Length......Page 134
10.3 The Topography of the Second-Degree Algebraic Numbers in a Discretized Complex z-Plane......Page 135
10.4 The Topography of the Third-Degree Algebraic Numbers in a Discretized Complex z-Plane......Page 137
10.5 The Topography of the Fourth-Degree Algebraic Numbers in a Discretized Complex z-Plane......Page 139
References......Page 143
11 Method for Adaptive Control of Technical States of Radio-Electronic Systems......Page 145
11.1 Introduction......Page 146
11.2 Related Work......Page 147
11.3.1 Complex Nature of the Transmitted Signals......Page 149
11.3.2 Creation of Three-Dimensional MI Model at the Touch Level......Page 150
11.3.3 Formation of an Ellipsoid of Working Capacity Considering FPR and FNR Errors......Page 154
11.4 Implementation of TS-RES Adaptive Control Method......Page 155
11.5 Conclusions......Page 157
References......Page 158

Citation preview

Intelligent Systems Reference Library 184

Margarita Favorskaya Lakhmi C. Jain   Editors

Advances in Signal Processing Theories, Algorithms, and System Control

Intelligent Systems Reference Library Volume 184

Series Editors Janusz Kacprzyk, Polish Academy of Sciences, Warsaw, Poland Lakhmi C. Jain, Faculty of Engineering and Information Technology, Centre for Artificial Intelligence, University of Technology, Sydney, NSW, Australia; KES International, Shoreham-by-Sea, UK; Liverpool Hope University, Liverpool, UK

The aim of this series is to publish a Reference Library, including novel advances and developments in all aspects of Intelligent Systems in an easily accessible and well structured form. The series includes reference works, handbooks, compendia, textbooks, well-structured monographs, dictionaries, and encyclopedias. It contains well integrated knowledge and current information in the field of Intelligent Systems. The series covers the theory, applications, and design methods of Intelligent Systems. Virtually all disciplines such as engineering, computer science, avionics, business, e-commerce, environment, healthcare, physics and life science are included. The list of topics spans all the areas of modern intelligent systems such as: Ambient intelligence, Computational intelligence, Social intelligence, Computational neuroscience, Artificial life, Virtual society, Cognitive systems, DNA and immunity-based systems, e-Learning and teaching, Human-centred computing and Machine ethics, Intelligent control, Intelligent data analysis, Knowledge-based paradigms, Knowledge management, Intelligent agents, Intelligent decision making, Intelligent network security, Interactive entertainment, Learning paradigms, Recommender systems, Robotics and Mechatronics including human-machine teaming, Self-organizing and adaptive systems, Soft computing including Neural systems, Fuzzy systems, Evolutionary computing and the Fusion of these paradigms, Perception and Vision, Web intelligence and Multimedia. ** Indexing: The books of this series are submitted to ISI Web of Science, SCOPUS, DBLP and Springerlink.

More information about this series at http://www.springer.com/series/8578

Margarita Favorskaya Lakhmi C. Jain •

Editors

Advances in Signal Processing Theories, Algorithms, and System Control

123

Editors Margarita Favorskaya Institute of Informatics and Telecommunications Reshetnev Siberian State University of Science and Technology Krasnoyarsk, Russia

Lakhmi C. Jain Faculty of Engineering and Information Technology Centre for Artificial Intelligence, University of Technology Sydney Broadway, NSW, Australia

ISSN 1868-4394 ISSN 1868-4408 (electronic) Intelligent Systems Reference Library ISBN 978-3-030-40311-9 ISBN 978-3-030-40312-6 (eBook) https://doi.org/10.1007/978-3-030-40312-6 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The research book is a continuation of our previous books which are focused on the recent advances in computer vision methodologies and technical solutions using conventional and intelligent paradigms. • Computer Vision in Control Systems—1, Mathematical Theory, ISRL Series, Volume 73, Springer-Verlag, 2015 • Computer Vision in Control Systems—2, Innovations in Practice, ISRL Series, Volume 75, Springer-Verlag, 2015 • Computer Vision in Control Systems—3, Aerial and Satellite Image Processing, ISRL Series, Volume 135, Springer-Verlag, 2018 • Computer Vision in Control Systems—4, Real Life Applications, ISRL Series, Volume 136, Springer-Verlag, 2018 • Computer Vision in Control Systems—5, Advanced Decisions in Technical and Medical Applications, ISRL Series, Volume XXX, Springer-Verlag, 2020 • Computer Vision in Control Systems—6, Advances in Practical Applications, ISRL Series, Volume XXX, Springer-Verlag, 2020 The book presents a very small sample of research work undertaken by the researchers of Russian Federation in the field of signal processing theories, algorithms and system control. The book is directed to the PhD students, professors, researchers and software developers working in the areas of digital video processing and computer vision technologies. We wish to express our gratitude to the authors and reviewers for their contribution. The assistance provided by Springer-Verlag is acknowledged. Krasnoyarsk, Russia Sydney, Australia

Margarita Favorskaya Lakhmi C. Jain

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Contents

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Signal Processing for Practical Applications Margarita N. Favorskaya and Lakhmi C. Jain 1.1 Introduction . . . . . . . . . . . . . . . . . . . . 1.2 Chapters Included in the Book . . . . . . . 1.3 Conclusion . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

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Polyphase Radar Signals with Zero Autocorrelation Zone and Their Compression Algorithm . . . . . . . . . . . . . . . . . . Roman N. Ipanov and Sergey M. Smolskiy 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Synthesis of the p-ary D-code and the Polyphase Coherent Additional Signal . . . . . . . . . . . . . . . . . . . . 2.3 Correlation Characteristics of Polyphase Coherent Additional Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The Compression Device of Polyphase Coherent Additional Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Output Power Amplifier Effects on Harmonic and Amplitude Modulated Signals Distortions . . . . . . . . . . . . . . . . . . . . . . . . . Yuri A. Bryukhanov and Kirill S. Krasavin 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Single-Ended Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Memoryless Amplifier . . . . . . . . . . . . . . . . . . . . 3.2.2 Amplifier with Memory . . . . . . . . . . . . . . . . . . . 3.3 Push-Pull Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Memoryless Amplifier . . . . . . . . . . . . . . . . . . . . 3.3.2 Amplifier with Memory . . . . . . . . . . . . . . . . . . .

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Distortion Types Separation of QAM-16 Signal . . . . . . . . . . Andrei E. Kiselnikov, Mikhail A. Dubov and Andrei L. Priorov 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Types of Distorting Effects . . . . . . . . . . . . . . . . . . . . . . 4.3 Signal Quality Assessment Metrics . . . . . . . . . . . . . . . . 4.4 Distortion Identification at the Transmitter Output . . . . . . 4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Threshold Effect Indicator Analysis for Template-Based Processing in Microwave Imaging . . . . . . . . . . . . . . . . . . . . . . . Aleksey S. Gvozdarev and Tatyana K. Artyomova 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Parameter Estimation Quality Bounds and Threshold Effect in Classical Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Approaches to the Problem of Threshold SNR Indication . . 5.4 Threshold Indication for Microwave Imaging Problem . . . . 5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Development of the Algorithm for Estimating the Spectral Correlation Function Based on Two-Dimensional Fast Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Timofey Shevgunov, Evgeniy Efimov, Vladimir Kirdyashkin and Tatiana Kravchenko 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Dyadic Cyclostationary Description . . . . . . . . . . . . . . . . . . 6.3 Two-Dimensional FFT Algorithm . . . . . . . . . . . . . . . . . . . 6.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Systems Analysis of Discrete Two-Dimensional Signal Processing in Fourier Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alexey V. Ponomarev 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Problems of Discrete Two-Dimensional Signal Processing in Fourier Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 System Analysis of the Nature of Origin, Sources, and Subject Areas of Applications of Discrete Two-Dimensional Signals . . . . . . . . . . . . . . . . . . . .

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The Main Directions of Scientific and Applied Research in the Field of Digital Spectral Processing of One-Dimensional and Two-Dimensional Signals . 7.3 Two-Dimensional Discrete Fourier Transform and Its Basis . . 7.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

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Sliding Spatial Frequency Processing of Discrete Signals . Olga V. Ponomareva, Alexey V. Ponomarev and Natalya V. Smirnova 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Direct Two-Dimensional Discrete Fourier Transform . . 8.3 Sliding Spatial-Frequency Processing . . . . . . . . . . . . . 8.3.1 First Method and Fast Algorithm 1 of HS+ Sliding Processing . . . . . . . . . . . . . . . . . . . . 8.3.2 Method of HS– Sliding Processing . . . . . . . . 8.3.3 Second Method and Fast Algorithm 2 of HS+ Sliding Processing . . . . . . . . . . . . . . . . . . . . 8.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Interpolation of Real and Complex Discrete Signals in the Spatial Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Olga V. Ponomareva, Alexey V. Ponomarev and Natalya V. Smirnova 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Direct and Inverse Two-Dimensional Discrete Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Algebraic Form of Direct and Inverse Two-Dimensional Discrete Fourier Transforms . . . . . 9.2.2 Matrix Form of Direct and Inverse Two-Dimensional Discrete Fourier Transform . . . . . . . . . . . . . . . . . . . 9.2.3 Fast Algorithms of Direct and Inverse Two-Dimensional Discrete Fourier Transform . . . . . 9.3 Interpolation Method of Two-Dimensional Discrete Real and Complex Signals in Spatial Domain . . . . . . . . . . . . . . . . 9.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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10 Topography of the z-Plane Discretized by Quantizing the Coefficients of the Canonical Form of Recursive Digital Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vladislav A. Lesnikov, Tatiana V. Naumovich, Alexander V. Chastikov and Alexander P. Metelyov 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 The Number-Theoretic Nature of the Zeros and Poles of IIR Digital Filters with Finite Word Length . . . . . . . . . . . . . . . . 10.3 The Topography of the Second-Degree Algebraic Numbers in a Discretized Complex z-Plane . . . . . . . . . . . . . . . . . . . . . 10.4 The Topography of the Third-Degree Algebraic Numbers in a Discretized Complex z-Plane . . . . . . . . . . . . . . . . . . . . 10.5 The Topography of the Fourth-Degree Algebraic Numbers in a Discretized Complex z-Plane . . . . . . . . . . . . . . . . . . . . 10.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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11 Method for Adaptive Control of Technical States of Radio-Electronic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . Pavel A. Budko, Alexey M. Vinogradenko, Alexey V. Mezhenov and Nina G. Zhuravlyova 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Mathematical Backgrounds: Representation of Control Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.1 Complex Nature of the Transmitted Signals . . . . . . 11.3.2 Creation of Three-Dimensional MI Model at the Touch Level . . . . . . . . . . . . . . . . . . . . . . . . 11.3.3 Formation of an Ellipsoid of Working Capacity Considering FPR and FNR Errors . . . . . . . . . . . . . 11.4 Implementation of TS-RES Adaptive Control Method . . . . . 11.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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About the Editors

Dr. Margarita Favorskaya is a Professor and Head of Department of Informatics and Computer Techniques at Reshetnev Siberian State University of Science and Technology., Russian Federation. Professor Favorskaya is a member of KES organization since 2010, the IPC member and the Chair of invited sessions of over 30 international conferences. She serves as an associate editor of Intelligent Decision Technologies Journal, International Journal of Knowledge-Based and Intelligent Engineering Systems, International Journal of Reasoning-based Intelligent Systems, a Honorary Editor of the International Journal of Knowledge Engineering and Soft Data Paradigms, Guest Editor, and Book Editor (Springer). She is the author or the co-author of 200 publications and 20 educational manuals in computer science. She co-edited several books for Springer. She supervised nine Ph.D. candidates to completion and presently supervising four Ph.D. students. Her main research interests are digital image and videos processing, remote sensing, pattern recognition, fractal image processing, artificial intelligence, and information technologies.

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About the Editors

Dr. Lakhmi C. Jain Ph.D., ME, BE (Hons), Fellow (Engineers Australia) is with the University of Technology Sydney, Australia, and Liverpool Hope University, UK. Professor Jain founded the KES International for providing a professional community the opportunities for publications, knowledge exchange, cooperation and teaming. Involving around 5000 researchers drawn from universities and companies world-wide, KES facilitates international cooperation and generate synergy in teaching and research. KES regularly provides networking opportunities for professional community through one of the largest conferences of its kind in the area of KES. http://www.kesinternational.org/organisation.php

Chapter 1

Signal Processing for Practical Applications Margarita N. Favorskaya and Lakhmi C. Jain

Abstract The chapter includes a brief description of chapters on recent advances in signal processing theories, algorithms and system control in different practical area such as polyphase radar signals with zero autocorrelation zone and their compression algorithm, output power amplifier effects on harmonic and amplitude modulated signals distortions, distortion types separation of QAM-16 signal, threshold effect indicator analysis for template-based processing in microwave imaging, development of the algorithm for estimating the spectral correlation function based on twodimensional fast Fourier transform, systems analysis of discrete two-dimensional signal processing in Fourier bases, sliding spatial frequency processing of discrete signals, interpolation of real and complex discrete signals in the spatial domain, topography of the z-plane discretized by quantizing the coefficients of the canonical form of recursive digital filter, and method for adaptive control of technical states of radio-electronic systems. Keywords Signal processing · Autocorrelation function · Amplifier · Signal-to-noise ratio · Fast Fourier transform · Complex parameter evaluation · Technical state of systems

M. N. Favorskaya (B) Institute of Informatics and Telecommunications, Reshetnev Siberian State University of Science and Technology, 31, Krasnoyarsky Rabochy ave., Krasnoyarsk 660037, Russian Federation e-mail: [email protected] L. C. Jain University of Technology Sydney, Ultimo, Australia e-mail: [email protected] Liverpool Hope University, Liverpool, UK © Springer Nature Switzerland AG 2020 M. Favorskaya and L. C. Jain (eds.), Advances in Signal Processing, Intelligent Systems Reference Library 184, https://doi.org/10.1007/978-3-030-40312-6_1

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M. N. Favorskaya and L. C. Jain

1.1 Introduction There is a tremendous interest among researchers in the development of new algorithms and applications in the field of signal processing. The availability of cheap computing power is also responsible for this interest among researchers and application engineers and scientists. We present a very small sample of research work undertaken by the researchers of Russian Federation in the field of signal processing theories, algorithms and system control. The contributions in this book do not represent the vast amount of work undertaken by researchers but we believe that this book will create enough interest among researchers to take this novel field much further. The contributions in the book are from radar, amplifiers, transmitters, microwave imaging, recursive digital filters, radio electronic systems, and so on. The following section presents brief abstracts of the chapters included in the book. It is followed by conclusion and references.

1.2 Chapters Included in the Book Chapter 2 studies a functionality of radars, which are installed on the quickly-moved aircraft-space carriers with the direct aperture synthesis. In recent years, the great attention is attracted to reduction of the detection possibility of radar stations by the means of the radio-electronic reconnaissance and by the self-guided anti-radar missiles [1]. The chapter provides the method of polyphase radar signal processing. The multi-channel compression device of this signal is studied. Chapter 3 analyses the effects of nonlinearity on the unmodulated harmonic and modulating signal of amplitude modulated signal distortions for single-ended and push-pull output power amplifiers. Nonlinear distortions of output power amplifiers are investigated by many authors, but in this chapter the processes in the singleended and push-pull output power amplifiers with memory and memoryless are considered. The estimates in different modes are obtained and can be used for the design of advanced signal processing systems. Chapter 4 describes a method of identification of QAM-16 signal distortion on the transmitter out. Information about the type of signal distortion can significantly reduce the time to adjust the radio device during its production [2]. The proposed method can be used as the basis for the system of the automatic measurement and adjustment of device parameters if the hardware platform supports MATLAB software package. In this chapter, the main criterion of signal quality is associated with modulation error ratio metric. The authors derive the rigorous wording of criteria using the language of Boolean algebra. Chapter 5 contributes in the problem of objects’ and signals’ parameters inference as one of the crucial problems in modern wireless and radio engineering in general and microwave imaging in particular. The task is in that for a given time interval (for which the parameters of the microwave imaging system setup are being constant: type

1 Signal Processing for Practical Applications

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and size of aperture, object and reference signals, etc.) the object ought to be found, whose parameters are under investigation, and which was illuminated by an exploring signal with known characteristics (time structure, bandwidth, carrier wavelength, polarization properties, etc.). More, the main problem is to find the joint estimates of all possible unknown object parameters: its geometry, position, electromagnetic properties, etc. The proposed hybrid estimator enables the threshold region detection with accuracy of 1 dB. Chapter 6 presents the algorithm for estimating spectral correlation function of a random process that is a valid bi-frequency description of the probabilistic properties of any wide-sense cyclostationary process and relates to its cyclic autocorrelation function via the Fourier transform. The proposed algorithm for estimating the spectral correlation function of wide-sense cyclostationary processes is the effective tool for revealing the cyclic frequencies exhibited by the process in case of a long finite-time observation of the available process [3, 4]. The algorithm is a simultaneous wideband estimator providing such coverage of the bi-frequency plane that is dense enough to avoid missing any components of spectral correlation function. Chapter 7 considers the common problems of 2D signals processing on the base of 2D discrete Fourier transform. It is shown that 2D version of the discrete canonical decomposition of random signals is a modification of the standard cyclic 2D correlation function of the original signal [5]. In this chapter, the theoretical and applied research on the development of new and improve existing methods and algorithms for 2D Fourier processing is carried out. Chapter 8 contains the proposed concept of sliding spatial-frequency processing of two-dimensional discrete signals on the basis of two-dimensional discrete Fourier transform [6, 7]. Based on the mathematical apparatus, the authors suggest the methods and two fast algorithms for horizontal sliding processing of two-dimensional discrete signals in the spatial-frequency domain. Such proposition allows to provide a real-time scale for obtaining 2D DFT coefficients using eliminating redundancy. Chapter 9 aims to develop the interpolation methods for real and complex discrete one-dimensional and two-dimensional signals in the spatial domain based on fast 2D Fourier transform [5]. The authors consider a relationship between the two concepts of digital signal processing given a finite number of samples: the interpolation of discrete signals and approximation of discrete signals. Also, this chapter provides the algebraic and matrix forms of direct and inverse two-dimensional discrete Fourier transform, as well as, the fast algorithms of direct and inverse twodimensional discrete Fourier transforms. Visual examples of the three-dimensional surface envelopes of the original and interpolated two-dimensional discrete signals with different parameters’ values are presented. Chapter 10 investigates the hypothesis that for digital filters of the higher orders, zeros and poles are points of algebraic numbers of the higher degrees. Thus, the study of the discrete structure of localization of zeros and poles in the z-plane is extended to complex algebraic numbers of a higher degree. This chapter continues the cycle of publications devoted to the new approach developed by the authors to infinite impulse response digital filters [8, 9]. Here, these results are extended to the fourth-degree algebraic numbers.

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Chapter 11 discusses a monitoring of technical state of complicated technical objects under different attacks and destabilizing factors, aging, and dispersion of technological parameters in complicated technical systems. The main idea is to eliminate the redundancy of measuring information under completeness of object control through communication channels and transmit not all measuring results but only messages about an exit of object parameters outside of the limits of the set admissions. In this chapter, new adaptive control method of autonomous radio-electronic equipment, which is based on estimation of their technical states, is developed. The complex control is based on the integration of indications of several types of sensors in universal automated system.

1.3 Conclusion The chapter provides a brief description of the chapters on signal processing with original algorithms, theories, and system control. The content of this book reflects a sample of recent research directions in the field, which will form the basis of future intelligent systems.

References 1. Ipanov, R.N., Baskakov, A.I., Olyunin, N., Ka, M.-H.: Radar signals with ZACZ based on pairs of D-code sequences and their compression algorithm. IEEE Signal Proc. Lett. 25(10), 1560–1564 (2018) 2. Kiselnikov, A., Dubov, M., Priorov, A.: Non-reference metrics and its application for distortion compensation. In: 21th Conference of Open Innovations Association, pp. 172–181 (2017) 3. Efimov, E., Shevgunov, T., Kuznetsov, Y.: Time delay estimation of cyclostationary signals on PCB using spectral correlation function, pp. 184–187. Baltic URSI Symposium, Poznan (2018) 4. Shevgunov, T.: A comparative example of cyclostationary description of a non-stationary random process. J. Phys.: Conf. Ser. 1163, 012037 (2019) 5. Ponomareva, O., Ponomarev, A., Ponomarev, V.: Evolution of forward and inverse discrete Fourier transform. In: IEEE East-West Design & Test Symposium, pp. 313–318 (2018) 6. Ponomarev, V.A., Ponomareva, O.V., Ponomarev, A.V.: Method for effective measurement of a sliding parametric Fourier spectrum. Optoelectron. Instrum. Data Process. 50(2), 1–7 (2014) 7. Ponomareva, O., Ponomarev, A., Ponomareva, N.: Window-presume parametric discrete Fourier transform. IEEE East-West Design & Test Symposium, pp. 364–368 (2018) 8. Lesnikov, V., Chastikov, A., Naumovich, T., Armishev, S.: Implementation of a new paradigm in design of IIR digital filters. In: 8th IEEE East-West Design and Test Symposium, St. Petersburg, pp. 156–159 (2010) 9. Lesnikov, V., Naumovich, T., Chastikov, A.: The topography of a third order IIR digital filter zeros and poles in the z-plane discretized due to the quantization of the direct form coefficients. In: 8th Mediterranean Conference on Embedded Computing. Budva, Montenegro, 4p (2019)

Chapter 2

Polyphase Radar Signals with Zero Autocorrelation Zone and Their Compression Algorithm Roman N. Ipanov and Sergey M. Smolskiy

Abstract The polyphase (p-phase, where p is the prime integer number) radar signal, which has an area of zero side lobes in a vicinity of the central peak of autocorrelation function, has been synthesized. It is shown that this signal represents a train from p coherent phase-code-shift keyed pulses, which are coded by complementary sequences of the p-ary D-code. The method of ensemble set formation of the p-ary D-code for signal synthesis is suggested. Correlation characteristics of the synthesized signal are discussed. The compression algorithm of this signal is considered including in its structure the combined algorithm of Vilenkin-Chrestenson and fast Fourier transform. Keywords Autocorrelation function · Complementary sequences · Polyphase signal · Pulse train · Vilenkin-Chrestenson functions · Zero autocorrelation zone

2.1 Introduction For accurate determination of the distance (range) and speed of a variety of smallsize space objects on the near-Earth orbit, for resolution of separate elements of complicate space objects and also for resolution of small-size objects on the Earth surface, it is necessary to use the wideband probing signals, which have high resolution on the slant range r = c/(2Fs ), where Fs is the signal spectrum width, and c is the radial speed. To obtain the high angular resolution θ of the Earth surface elements and targets located on this surface, radars are used, which are installed on the quickly-moved aircraft-space carriers with the direct aperture synthesis. High resolutions on the slant and transverse r⊥ = r0 θ ranges, where r0 is the slant R. N. Ipanov (B) · S. M. Smolskiy National Research University “Moscow Power Engineering Institute”, 14, Krasnokazarmennaya str., Moscow 111250, Russian Federation e-mail: [email protected] S. M. Smolskiy e-mail: [email protected] © Springer Nature Switzerland AG 2020 M. Favorskaya and L. C. Jain (eds.), Advances in Signal Processing, Intelligent Systems Reference Library 184, https://doi.org/10.1007/978-3-030-40312-6_2

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R. N. Ipanov and S. M. Smolskiy

range to observing resolution element, permit to obtained of two-dimensional target patterns in distance. Ensuring of the high angular resolution of small-size space objects or elements of complicate space objects is based on the effect of the inverse synthesis of the antenna aperture [1]. For resolution on Doppler frequency equaled to FD = 1/Ts , where Ts is the probing signal duration (time of coherent accumulation), the angular resolution θ = λ/(2V sin θ0 )FD is provided, where V is the ground speed of object motion, θ0 is the angle between the ground speed vector and the pointing direction. The transverse resolution is provided by turning of the target velocity vector with regard to the pointing direction and is realized by processing of the sequence of complex samples, which arrive from each target element resolved on the slant range. It follows from the above-mentioned that for providing of high resolutions on the slant r and transverse r⊥ ranges, it is necessary to use the probing signals with the wide spectrum and the long duration. We can find in world publications that for these purposes one can use the train of Linear-Frequency-Modulated (LFM) pulses with the high repetition frequency and also the “open” frequency-modulated coherent signals, which represents the train from M  1 rectangular radio pulses with Tp duration with the step of frequency variation from a pulse to a pulse of 1/Tp and with the of-duty factor q > 2 [2]. Nevertheless, as we know, the frequency-modulated and frequency-shift keyed signals have the “splay” mismatched function, which results in the ambiguity occurs in range. The ambiguity peaks are appeared on AutoCorrelation Function (ACF) of the train of LFM pulses. The theme of the probing signal choice is also relevant in connection with the resolution problem of echoes, which are overlapping in time and which amplitude changes in large range, i.e., in connection with the problem of weak echoes detection, which are closed by the side lobes of the strong echoes ACF. To suppress the side lobes of ACF echoes, one can apply the intra-pulse and inter-pulse weighting [3, 4]. However, at that, the spreading of the main ACF lobe occurs together with the loss in SNR. To solve of stated tasks, we can use the Phase-Code-Shift Keyed (PCSK) signals, which are free from shortcomings of frequency modulated and frequency shift keying signals. In [4–7], the radar PCSK signals are considered, which have the zero correlation zone in the region of the central peak of aperiodic ACF (Zero AutoCorrelation Zone—ZACZ). These signals represent the periodic sequence from M  1 coherent pulses coding (or phase-shift keyed) by the ensembles of complementary or orthogonal sequences. PCSK signals with ZACZ solve the problem of weak echo detection on the background of strong echoes. However, the relative ZACZ width of considered in [4–7] PCSK signals is ε = Z /L = (q − 1)/(q(M − 1) + 1)  1,

(2.1)

where Z is the ZACZ width, L is the signal duration [8]. In addition, at formation and processing of PCSK signal with the large number of pulses in the train, it is difficult enough to keep their coherence. The considered in

2 Polyphase Radar Signals with Zero Autocorrelation Zone …

7

[4–7] polyphase PCSK signals, for instance, Frank signals or P4, also have the large alphabet of phases equaled to the number of discretes in the pulse. In recent years, the great attention is attracted to reduction of the detection possibility of Radar Stations (RS) by the means of the radio-electronic reconnaissance and by the self-guided anti-radar missiles [9–11]. In RS with low probability of emission detection, the special measures are anticipated to increase of RS operation secrecy. The low spectral density of emission, variation of probing signal parameters according the random law, operation in the wide frequency band, and control of the emission power are among them. Applied wideband probing signals—LFM signals or bi-phase PCSK signals—do not provide RS operation secrecy. So, LFM signals can be easily recognized by means of reconnaissance on the phase variation speed; while bi-phase PCSK signals—are determined with the help of quadratic detection circuits. To a great extend, the emission secrecy can be increased by utilization of the polyphase PCSM signals [11]. Polyphase pulse signals can be formed by the wide set of p-ary codes and differ by the low spectral density and the low level of ACF side lobes. In this chapter, to solve the problems of high resolution of the variety of smallsize space objects on the near-Earth orbit, the separate elements of the complicated space object, as well as, the small-size objects on the Earth surface, the polyphase (p-phase, where p is the prime integer number) radar signal is synthesized, which has ZACZ. This signal represented the train from p coherent PCSK pulses coding by complementary sequences of the p-ary D-code. It has low pulse number p in the train, the small alphabet of phases equaled to p, and an approach to code formation allows usage of the fast transform algorithm for its compression in the matched filter. The chapter is organized as follows. The synthesis of the p-ary D-code and polyphase coherent additional signal is discussed in Sect. 2.2. Section 2.3 provides the correlation characteristics of polyphase coherent additional signals. Section 2.4 presents the compression device of polyphase coherent additional signals. Section 2.5 concludes the chapter.

2.2 The Synthesis of the p-ary D-code and the Polyphase Coherent Additional Signal p

Sequences {dn1 }, {dn2 }, . . . , {dni }, . . . , {dn }, (n = 1, 2, . . . , N ) of the length N = p k , where k ≥ 2 is the integer number, are called complementary [6, 12, 13] if  rm1

+ rm2

+ ···

+ rmi

+ ···

+ rmp

=

pN ; m = 0, 0; m = ±1, . . . , ±(N − 1),

(2.2)

N ∗i dni dn−m is the aperiodic ACF of the sequence {dni }, * is where rmi = rmi,i = n=m+1 the complex conjugation operation.

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R. N. Ipanov and S. M. Smolskiy

N / p sets of complementary (additional) sequences with the N length satisfying to Eq. 2.2 form a Matrix of p-phase Additional Sequences (MAS) with dimensions N × N , which is called the ensemble of Golay p-phase complimentary sequences [6, 12] or the p-ary D-code [13] (we introduce the generalized concept of Golay sequences and D-code for p > 2 [14]).  N   Let  D N = d˜i,n  be a matrix of p-ary D-codes, d˜i,n = 0, 1, . . . , p − 1, N = p k , 1

p if the prime number. Then MAS of the k order (dimensions N × N ) will have a form:    N 2π ˜   (2.3) D N = di,n 1 , di,n = exp j di,n . p  N  N j Let us call sequences Di1,N = di,n n=1 and D1,N = d j,n n=1 p-paired if (i − 1) p ⊕ ( j − 1) p = () p , i, j = 1, 2, . . . , N ,

(2.4)

where i, j are the numbers of sequences in D-code or numbers of MAS lines, (a) p is the number a in the p-ary form, ⊕ are the operation of adding modulo p,  = p k−1 . Notice that the p-paired sequences are complimentary, i.e., for them Eq. 2.2 holds true.  N Let D N be MAS (Eq. 2.3), and H N = h i,n 1 be a matrix of the system of Vilenkin-Chrestenson-Kronecker (VC-Kronecker) functions [15]. It is known that the system of VC-Kronecker is the multiplicative Abelian group [16]. Since the variety consisting of MAS lines is the adjacent class in the sub-group, elements of which are lines of VC-Kronecker matrix, and the first MAS line is the leader of the adjacent class, then we may write: DN = HN dN ,

(2.5)

  where d N = diag d1,1 , d1,2 , . . . , d1,n , . . . , d1,N is the diagonal matrix with elements from the first line of D N . At p = 2, VC-Kronecker matrix is transformed into Hadamard matrix [17]. It follows from Eq. 2.5 that to construct of MAS D N , it is necessary to form its first line D11,N . Elements of MAS first line are defined as follows [18]:

d1,y+1

 k−1 2π

= exp j yi+1 yli , p i=1

(2.6)

where y + 1 = n is the number of MAS column, (y) p = (yk yk−1 . . . yi . . . y1 ) is the number of MAS column in the p-ary form, yi = 0, 1, . . . , p − 1, y = 0, 1, . . . , p k − 1, li = 1, 2, . . . , i, i = 1, 2, . . . , k − 1, lk−1 = lk−2 = · · · = l2 .

2 Polyphase Radar Signals with Zero Autocorrelation Zone …

9

Adding in Eq. 2.6 is performed modulo p. This approach allows formation of γ = 2k−2 ensembles of D-code of k order. In [17], this approach of D-code formation at p = 2 is presented, and the example of D-code formation for k = 4 is discussed. Let us consider the example of formation of triple-ary D-code of k = 3 order with the code words length N = 33 = 27 allowing by approach (Eqs. 2.5–2.6) construction of γ = 23−2 = 2 ensemble of D-code. Now we form the first lines of two different matrices (ensembles) of D-code of the order k = 3. Remained lines of D-code matrix are obtained from the first line by the elementwise addition modulo p with appropriate lines of VC-Kronecker matrix. In this case, from Eqs. 2.3 and 2.6, we have the equations mentioned below. d˜1,y+1 =

(y)3 = y3 y2 y1

2



yi+1 yli = y2 yl1 + y3 yl2

i=1

y1 , y2 , y3 = 0, 1, 2 y = 0, 1, . . . , 26

i = 1, 2 l1 = 1 l2 = 1, 2 From this we obtain for two ensembles: d˜1,y+1 = y2 y1 + y3 y1 and d˜1,y+1 = y2 y1 + y3 y2 . At y = 0, 1, . . . , 26 for the first ensemble, we have the following first line of the D-code matrix: (0 0 0 0 1 2 0 2 1 0 1 2 0 2 1 0 0 0 0 2 1 0 0 0 0 1 2). For the second ensemble, the first line of the D-code matrix will have the following form: (0 0 0 0 1 2 0 2 1 0 0 0 1 2 0 2 1 0 0 0 0 2 0 1 1 0 2).  N  N j Sequences Di1,N = di,n n=1 and D1,N = d j,n n=1 are called adjacent if (i − 1) p ⊕ ( j − 1) p = (δ) p , i, j = 1, 2, . . . , N ,

(2.7)

where i, j are numbers of sequences in the D-code or number of MAS lines, δ = plk−1 −1 , lk−1 = 1, 2, . . . , k − 1 from Eq. 2.6. Adjacent sequences are also complimentary and for them Eq. 2.2 is true. Let us refer as the polyphase Coherent Additional Signal (CAS) of the train of p PCSK pulses encoded by p-paired or adjacent sequences of the D-code [14]. The analytical expression of the Complex Envelope (CE) of CAS has a form:

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˙ = S(t)

p N



S0 (t − (n + (i − 1)N q − 1)T0 )di,n ,

(2.8)

i=1 n=1



1 if(n − 1)T0 ≤ t ≤ nT0 is the envelope of the n-th 0 otherwise  N discrete of CAS, T0 is the discrete duration, q ≥ 2 is the off-duty factor, di,n 1 = Di1,N are elements of i-th p-paired sequence. CE of CAS in the vector form will have the following form: where S0 (t − (n − 1)T0 ) =

p S1,N (( p−1)q+1) = D11,N 01,N (q−1) D21,N 01,N (q−1) . . . Di1,N 01,N (q−1) . . . D1,N , (2.9)

where 01,N (q−1) = 01 02 . . . 0n . . . 0 N (q−1) is the zero vector-line with length N (q − 1).

2.3 Correlation Characteristics of Polyphase Coherent Additional Signals An analysis of CAS correlation characteristics is performed in [14, 17]. The aperiodic j cross-correlation function of sequences {dni } and {dn } is defined as: rmi, j =

N

∗j

dni dn−m at i = j; m = 0, ±1, . . . , ±(N − 1),

(2.10)

n=m+1 i, j

where rm = 0 at m = 0, because the complimentary sequences built according to Eqs. (2.5–2.6), are orthogonal. In the vector form, ACF of the polyphase CAS will have a form [14]: s R1,2N (( p−1)q+1)−1

 p−( p−1)

=

i+ p−1,i

R1,2N −1 . . .

i=1



p

i=1

p−1

i,i+1 R1,2N −1 01,N (q−2) 0

i=1



i R1,2N −1 01,N (q−2) 0

p−2

i,i+2 R1,2N −1

i=1

p− j

...

i=1





p−( p−1) i,i+ j R1,2N −1

01,N (q−2) 0 . . .

i,i+ p−1 R1,2N −1

, (2.11)

i=1



where 01,N (q−2) = 01 02 . . . 0n . . . 0 N (q−2) is the zero vector-line with the length N (q − 2),

2 Polyphase Radar Signals with Zero Autocorrelation Zone …

11

  i, j i, j i, j i, j i, j i, j i, j i, j R1,2N −1 = r−N +1 r−N +2 . . . r−1 r0 r1 . . . rmi, j . . . r N −2 r N −1 . i, j

i For i = j, R1,2N −1 = R1,2N −1 and according to Eq. 2.2: p

i R1,2N −1 = (0−N +1 0−N +2 . . . pN . . . 0 N −2 0 N −1 ),

(2.12)

i=1 i, j

and at i = j, r0 = 0. It follows from Eqs. 2.11 and 2.12 that ZACZ width (from both sides of the central peak of ACF) of the polyphase CAS is equaled to 2Z = 1 + N (q − 2) + 2N − 1 − 1 + N (q − 2) + 1 = 2N (q − 1), and taking into account the discrete duration T0 : Z = N T0 (q − 1).

(2.13)

The relative width of ZACZ is defined as: ε = Z /L T0 = (q − 1)/(( p − 1)q + 1),

(2.14)

where L = N (( p − 1)q + 1) is the discrete number in CAS. It follows from Eq. 2.14 that 1/(2 p − 1) ≤ ε < 1/( p − 1),

(2.15)

and at p = 2, 1/3 ≤ ε < 1 [17]. In Eq. 2.1, M ≥ N = p k , k ≥ 2 is the integer number, therefore, for signals considered in [4–7], ε  1. The polyphase CAS can be considered as the signal formed by the sequence from ZACZ-ensemble [8, 17] with parameters: Z AC Z (J, L , Z ), where J = γ Np is the number of sequences in the ensemble. The set of sequences forming p CAS and formed from the adjacent sets of p-paired sequences of D-code can be considered as ZCZ-ensemble [17] with parameters: ZC Z ( p, L , Z ). Figures 2.1 and 2.2 show, relatively, a part of the two-dimensional ambiguity function |R(τ, F)| of the three-phase CAS with the number of discretes of N = 243 in the pulse and with the off-duty factor q = 3 and its section by the plane F = 0, i.e., ACF of CE of CAS at complete filter matching with echoes in frequency. The width of CAS ZACZ with given parameters in relative units is Z /T0 = 486. From Fig. 2.1 it is seen that in the region of the central peak, the ambiguity function

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Fig. 2.1 Ambiguity function of the three-phase CAS

Fig. 2.2 Autocorrelation function of the three-phase CAS

has the clearly expressed rectangular region of zero correlation along the whole frequency axis F at N − 1 < |τ/T0 | ≤ N (q − 1), which is caused by a presence of the vector 01,N (q−2) in Eq. 2.11. The dimensions of this region do not depend on the law of the phase-shift keying and on mismatching in frequency, but depend only on the off-duty factor q. The region of zero correlation at 0 < |τ/T0 | ≤ N − 1 near the central peak of ACF, which is caused by the property of complementary sequences (Eq. 2.2), takes a place only at complete filter matching with the echoes in frequency. The ambiguity function section of CAS by the plane τ = 0 has the envelope of the form |sin x/x| with the main lobe width on the zero level 2/(N T0 ) and the internal comb structure. The spectrum combs are spaced from each other in F by the value 1/(q N T0 ). The comb width on the zero level is 2/( pq N T0 ), and the total number of combs within the main lobe of the amplitude-frequency spectrum envelope of the

2 Polyphase Radar Signals with Zero Autocorrelation Zone …

13

Fig. 2.3 Section of ambiguity function of the three-phase CAS by the plane F = 0.3/ pq N T0

CAS CE square is equaled to 2q − 1. The side lobes with the width 1/( pq N T0 ) on the zero level occur between combs and the total number of side lobes is equaled to p − 2. ZACZ exists only at complete, filter matching with echoes in Doppler frequency [14]. At mismatch F in the frequency in ZACZ near the main ACF peak, the side lobed appear, the maximal from them is compared in the level with the maximal side lobe outside ZACZ at F = 0.3/ pq N T0 . Figure 2.3 shows the ambiguity function section by the plane F = 0.3/ pq N T0 of the three-phase CAS with the same parameters. CAS is assumed to use at radar target tracking in resolution modes for accurate measurements (specification) of Doppler frequency, when the target rough estimation is already known from the preliminary target detection. At that, the compression device for CAS should be multi-channel in Doppler frequency with the necessary channel width.

2.4 The Compression Device of Polyphase Coherent Additional Signals For compression of the coherent pulse sequence, the correlation-filtering processing is usually used, at which the reflected signal modulation is first removed and then, with the help of Fast Fourier Transform (FFT) and Doppler frequency is defined [4, 6]. The structural circuit of the compression device for polyphase CAS is shown in Fig. 2.4 and represents the equivalent structural diagram of the matched filter of

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R. N. Ipanov and S. M. Smolskiy

Fig. 2.4 Structural diagram of the single-channel compression device of CAS

polyphase CAS at the known Doppler frequency or the equivalent structural diagram of the matched filter in the single frequency channel. The compression device consists of the input register on N memory cells, the processor of the discrete D-transform with N inputs and N outputs, the switching block, p − 1 similar shift registers on qN memory cells and p − 1 similar summation units of complex number, where q is the off-duty factor, N = p k is the D-code length. At p = 2, we obtain the compression device of bi-phase (binary) CAS. The basing element of this device is the processor of discrete D-transform (the processor DT-D), which operation algorithm is described by the following mathematical expression: T , G N ,1 = D N S1,N

(2.16)

where S1,N is the vector of input signal samples of the discrete D-transform, T is the operation of the vector transposition. Substituting Eq. 2.5 in Eq. 2.16, we obtain: T . G N ,1 = H N d N S1,N

(2.17)

It is known that VC-Kronecker matrix can be factorized by Good method [19], i.e., the discrete D-transform (Eq. 2.17) can be reduced to FFT in the basis of VCKronecker function system (FTVC), which has the form: T , G N ,1 = Ck N Ck−1 N . . . C jN . . . C1 N d N S1,N

Ck N = E p ⊗ 1 p ⊗ · · · ⊗ 1 p , Ck−1 N = 1 p ⊗ E p ⊗ · · · ⊗ 1 p , ... C jN = 1 p ⊗ · · · ⊗ E p ⊗ · · · ⊗ 1 p , ... C1 N = 1 p ⊗ · · · ⊗ 1 p ⊗ E p    k

(2.18)

2 Polyphase Radar Signals with Zero Autocorrelation Zone …

15

Fig. 2.5 Structural diagram of the single-channel compression device of CAS with FTVC

where ⊗ is the operation of Kronecker product, 1 p is the unitary matrix with dimensions p × p, E p is the matrix of Discrete Exponential Functions (DEF) with dimensions p × p. From Eq. 2.18, it follows that DT-D processor in the diagram in Fig. 2.4 can be replaced by FTVC processor with addition of weight coefficients (the matrix d N in expression) in the processor input, which are elements of the first line of MAS D N . Then the structural diagram of the compression device of the polyphase CAS will have the form presented in Fig. 2.5. The switching block performs connection of p from its N inputs with p outputs according to Eq. 2.4 or Eq. 2.7, i.e., in accordance with the line numbers, in which the p-paired or adjacent D-codes are situated. For p = 2, the compression algorithm of polyphase CAS presented in the form of the structural diagram in Fig. 2.5 is transformed into the compression algorithm of binary CAS, and FTVC processor is transformed into the processor of the fast Walsh transform. In [20], the multi-channel compression device for CAS is described, which allows simultaneously remove the modulation of polyphase pulse signals encoded by complimentary sequences and determine Doppler frequency in restricted Doppler frequency range according to preliminary target detection. This device consists of the processor of fast D-transform of Fourier (FT-D-FK , K is a number of used frequency channels), using the combination of FFT algorithms in basis-matrices of additional sequences and DEF by means of bit-by-bit multiplication of each MAS line with dimensions of N × N . MAS matrix here is the matrix of pulse characteristics of CAS pulses. The block matrix with dimension N K × N obtained at that represents the set of matrices of pulse characteristics on K different frequency, i.e., lines of DEF matrix play the role of frequency channels. In FT-D-FK algorithm, MAS matrix itself is factorized. The multi-channel compression device of CAS described in [20] can be built on the base of FTVC using Eq. 2.5. The structural diagram of such a device is presented in Fig. 2.6. The controllable local oscillator of the radar receiving device, according to rough estimation of Doppler frequency F D obtained in the mode of target detection, retunes its frequency so as the value F D falls in the frequency range, which is covered by frequency channels of the compression device of CAS. To CAS compression, the 



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R. N. Ipanov and S. M. Smolskiy

Fig. 2.6 Structural diagram of the multi-channel compression device of CAS with FTVC-FK

processor of fast Vilenkin-Chrestenson-Fourier (FTVC-FK , K is a number of used frequency channels) using of combination of FFT algorithms in the basis of VCKronecker and DEF function system by means of the bit-by-bit multiplication of each line of VC-Kronecker matrix with dimension N × L, L = N (( p − 1)q + 1) (the matrix lines are prolonged by repeating of each element or zero padding), by each from K lines of DEF matrix with dimension L × L. CE of the signal reflected from the target can be written as:     ˙ exp − j2π FˆD t , S˙ t, FˆD = S(t) ˙ is CAS CE (Eq. 2.8). The signal is fed from ADC outputs in the quadratic where S(t) channels to the input shift register of the compression device. Having transferred from analog quantities to discrete ones, i.e., at t → tn = (n − 1)T0 , F D → F Dk = (k − 1)F, F = 1/L T0 is the mismatch between frequency channels, n = 1, 2, . . . , L, k = 1, 2, . . . , K , we obtain the echo CE in the discrete form:   2π ˙ k) = S(n) ˙ (n − 1)(k − 1) . S(n, exp − j L 



Hence, DEF matrix should have the dimension L × L. Because MAS has N columns, and a number of MAS and DEF columns should be equal, the lines of MAS matrix and CAS pulses need to be prolonged, for example, owing to repeat of each samples by L/N = ( p−1)q +1 times or to use zero padding. Then the discrete D-transform-Fourier (DT-D-FK ) has a form:

T T T G K N ,1 = D N ,L E1L E2L . . . Ek L . . . E K L S 1,L = D N ,L E K L ,L S 1,L ,

(2.19)

2 Polyphase Radar Signals with Zero Autocorrelation Zone …

17

where 1,L is the sample vector of the prolonged  input signal, Ek L =  S 0(k−1) W 1(k−1) . . . W (n−1)(k−1) . . . W (L−1)(k−1) , W = exp(− j 2π/L) is a diag W diagonal matrix with elements from the k-th line of DEF matrix, which is included in the structure of the block matrix E K L ,L , D N ,L is MAS with prolonged lines. Taking Eq. 2.5 into consideration, we obtain from Eq. 2.19 the discrete VilenkinChrestenson-Fourier transform (DTVC-FK ): T T = H N ,L E K L ,L d L S 1,L , G K N ,1 = H N ,L d L E K L ,L S 1,L

(2.20)

 1  is where H N ,L is the VC-Kronecker matrix with prolonged lines, d L = diag D 1,L the diagonal matrix with elements from the first lines of D N ,L matrix. From [19], we know that column repeating of VC-Kronecker matrix with dimension N × N , N = p k , pl times is equivalent to line decimation of VC-Kronecker matrix with dimension L × L, where L = p k+l to the rectangular matrix with dimension N × L. In other words, in Eq. 2.20, H N ,L matrix can be replaced by VCKronecker matrix with dimension L × L, we can factorize it, and necessary values of the signal spectrum can be obtained from the known decimated line numbers. Thus, from Eq. 2.20 we have the expression for FTVC-FK : T T = Ck+l L Ck+l−1L . . . C jL . . . C1L E K L ,L d L S 1,L , (2.21) G K L ,1 = H L E K L ,L d L S 1,L

where C jL is the weakly-filled matrix from Good factorization algorithm (Eq. 2.18), j = 1, 2, . . . , k + l. At p = 2, FTVC-FK transforms into the fast Walsh-Fourier (FTW-FK ) and L = N (q + 1) [8]. To achieve the maximal FFT effectiveness, DEF matrix dimension should be equal to the power of 2. For this, we introduce the quantity  l = log2 (q + 1) , where x is the operation of the number x rounding to the larger value. Then L = 2l N = 2k+l . The rectangular matrix in Eq. 2.20 has a view:  N ,L    H N ,L = h i,(m−1)/2l +1 

i=1,m=1

,

 N where h i,n 1 = H N is the Hadamard matrix with dimension N × N , x is the operation of integer part extraction of the number x, which is obtained from Hadamard matrix H L with dimension L × L in Eq. 2.21 be means of its lines decimation. The diagonal matrix in Eqs. 2.20 and 2.21 is   d L = diag d1,1 d1,1 . . . d1,(m−1)/2l +1 . . . d1,N d1,N , m = 1, 2, . . . , L ,  N where d1,n 1 = D11,N . The processor FTW-FK has NK outputs (decimated lines). First N outputs represent the result of multiplication of the pulse characteristics matrix (MAS) by the processor input signal samples in the first frequency channel, the second N outputs—in the second frequency channel and so on, the last N outputs—in the K-th frequency channel. The switching block in each frequency channel performs

18

R. N. Ipanov and S. M. Smolskiy

the connection of two from its N inputs with two outputs in accordance with the lines number, in which the paired or adjacent sequences of D-code are located. In each channel adders, the summation of ACF samples of MAS pulses is performed owing to the samples delay of the one from ACF in the shift register by the repetition period of pulses qL. According to the number of the threshold device (TD in Fig. 2.6) (k = 1, 2, . . . , K ), in which the threshold is exceeded, Doppler frequency shift (F Dk = (k − 1)F) is determined. 

2.5 Conclusions The method of polyphase radar signal with ZACZ is offered in this chapter. At that, this signal represents the train of p PCSK pulses encoded by p-ary complementary sequences and is called the coherent additional signal. ZACZ takes place only at complete matching of the filter with echoes in Doppler frequency. At mismatch in frequency, the level of ACF main peak decreases and side lobes appear in ZACZ. The multi-channel compression device of this signal is studied. It is shown that the method of D-code formation allows utilization of algorithms of the fast transform for signal compression in the matched filter. Acknowledgements Investigation is performed at financing from the Russian Scientific Fund grant (the project No 17–19–01616).

References 1. Wu, H., Delisle, G.Y.: Precision tracking algorithms for ISAR imaging. IEEE Trans. Aerosp. Electron. Syst. 32(1), 243–254 (1996) 2. Wehner, D.R.: High Resolution Radar. Artech House, Norwood (1994) 3. Akbaripour, A., Bastani, M.H.: Range sidelobe reduction filter design for binary coded pulse compression system. IEEE Trans. Aerosp. Electron. Syst. 48(1), 348–359 (2012) 4. Mozeson, E., Levanon, N.: Removing autocorrelation sidelobes by overlaying orthogonal coding on any train of identical pulses. IEEE Trans. Aerosp. Electron. Syst. 39(2), 583–603 (2003) 5. Sivaswamy, R.: Digital and analog subcomplementary sequences for pulse compression. IEEE Trans. Aerosp. Electron. Syst. AES 14(2), 343–350 (1978) 6. Levanon, N., Mozeson, E.: Radar Signals. Wiley, Hoboken (2004) 7. Chebanov, D., Lu, G.: Removing autocorrelation sidelobes of phase-coded waveforms. In: 2010 IEEE Radar Conference, Washington, May 2010, pp. 1428–1433. IEEE, New York (2010) 8. Ipanov, R.N., Baskakov, A.I., Olyunin, N., Ka, M.-H.: Radar signals with ZACZ based on pairs of D-code sequences and their compression algorithm. IEEE Signal Proc. Lett. 25(10), 1560–1564 (2018) 9. Wang, H., Diao, M., Gao, L.: Low probability of intercept radar waveform recognition based on dictionary leaming. In: 2018 10th International Conference on Wireless Communications and Signal Processing (WCSP), Hangzhou, October 2018, pp. 1–6. IEEE, New York (2018)

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10. Zhang, M., Liu, L., Diao, M.: LPI radar waveform recognition based on time-frequency distribution. Sensors 16(10), 1682 (2016) 11. Carlson, E.J.: Low probability of intercept (LPI) techniques and implementations for radar systems. In: Proceedings of the 1988 IEEE National Radar Conference, Ann Arbor, April 1988, pp. 56–60. IEEE, New York (1988) 12. Durai, R.R., Suehiro, N., Han, C.: Complete complementary sequences of different length. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E90-A(7), 1428–1431 (2007) 13. Welti, G.: Quaternary codes for pulsed radar. IRE Trans. Inf. Theor. 6(3), 400–408 (1960) 14. Ipanov, R.N.: Polyphase coherent complemented signals. J. Radio Electron. http://jre.cplire. ru/jre/jan17/14/abstract_e.html (2017). Accessed 23 Jan 2017 15. Chrestenson, H.F.: A class of generalized Walsh functions. Pacific J. Math. 5, 17–31 (1955) 16. Helm, H.A.: Group codes Walsh functions. IEEE Trans. Electromagn. Compat. EMC 13(3), 78–83 (1971) 17. Ipanov, R.N.: Pulsed phase-shift keyed signals with zero autocorrelation zone. J. Commun. Techn. Electron. 63(8), 895–901 (2018) 18. Ipanov R.N.: The Formation Method of the Ensemble Variety of p-ary D-codes. RU Patent 2,670,773, 25 October 2018 19. Zhou, M., Shi, X., Liu, Z.: Chrestenson transform and its relations with Fourier transform. In: 2015 Third International Conference on Robot, Vision and Signal Processing (RVSP), Kaohsiung, November 2015, pp. 212–215. IEEE, New York (2016) 20. Ipanov R.N.: Device of Digital Processing of Polyphase Additional Phase-Codes-Hift Keyed Signals. RU Patent 2,647,632, 16 March 2018

Chapter 3

Output Power Amplifier Effects on Harmonic and Amplitude Modulated Signals Distortions Yuri A. Bryukhanov and Kirill S. Krasavin

Abstract This chapter considers the nonlinear distortions of harmonic signal and Amplitude Modulated (AM) signal on the passage through single-ended and pushpull nonlinear power amplifiers with memory and without it. The amplifier characteristic is preset by the cubic polynomial. We consider two operating modes of amplifier for harmonic signal: cutoff mode and non-cutoff mode. Demodulation of AM signal is performed using a coherent quadrature amplitude detector. The distortion estimate based on Total Harmonic Distortion (THD). Dependence of THD versus cutoff angle for harmonic signal and THD versus amplitude sensitivity m for AM signal are calculated. Keywords Amplifier · Power · Nonlinearity · Memoryless · Memory · Single-ended · Push-pull · Distortions · Amplitude modulation

3.1 Introduction Nonlinear radio transmitter effects usually occur in the output power amplifier, which is connected to the transmitting antenna. This amplifier should have a high efficiency, thus, most of electronic device characteristic ic (vBE ) should be used. In this case, the input signal waveform is different from the output signal waveform that is how the nonlinear distortions are appeared. Nonlinear distortions of output power amplifiers are investigated in [1–7]. The goal is to analyze effects of nonlinearity on the unmodulated harmonic and modulating signal of AM signal distortions for single-ended and push-pull output power amplifiers. It’s assumed that the characteristic ic (vBE ) of transistor amplifier normalized by the transfer characteristic steepness S is defined as [7]: Y. A. Bryukhanov (B) · K. S. Krasavin Demidov Yaroslavl State University, 14, Sovetskaya Str., Yaroslavl 150003, Russian Federation e-mail: [email protected] K. S. Krasavin e-mail: [email protected] © Springer Nature Switzerland AG 2020 M. Favorskaya and L. C. Jain (eds.), Advances in Signal Processing, Intelligent Systems Reference Library 184, https://doi.org/10.1007/978-3-030-40312-6_3

21

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Y. A. Bryukhanov and K. S. Krasavin

 i c (v B E ) = i c (u) =

u − 13 u 3 if 0 ≤ u ≤ 1 , 0 if − 1 ≤ u < 0

(3.1)

where u = U0 + u s , U0 is the offset voltage, u s is the input signal. The distortions are estimated based on THD [8]:   −1  K     C˙ k  × 100%, ˙ T H D = Ck k=2

(3.2)

  where C˙ k  are the spectral components amplitudes of output amplifier signal u out (t). Under the input unmodulated signal, the input signal is defined as u s = U cos(ωt + ϕ) = U cos ξ

(3.3)

  and C˙ k  are the harmonics of frequency ω. If the input amplifier signal is the amplitude modulated signal, then u s = U (1 + m cos t) cos(ωc t + ϕ) = U (1 + m cos γ) cos ξ,

(3.4)

is the where m = U U   amplitude sensitivity,  is the frequency of modulating signal, then in Eq. 3.2 C˙ k  are the harmonic amplitudes of frequency . In the case of without cutoff, the collector current is defined by direct substitution of the input signal expression in Eq. 3.1. In cutoff mode, the function i c (u) is represented by a Fourier series of cos ξ [8] because of its symmetricity with respect to ξ under the input harmonic signal (Eq. 3.3): i c (ξ) = I0 +

K 

Ik cos kξ,

(3.5)

k=1

where 1 I0 = 2π

π −π

1 i c (ξ)dξ Ik = π

π i c (ξ) cos kξ dξ,

(3.6)

−π

and for input AM signal, the function i c (u) is represented by a Fourier series of cos γ because of its symmetricity with respect to γ according to Eqs. 3.5–3.6 with ξ replaced by γ in them. The modulating signal is recovered from the output voltage of the amplifier using a coherent quadrature detector to estimate the distortion [9]. The output voltage of the detector is expressed by: U (t) = u outc (t) cos ξ + u outs (t) sin ξ − U,

(3.7)

where u outc (t) is the output amplifier voltage, u outs (t) is the u outc (t) shifted by 90°.

3 Output Power Amplifier Effects on Harmonic and Amplitude …

23

The function U (t) is represented by a Fourier series of cos γ to find the spectral composition of the modulated signal: U (γ) = C0 +

K 

Ck cos kγ,

(3.8)

k=1

where 1 C0 = 2π

π −π

1 U (γ)dγ Ck = π

π U (γ) cos kγ dγ.

(3.9)

−π

We consider the processes in the single-ended and push-pull output power amplifiers with memory and memoryless. The chapter is organized as follows. Section 3.2 presents a discussion of singleended amplifier, while push-pull amplifier is considered in Sect. 3.3. Conclusions are formulated in Sect. 3.4.

3.2 Single-Ended Amplifier Hereinafter, a mathematical theory of memoryless amplifier is given in Sect. 3.2.1. Section 3.2.2 presents a mathematical theory of amplifier with memory.

3.2.1 Memoryless Amplifier In the case of memoryless amplifier, the transistor load is resistive R L . We first consider an operation mode of transistor with and without cutoff i c under unmodulated harmonic signal (Eq. 3.3). In operation mode without cutoff, the offset voltage value U 0 is of 0.5. Since the value of u B E is in the range [0, 1], the value of input signal amplitude is in the range [0, 0.5]. Wherein, the normalized collector current is defined by direct substitution of Eq. 3.3 in Eq. 3.1: i c (ξ) =

3 

ak cos kξ,

k=0

where     1 1 1 a1 = U 1 − U02 − U 2 a0 = U0 1 − U02 − U 2 3 2 4

(3.10)

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Y. A. Bryukhanov and K. S. Krasavin

1 1 a2 = − U 0 U 2 a3 = − U 3 . 2 12

(3.11)

The harmonic amplitudes in Eq. 3.2 are given by Ck = ak R L . The calculated results of THD are presented in Table 3.1. The dependence of THD (percent) versus the input signal amplitude U is shown in Fig. 3.1. The calculations show that THD comes to 17.63% at U = 0.5. In operation mode with cutoff, the offset voltage value is less than 0.5 and the input signal amplitude is given by U = 1 − U 0 . Since the value of is in the range [0, 1], the value of input signal amplitude is in the range [0, 0.5]. The cutoff angle is θ = arccos − UU0 [8]. The function i c (ξ) is represented by a Fourier series Eq. 3.5 of Eq. 3.6. The harmonic amplitudes in Eq. 3.2 are given by Ck = Ik R L . The calculated results of THD are presented in Table 3.2. The calculations show that THD equals 39.57 and 18.59% as θ ∈ {90°, 120°}, respectively. The decrease of distortions as θ increases from 90° to 140° is due to a decrease of the parasitic harmonics level. And minor THD increases as θ increases from 140° to 180° is due to a relative growth of the 2nd harmonic. In operation mode without cutoff under AM signal, the normalized collector current is defined by direct substitution of Eq. 3.4 in Eq. 3.1: i K (γ, ξ) = b0 + b1 cos γ + b2 cos 2γ + b3 cos ξ + b4 cos(ξ ± γ) + b5 cos(ξ ± 2γ) + b6 cos(ξ ± 3γ) + b7 cos 2ξ + b8 cos(2ξ ± γ) + b9 cos(2ξ ± 2γ) + b10 cos 3ξ + b11 cos(3ξ ± γ) + b12 cos(3ξ ± 2γ) + b13 cos(3ξ ± 3γ), (3.12) where 

2 b1 = −U b0 = U0 1 − 13 U02 − 21 U 2 1 + 21 m 2 0U m

 b3 = U 1 − U02 − 41 U 2 1 + 23 m 2 b2 = − 41 U0U 2 m 2

3 U 3 m 2 b4 = 21 U m 1 − U02 − 43 U 2 1 + 41 m 2 b5 = − 16

1 1 3 3 . b6 = − 32 U m b7 = − 2 U0 U 2 1 + 21 m 2 1 2 2 b = − U U m b8 = − 21 U0 U 2 m 9 8 0



1 b11 = − 18 U 3 m 1 + 41 m 2 b10 = − 12 U 3 1 + 23 m 2 1 1 U 3m2 b13 = − 96 U 3m3 b12 = − 16 (3.13) 1 It’s assumed that U 0 = 0.5, U = 2(1+m) . The harmonic amplitudes of the amplifier output are given by Uk = bk R L . The amplifier output spectrum at m = 1 is shown in Fig. 3.2. In coherent quadrature amplitude detector, the frequency of each spectral component from the amplifier output is reduced by the value of ωn according to Eq. 3.7. The detector output spectrum at m = 1 (excluding DC) is shown in Fig. 3.3.

0.1

3.34

U

THD, %

5.01

0.15 6.70

0.2 8.41

0.25

Table 3.1 Calculated value of THD for the unmodulated harmonic signal 10.14

0.3 11.92

0.35

13.74

0.4

15.64

0.45

17.63

0.5

3 Output Power Amplifier Effects on Harmonic and Amplitude … 25

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Y. A. Bryukhanov and K. S. Krasavin

Fig. 3.1 THD versus U without cutoff

In the spectrum of the demodulated signal, in addition to the first and second harmonics of the modulated signal there are high-frequency components. The calculated results of THD are presented in Table 3.3. The dependence of THD (percent) versus the amplitude sensitivity m is shown in Fig. 3.4. The calculations show that THD equals 49.51% and 32.74% as m ∈ {0.5, 1}, respectively. The decrease of distortions as m increases is due to the relative increase of the modulating signal amplitude compared with the parasitic harmonics level (in proportion to m). In operation mode with cutoff, the collector current is expressed by Eq. 3.1 and output voltage of the amplifier is Uout = i k (t)R L . The calculations show that THD of the modulating signal equals 127.90% and 95.38% at θ = 90° as m ∈ {0.5, 1}, respectively. This is due to the high even harmonics level of the modulated signal at the detector output for the given θ.

3.2.2 Amplifier with Memory In the case of amplifier with memory, the transistor load is parallel RLC circuit. The complex impedance of which is expressed as [10]: Z ( jω) = Re

2α(2α + jω) , (α + jω)2 + ω2d

90

39.57

θ, °

THD, %

30.56

100 23.65

110 18.59

120

Table 3.2 Calculated value of THD for the unmodulated harmonic signal 15.84

130 15.30

140 16.30

150

17.54

160

18.28

170

18.43

180

3 Output Power Amplifier Effects on Harmonic and Amplitude … 27

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Y. A. Bryukhanov and K. S. Krasavin

Fig. 3.2 Output spectrum of the amplifier

Fig. 3.3 Output spectrum of the detector

where Re is the equivalent resistance at resonance, α is the attenuation of RLC circuit, ωd = ω20 − α2 is the damped natural frequency, ω0 is the resonance frequency. We consider the operation mode of the transistor with cutoff and without it under unmodulated harmonic signal (Eq. 3.3). The value of U 0 and U is given as in the memoryless amplifier.

0.1

203.60

m

THD, %

102.00

0.2 71.71

0.3 57.65

0.4

Table 3.3 Calculated value of THD for the modulating signal 49.51

0.5 44.09

0.6 40.17

0.7

37.15

0.8

34.73

0.9

32.74

1.0

3 Output Power Amplifier Effects on Harmonic and Amplitude … 29

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Y. A. Bryukhanov and K. S. Krasavin

Fig. 3.4 THD of modulating signal versus m

In operation mode without cutoff, the collector current i c (ξ) is defined by Eqs. 3.10–3.11. The harmonic amplitudes in Eq. 3.2 are given by Ck = ak Z (ωk ), where Z (ωk ) is the value of amplifier frequency response normalized by k0 = S · Re . The calculated results of THD are presented in Table 3.4. The dependence of THD (percent) versus the input signal amplitude U for Q ∈ {5, 10} is shown in Fig. 3.5, where Q = ω2α0 is the quality factor. The calculations show that THD equals 2.08 and 1.16% at U = 0.5 as Q ∈ {5, 10}, respectively. In operation mode with cutoff, the function i c (ξ) is defined by Eqs. 3.5–3.6. The harmonic amplitudes in Eq. 3.2 are given by Ck = Ik Z (ωk ). The calculated results of THD are presented in Table 3.5. The dependence of THD (percent) versus the cutoff angle θ (degrees) for Q ∈ {5, 10} is shown in Fig. 3.6. The calculations show that THD equals 4.6 and 2.4% at θ = 90° as Q ∈ {5, 10}, respectively. In operation mode without cutoff under AM signal, the normalized collector current is defined by Eqs. 3.12–3.13. The collector current spectrum is similar to that shown in Fig. 3.1. The harmonic amplitudes of the amplifier output are given by U˙ k = bk Z ( jωk ). The harmonic amplitudes C k of the demodulated signal U (t) and 1 THD of the modulating signal are calculated by Eqs. 3.8–3.9 at U 0 = 0.5, U = 2(1+m) . The calculated results of THD are presented in Table 3.6. The dependence of THD (percent) versus the amplitude sensitivity m for Q ∈ {5, 10} is shown in Fig. 3.7. The calculations show that THD comes to 6.66% and 5.32 at m = 1 as Q ∈ {5, 10}, respectively.

0.43

0.22

10

0.1

U

5

Q

0.33

0.65

0.15 0.44

0.87

0.2 0.56

1.09

0.25

Table 3.4 Calculated value of THD for the unmodulated harmonic signal

0.67

1.32

0.3 0.79

1.55

0.35

0.91

1.78

0.4

1.03

2.02

0.45

1.16

2.08

0.5

3 Output Power Amplifier Effects on Harmonic and Amplitude … 31

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Y. A. Bryukhanov and K. S. Krasavin

Fig. 3.5 THD versus U without cutoff

In operation mode with cutoff, the collector current is defined by Eq. 3.1. It’s represented by a Fourier series Eq. 3.5 of Eq. 3.6 with ξ replaced by γ in them. The harmonic amplitudes of the amplifier output are given by U˙ k = Ik Z ( jk ). The calculations show that THD of modulating signal U (t) equals 19.90 and 15.99% at m = 1 as Q ∈ {5, 10}, respectively.

3.3 Push-Pull Amplifier A push-pull amplifier is used to provide high efficiency and to improve the output power of transistor. The transistor is operated with the collector current i c cutoff. The simplified circuit diagram of amplifier is shown in Fig. 3.8. Hereinafter, Sect. 3.3.1 provides a description about memoryless amplifier. Amplifier with memory is considered in Sect. 3.3.2.

3.3.1 Memoryless Amplifier In the case of unmodulated harmonic signal us (Eq. 3.3) with cutoff, the collector current of each transistor i ci is defined by Eq. 3.1. The output voltage of amplifier is defined by Eq. 3.14, where u outi (t) = i ci (t)R L .

4.61

2.35

10

90

θ, °

5

Q

1.68

3.40

100

1.11

2.19

110 0.71

1.73

120 0.61

1.20

130

Table 3.5 Calculated value of THD for the unmodulated harmonic signal

0.78

1.53

140 0.99

1.94

150

1.13

2.22

160

1.20

2.36

170

1.21

2.38

180

3 Output Power Amplifier Effects on Harmonic and Amplitude … 33

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Y. A. Bryukhanov and K. S. Krasavin

Fig. 3.6 THD versus θ

u out (t) = u out1 (t) − u out2 (t)

(3.14)

The harmonic amplitudes of Eq. 3.2 are defined by Eqs. 3.5–3.6 with i c , I k replaced by u out (t), and C k . The calculated results of THD are presented in Table 3.7. The dependence of THD (percent) versus the cutoff angle θ (degrees) is shown in Fig. 3.9. The calculations show that THD equals 11.11% at θ = 90°, which is in 3.56 times less than for a single-ended amplifier with the same cutoff angle. This is due to the even-order harmonic at the output of push-pull amplifier are cancelled [1]. The relative level of odd-order harmonics takes the greatest value at θ = 120° and gradually decreases with increasing cutoff angle from the above value. In operation mode with cutoff under AM signal (Eq. 3.4), the normalized collector current of each transistor is defined by Eq. 3.1. The output amplifier voltage is given by Eq. 3.14. The demodulated signal U (t) of Eq. 3.7 is represented by a Fourier series Eq. 3.8 of Eq. 3.9. The calculations of THD are performed by MATLAB software with θ = 90°, U 0 1 = 0, U = 1+m according to Eq. 3.2. The calculated results of THD are presented in Table 3.8. The dependence of THD (percent) versus the amplitude sensitivity m is shown in Fig. 3.10. The calculations show that THD equals 24.35 and 16.16% at m ∈ {0.5, 1}, which is in 5.25 and 5.9 times less than for single-ended amplifier at cutoff angle θ is 90°.

36.77

29.10

10

0.1

m

5

Q

14.85

18.76

0.2

10.66

13.44

0.3 8.73

11.00

0.4

Table 3.6 Calculated value of THD for the modulating signal

7.62

9.58

0.5 6.88

8.64

0.6 6.35

7.96

0.7

5.93

7.43

0.8

5.60

7.00

0.9

5.32

6.66

1.0

3 Output Power Amplifier Effects on Harmonic and Amplitude … 35

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Y. A. Bryukhanov and K. S. Krasavin

Fig. 3.7 THD of modulating signal versus m

Fig. 3.8 Simplified circuit diagram of push-pull amplifier

3.3.2 Amplifier with Memory Consider the effects of unmodulated harmonic and amplitude-modulated signals then the transistor are operated with collector current cutoff. The collector current of each transistor i ci under harmonic signal is determined as in the memoryless amplifier and the harmonic amplitudes of output amplifier voltage Eq. 3.14 is defined by Eq. 3.15, where Ck = Iki Z (ωk ). C k = C k1 − C k2

(3.15)

90

11.11

θ, °

THD, %

17.55

100 19.33

110 17.69

120

Table 3.7 Calculated value of THD for the unmodulated harmonic signal 14.49

130 10.69

140 7.18

150

4.52

160

3.24

170

3.03

180

3 Output Power Amplifier Effects on Harmonic and Amplitude … 37

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Y. A. Bryukhanov and K. S. Krasavin

Fig. 3.9 THD versus θ in push-pull amplifier

The calculated results of THD are presented in Table 3.9. The dependence of THD (percent) versus the cutoff angle θ (degrees) is shown in Fig. 3.11. The calculations show that THD equals 5.29 and 2.92% at θ = 90° and Q ∈ {5, 10}, which is in 2.1 and 3.8 times less than for memoryless push-pull amplifier. It’s also is in 1.15 and 1.24 times more than for single-ended amplifier with memory at cutoff angle θ is 90°. In operation mode with cutoff under AM signal (Eq. 3.4), the normalized collector current of each transistor is represented by a Fourier series Eq. 3.5 of Eq. 3.6 with ξ replaced by γ in them. The complex amplitudes of spectral components in the output voltage of ith transistor is defined by U˙ ki = Iki Z ( jk ). Therefore, the complex amplitudes of the output amplifier voltage are from Eq. 3.15 with C replaced by U˙ . The harmonic amplitudes of the demodulated signal U (t) and THD of the modulating signal are calculated by Eqs. 3.8–3.9 at U 0 = 0 (this corresponds to a 1 . The calculated results of THD are presented in cutoff angle θ of 90), U = 1+m Table 3.10. The dependence of THD (percent) versus the amplitude sensitivity m for Q ∈ {5, 10} is shown in Fig. 3.12. The calculations show that THD equals 8.21 and 7.04% at m = 1 and Q ∈ {5, 10}, which is in 2.4 and 2.27 times less than for a single-ended amplifier with the same cutoff angle.

0.1

188.80

m

THD, %

70.71

0.2 41.32

0.3 29.89

0.4

Table 3.8 Calculated value of THD for the modulating signal 24.35

0.5 21.23

0.6 19.27

0.7

17.92

0.8

16.93

0.9

16.16

1.0

3 Output Power Amplifier Effects on Harmonic and Amplitude … 39

40

Y. A. Bryukhanov and K. S. Krasavin

Fig. 3.10 THD of modulating signal versus m in push-pull amplifier

3.4 Conclusions The analysis of nonlinear distortion in a single-ended memoryless amplifier showed that the distortions are proportional to the input voltage amplitude and THD comes to 17.63% with voltage value of 0.5 and without cutoff under unmodulated harmonic signal. In operation mode with cutoff angle of 120°, the distortions increases to 18.59% and THD comes to 39.57 at θ = 90°. The distortions of modulating signal decreases as the amplitude sensitivity m of AM signal increases (without cutoff) and THD is 32.74% at m = 1. In operation mode with cutoff angle of 90°, THD is 95.38%. In a single-ended amplifier with memory as Q = 5 and without cutoff, the distortions of harmonic signal decreases to 2.08%. The distortions of modulating signal decreases to 6.66% at the same conditions and m = 1. THD values are equaled to 4.61% and 19.9%. In a push-pull memoryless amplifier, the distortions of harmonic signal decrease to 1.59 times. This is due to the absence of an even-order harmonic signal at the amplifier output. The distortions of modulating signal (m = 1) decrease to 1.97 times than for a single-ended amplifier in the mode without the collector current cutoff and 5.74 times than for a single-ended amplifier with same cutoff angle. In a push-pull amplifier with memory at θ = 90° as Q = 5, THD of harmonic signal decreases to 2.1 times than for a single-ended memoryless amplifier, and THD of modulating signal (m = 1) decreases to 2.02 times. The results of this chapter can be used for the design of signal processing systems.

5.29

2.92

10

90

θ, °

5

Q

4.45

8.08

100

4.94

8.97

110 4.61

8.36

120 3.78

6.86

130

Table 3.9 Calculated value of THD for the unmodulated harmonic signal

2.77

5.02

140 1.83

3.31

150

1.16

2.10

160

0.85

1.54

170

0.80

1.44

180

3 Output Power Amplifier Effects on Harmonic and Amplitude … 41

19.98

15.88

10

0.1

m

5

Q

8.39

10.17

0.2

7.60

9.00

0.3 7.53

8.85

0.4

Table 3.10 Calculated value of THD for the modulating signal

7.51

8.79

0.5 7.46

8.82

0.6 7.38

8.62

0.7

7.28

8.50

0.8

7.16

8.36

0.9

7.04

8.21

1.0

42 Y. A. Bryukhanov and K. S. Krasavin

3 Output Power Amplifier Effects on Harmonic and Amplitude …

Fig. 3.11 THD versus θ in push-pull amplifier

Fig. 3.12 THD of modulating signal versus m in push-pull amplifier

43

44

Y. A. Bryukhanov and K. S. Krasavin

References 1. Mamonkin, I.G.: Amplifying Systems, 2nd edn. Svyaz, Moscow (in Russian) (1977) 2. Liang, C.-P., Jong, J.-H., Stark, W.E., East, J.R.: Nonlinear amplifier effects in communications systems. IEEE Trans. Microwave Theory Techniq. 47(8), 1461–1466 (1999) 3. Jantunen, P., Gamez, G., Laakso, T.: Measurements and modelling of nonlinear power amplifiers. 6th Nordic Signal Processing Symposium, Espoo, Finland, pp. 328–331 (2004) 4. Lin, F.-L., Chen, S.-F., Chuang, H.-R.: Effects of RF-circuit nonlinear distortion on digitally modulated signals in wireless communication. Microwave J. 43(9), 126–138 (2000) 5. Smirnov, A.V.: Analysis of AM-PM distortion phenomena in high-efficiency PA applications. Elektrosvya 4, 61–64 (in Russian) (2016) 6. Aschbacher, E., Rupp, M.: Modelling and identification of a nonlinear power-amplifier with memory for nonlinear digital adaptive pre-distortion. In: 4th IEEE Workshop on Signal Processing Advances in Wireless Communications, Rome, Italy, pp. 658–662 (2003) 7. Kharchenko, V., Grekhov, A., Ali, I.: Influence of nonlinearity on aviation satellite communication channel parameters. Proc. Natl. Aviat. Univ. 4(65), 12–21 (2015) 8. Baskakov, S.I.: Radiotechnical Circuits and Signals, 4th edn. Moscow, LENAND (in Russian) (2016) 9. Fomin, N.N., Buga, N.N., Glovin, O.V.: Radiotechnical Systems, 3rd edn. Goryachaya liniya, Telekom, Moscow (in Russian) (2007) 10. Zolotarev, I.D.: Non-stationary Processes in Resonant Amplifiers of Phase-Pulse Measuring Systems. Nauka, Novosibirsk (in Russian) (1969)

Chapter 4

Distortion Types Separation of QAM-16 Signal Andrei E. Kiselnikov, Mikhail A. Dubov and Andrei L. Priorov

Abstract The spectral efficiency and signal quality are the main requirements for modern communication devices. This chapter describes a method of identification of QAM-16 signal distortion on the transmitter out. This method is based on the error vector magnitude analysis. The proposed method is compatible with all up-todate measurement equipment. In addition, the features of modeling different signal distortion examined in this chapter. The method of reconstruction of real power amplifier devices characteristics is implemented in MATLAB Simulink. It allows to estimate the influence of non-linear amplifier distortion. Keywords Error vector magnitude · Quadrature modulation · Nonlinear-distortion · Distortion identification

4.1 Introduction There are many methods for assessing a quality of the transmitted signal and noise immunity of the communication system as a whole but the final quality criteria in digital information transmission systems are Bit Error Ratio (BER) or Symbol Error Ratio (SER). Consequently, for assessing a quality of the receiving device, the most effective technique is such technique, which is based on transmitting reference sequences and counts a number of errors providing that a signal of a given level is fed to the input of the receiver. In the case of assessing the signal quality at the transmitter output, the use of an error-based technique is irrational: since even if the signal constellation is distorted, A. E. Kiselnikov · M. A. Dubov · A. L. Priorov (B) P.G. Demidov, Yaroslavl State University, 14, Sovetskaya st., Yaroslavl 150003, Russian Federation e-mail: [email protected] A. E. Kiselnikov e-mail: [email protected] M. A. Dubov e-mail: [email protected] © Springer Nature Switzerland AG 2020 M. Favorskaya and L. C. Jain (eds.), Advances in Signal Processing, Intelligent Systems Reference Library 184, https://doi.org/10.1007/978-3-030-40312-6_4

45

46

A. E. Kiselnikov et al. EVM/MER measurement

Fig. 4.1 EVM measurement diagram

Test data

Transmitter

Load

Measurer EVM/MER

the ratio of the bit energy to the spectral density of the noise power remains high. For assessing the quadrature signals’ quality with digital manipulation, the most common and effective methods are related to measuring Error Vector Magnitude (EVM) and coefficient of Modulation Error Ratio (MER). The schematic representation of the measurement technique is presented in Fig. 4.1. In their pure form, these methods give information only about the degree of signal distortion and, accordingly, about the noise immunity of the transmission system, but they do not provide information about the cause of signal distortion, i.e. they do not identify it. This chapter proposes a modified method of assessing the signal quality with QAM-16 modulation, which allows not only to estimate the noise immunity of the information transmission system but also to determine the type of distortion. Information about the type of signal distortion can significantly reduce the time to adjust the radio device during its production. The proposed method can be used as the basis for the system of the automatic measurement and adjustment of device parameters if the hardware platform supports the MATLAB software package. The same approaches for lower-order modulation are well described in [1]. It should be noted that another area of application may be automated monitoring of radio links. The remainder of the chapter is organized as follows. Section 4.2 gives an overview of the types of distorting effects. Section 4.3 presents the most common quality assessment metrics of the digital signals. Section 4.4 provides a detailed description of the proposed method and results of the experimental studies. Section 4.5 concludes the chapter.

4.2 Types of Distorting Effects In this section, we consider various types of distorting effects for QAM-16 signal, their influence on noise immunity, and modeling features in MATLAB Simulink software package. First of all, we will consider the nonlinear distortion of the signal by the amplifier shown in Fig. 4.2, since QAM signals have a high peak factor and a large scatter of symbols in energy, as a result of which the linearity of the amplification characteristic is one of the most important parameters in the design of the transmitting part of the device [2].

4 Distortion Types Separation of QAM-16 Signal Fig. 4.2 Nonlinear QAM signal distortion

47 Q 3

1

12

8

13

9

1

3

I

In a computer simulation of this situation, the most appropriate solution is to imitate the amplifying characteristic of a real amplifier in the model. Let’s set a formal task: it is necessary to create a model of the very common amplifying module Polyfet RF Devices MLCQ-02 [3] in MATLAB Simulink using approximation by a cubic polynomial. Figure 4.3 shows its amplification characteristic for the operating frequency of 512 MHz. In this simulation environment with a block amplifier, our task is to adjust it so that it corresponds to the behavior of the real amplifier as much as possible under the condition that the approximation is used by a cubic polynomial.

Fig. 4.3 Polyfet MLCQ-02 characteristics

48

A. E. Kiselnikov et al.

At the first stage, it is necessary to form an array of control points using the technical documentation for the amplifying module. The following values should be recorded at these units: the input power in Watts, the output power in Watts, and the gain in dB. These three arrays will be the source data for building the amplifier model. At the second stage, it is necessary to consider how the approximation of the amplification characteristic takes place using a cubic polynomial in the modeling environment. It is worth noting that the cubic polynomial approximation does not take into account the phase distortion of a signal in the amplifying tract but the documentation on the amplification module does not provide information on the phase distortion. Therefore, phase distortion is not a significant omission when creating its model. First, the normalization factor f is calculated, which is used to normalize the input signal. The basis for calculating the normalization factor is the Third-order Input Inception Point (IIP3) [4], which sets the normalization scale in dBm:  f =

3 = I I P3[W ]



3 . 10(I I P3[d Bm]−30)/10

(4.1)

We normalize and truncate the array of source data. All values greater than 1 is truncated (Eq. 4.2), where PSC AL E D is the normalized power, PI N is the input power. PSC AL E D = PI N ∗ f ; (PI N ∗ f > 1) = 1

(4.2)

At the next stage, it is necessary to apply a nonlinear amplitude transform from the array, in which the input power values are stored according to Eq. 4.3. FAM/AM = U −

U3 3

(4.3)

After applying the conversion, it is necessary to bring the values of elements of the array of input power to the original values by dividing on the normalization factor and multiplying on the gain G, which is one of the initial parameters:  PM DL =

PSC AL E D − f

3 PSC AL E D 3

 G

10 10 .

(4.4)

Having received the amplifier characteristic of the amplifier model, it is necessary to evaluate its accuracy and select the optimal one. For evaluation, it is proposed to use the least-squares method. Let us focus on the fact that all the approximation parameters are determined by the chosen value. Therefore, it is necessary to carry out a cycle of calculations, gradually shifting this parameter by a small step. When the value repeatedly exceeds the maximum value from the array of input power values, the sum of the smallest squares will monotonously grow, and the experiment can

4 Distortion Types Separation of QAM-16 Signal

49

be stopped. Figure 4.4 shows the block diagram of the algorithm, with which the calculations were carried out. As a result of the calculations done, all the necessary data were obtained to create an amplifier model that corresponds to the real amplifier module. After this stage, one can proceed directly to the creation of the Simulink model. Figure 4.5 shows the graphs of the amplification characteristic from the documentation for MLCQ-02 and the characteristic that will be implemented in the model as a result of entering the calculated parameters. It should be noted that on the given graph, the power scale is shown on a linear scale, therefore, at first glance, they differ greatly from Fig. 4.3. Start

Source data: Input power values array Output power values array Gain coefficients array

Calculating parameters for current IIP3

Saving the LMS sum of the IIP3

IIP3+step

no

LMS of IIP3>>MAX(Pin)

yes Searching of least squares summ and according IIP3 value

End

Fig. 4.4 Flow chart of gain characteristic approximation process

50

A. E. Kiselnikov et al. Amplification characteristicMLCQ-02

60

Output power, Wt

50 40 30 20 10 0

15 10 Input power, mWt

5

20

25

Fig. 4.5 Amplification characteristic of MLCQ-02 (solid line), amplification characteristic of the device model (dashed line)

Let us give an example of QAM-16 signal distortion with the help of the obtained model using the following initial data: the average power of the signals fed to the amplifier is 5 and 15 mW, the magnitude of the error vector calculated on the frame by the character size will be used as an estimate. Figure 4.6 shows diagrams of QAM-16 signal constellations for various levels of input signal power applied to the model of the amplifier module. Let’s now consider such distortions as the imbalance of quadratures and offset of the signal constellation. In order to obtain their mathematical models, we present an analytical record of QAM-16 signal provided by Eq. 4.5, where S i (t) is the current position of the signal vector, i.e. the symbol of the signal constellation currently transmitted, Ak I , Al Q are its coordinates, f s is the carrier frequency of the signal, T is the symbol period, g0 (t) is the normalization function. Let us introduce into the analytical record the signal of the distortion model—the shift of the signal constellation and the quadrature imbalance. Q

(a)

Q

(b)

3

3

1 -3

-1

1 1

-1

-3

-3

3

I

-1

1

-1

3

I

-3

Fig. 4.6 QAM-16 at the output of the amplifier model with different input power: a 5 mW, b 15 mW

4 Distortion Types Separation of QAM-16 Signal Q

(a)

51 Q

(b) 3

3

-1

1

-1

I

1

1 -3

-3

3

I

-3

-1

1

3

-1

-3

Fig. 4.7 Distorted QAM-16 signal constellations: a offset, b phase unbalance of quadrature

S i (t) = g0 (t)Ak I cos(2π f s t) + jg0 (t)Al Q sin(2π f s t) √ k, l ∈ 1 . . . M;  √  Ak I , Al Q = (2m − 1 − M)  2 ; t ∈ [0, T ] T g0 = 0; t ∈ / [0, T ]

(4.5)

Accordingly, we consider Eqs. 4.6–4.7, where o f s I i o f s Q are the offset along the quadrature axes, ϕ I M B is the phase offset. 

S i (t) = g0 (t)(Ak I + o f s I ) cos(2π f s t) + jg0 (t) Al Q + o f s Q sin(2π f s t)

(4.6)

S i (t) = g0 (t)Ak I cos(2π f s t + ϕ I M B ) + jg0 (t)Al Q sin(2π f s t − ϕ I M B )

(4.7)

Figure 4.7 shows graphic illustrations of distortion. A special block is provided for modeling the distortion data in the specified software package, due to which the process of introducing the distortion data into the data model is not of interest [5].

4.3 Signal Quality Assessment Metrics For digital signals, the natural measures of quality are ratios E b /N0 or E s /N0 , which respectively show the ratio of the energy of a bit E b or symbol E s to the spectral density of the noise power N0 . These metrics are historically the first, as a result of which they do not fully reflect the specifics of quadrature signals but are formally applicable for their evaluation. The most appropriate for quadrature signals is the measure of the deviation of the constellation points from their original positions [6]. The main metrics that are the most common are considered below.

52

A. E. Kiselnikov et al.

Fig. 4.8 Error vector Q

I

MER is the ratio of the average power of a symbol to the average power of an error expressed in dB [7]. Figure 4.8 shows an example of the deviation due to the impact of the noise of the received vector from the signal constellation; here it is the − → − → error vector e, S , S  are the original and the distorted vector, respectively. This metric can be calculated for a frame consisting of N characters or for a single character. We give the analytical records for each case (Eq. 4.8), where Ik and Q k − → are the projections on the in-phase and quadrature axis of the vector S k , and the squared modulus of the error vector can be represented by (Eq. 4.9). N

M E R = 10 lg

2 2 k=1 Ik + Q k n ek |2 k=1 |

 (4.8)

2

2

|ek |2 = Ik − Ik + Q k − Q k

(4.9)

For a single channel character, the averaged metric will take a form represented by Eq. 4.10. L

M E Rk = 10 lg

i=1

L

Iki2 + Q 2ki

i=1



|eki |2

(4.10)

In this chapter, the main criterion of signal quality is associated with MER metric—EVM as a relative modulation error. This is a ratio of the standard deviation of the error vector to the average amplitude of the quadrature signal, expressed in percent [8]. We write its analytical representation in the same way for the frame and average metric for a single symbol using Eqs. 4.11–4.12, respectively.

 N  |ek |2 E V M =  N k=1

2  · 100% 2 k=1 Ik + Q k

(4.11)

4 Distortion Types Separation of QAM-16 Signal

53

 L  |eki |2 E V Mk =  L i=1

2  · 100% 2 i=1 Iki + Q ki

(4.12)

4.4 Distortion Identification at the Transmitter Output As already noted, all the above metrics in their pure form provide only information on the degree of signal distortion and, therefore, on the noise immunity of the communication system, but they do not provide information on the nature of the distortion [9]. Obviously, in the case of calculations on a sufficiently large frame, the same values of EVM can be obtained by introducing various distortions into the signal (Fig. 4.9). Large frame is understood to be a sequence of data, during transmission of which all symbols of the signal constellation are encountered with approximately equal probability, and if, as a result of introducing distortion when transmitting different symbols, the error vector is different, but no one of the vectors is comparable in magnitude with total frame error: |emax |2 

L 

|ei |2 .

(4.13)

i=1

To identify the signal distortion, it is proposed to calculate the average metric for each individual symbol within a large frame or transmit frames consisting of data defining one symbol. We will conduct the following experiment. We will use the technique for modeling the nonlinear distortion of the power amplifier, which was given earlier and create a control script in MATLAB software package. This script will run the Simulink model Q

Q

3

3

1 -3

-1

1 1

-1

-3

3

-1

I

-3

Fig. 4.9 Two possible constellations with the same EVM 26%

1

-1

-3

3

I

54

Symbol

A. E. Kiselnikov et al.

QAM-16

DistorƟon Amplifier

EVM calculaƟon

Results collecƟng

Noise

Fig. 4.10 Scheme of modeling various distortions

Fig. 4.11 Diagram of character-by-character change of EVM depending on the degree of distortion for: a phase unbalance, b non-linear distortion

and determine its parameters: signal power supplied to the amplifier, message transfer rate, etc. Its other function is to transfer data to the input of models, i.e. determine what the frame length will be and on what sequence characters will be calculated metric evaluation of signal quality [10, 11]. Figure 4.10 shows the structural scheme of the described model. Let’s calculate successively the averaged value of the error vector for each symbol of the signal constellation. The length of the frame, for which the calculation is performed, is equal to 4 × 107 bits (107 symbols). The diagram below (Fig. 4.11) shows the simulation results. The vertical axis represents the error vector expressed as a percentage along the frontal axis provided the number of the symbol reflecting its position in the signal constellation. Depending on the graph, the third axis reflects either the quadrature imbalance angle or power of the signal supplied to the power amplifier. The presented diagrams show obvious differences in the behavior of the error vector depending on the type of signal distortion. There is a need to formulate criteria, on the basis of which it is possible to distinguish the distortions. To begin with, we distinguish the characteristic features of the behavior of the error vector for each type of distortion. We define the symbols, for which the magnitude of the error vector will be maximum or minimum, as well as, the characteristic relations between them [12, 13]. In the case of phase imbalance, as shown in Fig. 4.11, the characters with numbers 2, 7, 8, 13 are subjected to the greatest distortion. The numbering of characters is presented in Fig. 4.12. Symbols 3, 6, 9, 12 are slightly less distorted.

4 Distortion Types Separation of QAM-16 Signal Fig. 4.12 Signal constellation QAM-16

55 Q 0

4

12

8

13

9

3

1

5 1

-3

-1

1

3

3

7

15

11

14

10

-1

2

6

-3

I

Fig. 4.13 Chart of the character EVM depending on the degree distortion

In the case of nonlinear distorting effect, the degree of distortion is determined by the energy of the symbol: the symbols 0, 2, 8, 10 and symbols 5, 7, 13, 15 will be most distorted and the smallest distortion, respectively. Note that with nonlinear distortion, symbols that have the same energy are distorted equally regardless of their position in the signal constellation, in contrast to the phase unbalance or amplitude unbalance (Fig. 4.13). When the amplitude imbalance is entered by compressing the quadrature axis, the symbols 4, 12, 6, 14 are maximally distorted. This is because the quadrature component prevails during the formation of the symbol and carries most of its energy. We write rigorous wording of criteria using the language of Boolean algebra (Eqs. 4.14– 4.17), where EVM(S i ) is the error vector magnitude calculated for defined channel symbol S i . N on Linear   max(E V M(S5 ), E V M(S13 ), E V M, E V M(S15 )) = −min(E V M(S5 ), E V M(S13 ), E V M(S7 ), E V M(S15 )) ≤    ⎞ ⎛ E V M(S1 ), E V M(S3 ), E V M(S4 ), E V M(S6 ), max ⎟ ⎜ E V M(S9 ), E V M(S11 ), E V M(S12 ), E V M(S14 ) ⎟   ∩⎜ ⎠ ⎝ E V M(S1 ), E V M(S3 ), E V M(S4 ), E V M(S6 ), ≤ −min E V M(S9 ), E V M(S11 ), E V M(S12 ), E V M(S14 )

56

A. E. Kiselnikov et al.



 max(E V M(S0 ), E V M(S2 ), E V M(S8 ), E V M(S10 )) ∩ −min(E V M(S0 ), E V M(S2 ), E V M(S8 ), E V M(S10 )) ≤  ⎞ ⎛ max(E V M(S5 ), E V M(S13 ), E V M(S7 ), E V M(S15 )) +    ⎠ ∩⎝ E V M(S1 ), E V M(S3 ), E V M(S4 ), E V M(S6 ), ≤ min E V M(S9 ), E V M(S12 ), E V M(S14 )   ⎛ ⎞ E V M(S1 ), E V M(S3 ), E V M(S4 ), E V M(S6 ), max + ⎠ ∩⎝ E V M(S9 ), E V M(S12 ), E V M(S14 ) ≤ min(E V M(S0 ), E V M(S2 ), E V M(S8 ), E V M(S10 ))

(4.14)

The signal is a subject to non-linear distortion if all symbols with the same energy have the same error vector, and, the greater the energy of the symbol, the more distorted it is. A logical expression describing this criterion is given by Eq. 4.14. The signal is subject to phase unbalance of quadratures if the symbols located along the diagonals of the constellation are equally distorted, and the symbols of one of the diagonals are distorted more than all others, and the opposite—less than all others. Symbols located in the quadrants of the most distorted diagonal are subject to distortion more strongly than symbols that are in the quadrants of the opposite diagonal. Let us write a logical expression describing the criterion of phase unbalance at a dull angle between the quadrature axes; we note that the generalized criterion will look like a logical sum of a case with the acute and obtuse angles between quadratures (Eq. 4.15). PhaseImb   max(E V M(S2 ), E V M(S7 ), E V M(S13 ), E V M(S8 )) = −min(E V M(S2 ), E V M(S7 ), E V M(S13 ), E V M(S8 )) ≤    max(E V M(S3 ), E V M(S6 ), E V M(S9 ), E V M(S12 )) ∩ −min(E V M(S3 ), E V M(S6 ), E V M(S9 ), E V M(S12 )) ≤    max(E V M(S1 ), E V M(S4 ), E V M(S14 ), E V M(S11 )) ∩ −min(E V M(S1 ), E V M(S4 ), E V M(S14 ), E V M(S11 )) ≤    max(E V M(S0 ), E V M(S5 ), E V M(S15 ), E V M(S10 )) ∩ −min(E V M(S1 ), E V M(S4 ), E V M(S14 ), E V M(S11 )) ≤    (max(E V M(S0 ), E V M(S5 ), E V M(S15 ), E V M(S10 )) + ) ∩ ≤ min(E V M(S1 ), E V M(S4 ), E V M(S14 ), E V M(S11 ))   (max(E V M(S1 ), E V M(S4 ), E V M(S14 ), E V M(S11 )) + ) ∩ ≤ min(E V M(S3 ), E V M(S6 ), E V M(S9 ), E V M(S12 ))   (max(E V M(S3 ), E V M(S6 ), E V M(S9 ), E V M(S12 )) + ) (4.15) ∩ ≤ min(E V M(S2 ), E V M(S7 ), E V M(S13 ), E V M(S8 )) The signal is a subject to a magnitude imbalance of quadratures if the symbols located along the diagonals of the constellation are the same distorted, four symbols

4 Distortion Types Separation of QAM-16 Signal

57

located closer to the axis, along which compression occurs, are distorted most of all, and the remaining four symbols are distorted least of all. Equation 4.16 is a record of the criterion of compliance with amplitude unbalance in the case of compression along the quadrature axis. MagImb   max(E V M(S4 ), E V M(S12 ), E V M(S6 ), E V M(S14 )) = −min(E V M(S4 ), E V M(S12 ), E V M(S6 ), E V M(S14 )) ≤    ⎞ ⎛ E V M(S0 ), E V M(S5 ), E V M(S15 ), E V M(S10 ), max ⎟ ⎜ E V M(S2 ), E V M(S7 ), E V M(S13 ), E V M(S8 ) ⎟   ∩⎜ ⎠ ⎝ E V M(S0 ), E V M(S5 ), E V M(S15 ), E V M(S10 ), ≤ −min E V M(S2 ), E V M(S7 ), E V M(S13 ), E V M(S8 )   max(E V M(S1 ), E V M(S3 ), E V M(S9 ), E V M(S11 )) ∩ −min(E V M(S1 ), E V M, E V M(S9 ), E V M(S11 )) ≤  ⎞ ⎛ (max(E V M(S1 ), E V M(S3 ), E V M(S9 ), E V M(S11 )) + )   ∩⎝ E V M(S0 ), E V M(S5 ), E V M(S15 ), E V M(S10 ), ⎠ ≤ min E V M(S2 ), E V M(S7 ), E V M(S13 ), E V M(S8 )   ⎞ ⎛ E V M(S0 ), E V M(S5 ), E V M(S15 ), E V M(S10 ), max +  ⎠ ∩⎝ E V M(S2 ), E V M(S7 ), E V M(S13 ), E V M(S8 ) ≤ min(E V M(S4 ), E V M(S12 ), E V M(S6 ), E V M(S14 )) (4.16) The signal constellation is shifted if the degree of distortion of a symbol is inversely proportional to its energy, this distortion exactly the opposite repeats the compression nonlinear distortion considered earlier. The logical expression describing this distortion is given by Eq. 4.17. O f set   max(E V M(S5 ), E V M(S13 ), E V M(S7 ), E V M(S15 )) = −min(E V M(S5 ), E V M(S13 ), E V M(S7 ), E V M(S15 )) ≤    ⎞ ⎛ E V M(S1 ), E V M(S3 ), E V M(S4 ), E V M(S6 ), ⎟ ⎜ max E V M(S9 ), E V M(S11 ), E V M(S12 ), E V M(S14 ) ⎟   ∩⎜ ⎠ ⎝ E V M(S1 ), E V M(S3 ), E V M(S4 ), E V M(S6 ), ≤ −min E V M(S9 ), E V M(S11 ), E V M(S12 ), E V M(S14 )   max(E V M(S0 ), E V M(S2 ), E V M(S8 ), E V M(S10 )) ∩ −min(E V M(S0 ), E V M(S2 ), E V M(S8 ), E V M(S10 )) ≤  ⎞ ⎛ max(E V M(S0 ), E V M(S2 ), E V M(S8 ), E V M(S10 )) +    ⎠ ∩⎝ E V M(S1 ), E V M(S3 ), E V M(S4 ), E V M(S6 ), ≤ min E V M(S9 ), E V M(S11 ), E V M(S12 ), E V M(S14 )

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⎛



E V M(S1 ), E V M(S3 ), E V M(S4 ), E V M(S6 ), E V M(S9 ), max ∩⎝ E V M(S11 ), E V M(S12 ), E V M(S14 ) ≤ min(E V M(S5 ), E V M(S13 ), E V M(S7 ), E V M(S15 ))



⎞ + ⎠ (4.17)

Let us consider in more detail the algorithm of the system for assessing the signal quality at the transmitter output (Fig. 4.14). First, you need to create a package of test data. Features of its formation were mentioned at the beginning of this section. Second, data is transmitted using the device under test, and the error vector is calculated using the reference signal. If the average error vector does not exceed the specified value, the signal quality is considered satisfactory otherwise, an attempt is made to identify the distortion if this fails and an error message is an output. Having determined all the necessary criteria for recognizing the main types of distortion and the general principles of the identification system, it is necessary to evaluate the effectiveness of their work. For Polyfet MlCQ-02 amplification characteristic above, the criterion works successfully starting with the average power supplied to the amplifier and is equaled to 0.03 dBm. The phase imbalance is detected

Start

Preparing of test data

Sending test data

EVM calculation

no

EVM < threshold

yes

Output

EVM analysis

no

Detected unidentified distortion

Distortion identified

yes

Output

Fig. 4.14 Block diagram of the system for distortions identification

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Fig. 4.15 Diagram of work areas for distortion identification criteria

starting from the amplitude unbalance. The threshold is a ratio of quadratures equal to shifting the signal constellation from half the minimum energy of the symbol. Figure 4.15 shows a diagram of the performance of the criteria with different measurement errors and noise temperature of the amplifier equaled to 290 K. It is worth noting the strong dependence of identifying phase and amplitude unbalance on the measurement error introduced into the algorithm. Also, it is important that despite the cumbersome notation of logical expressions describing the criteria for recognition of distortion they can be succinctly and effectively implemented using programming languages.

4.5 Conclusions A method is proposed for identifying the distortion at the transmitter output when analyzing the QAM-16 modulated signal, which is based on the calculation of the error vector. Identification criteria for the main types of distortions are given, complex modeling and evaluation of the efficiency of the proposed system are carried out, as a result of which the sensitivity of the proposed method to the accuracy of measuring the error vector is revealed. The method of transfer of the amplification characteristic to the MATLAB-Simulink environment is also given, which allows to create the models of amplifiers as close as possible to their physical prototypes in this environment.

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Acknowledgements The reported study was supported by the ETMC Exponenta (Russian The MathWorks department).

References 1. Kiselnikov, A., Dubov, M., Priorov, A.: Non-reference metrics and its application for distortion compensation. In: 21th Conference of Open Innovations Association, pp. 172–181 (2017) 2. Application note: 802.11ac Transmitter Modulation Testing. Available at: https://www. mathworks.com/help/wlan/examples/802-11ac-transmitter-modulation-accuracy-andspectral-emission-testing.html. Accessed 06 July 2019 3. Device datasheet: Amplifier Module Polyfet RF Devices MLCQ-02. Available at: http://www. polyfet.com/module/mlcq02.PDF. Accessed 06 July 2019 4. Kundert, K.: Accurate and Rapid Measurement of IP 2 and IP 3. Available at: https://www. researchgate.net/scientific-contributions/11208128_Ken_Kundert. Accessed 06 July 2019 5. Software Documentation: Block Distortion in the Environment MATLAB Simulink. Available from: https://www.mathworks.com/help/comm/ref/iqimbalancecompensator.html. Accessed 06 July 2019 6. Mendosa, O.: Measuring of EVM for 3G Receivers. Master Thesis Chalmers University of Technology, Gothenburg, Sweden (2002) 7. Measurement Guidelines for DVB Systems. ETSI TR101 290 (2001) 8. Application Note: Modulation Error Ratio (MER) and Error Vector Magnitude (EVM). National instruments. Available from: http://www.ni.com/white-paper/3652/en/. Accessed 06 July 2019 9. Georgiadis, A.: Gain, phase imbalance, and phase noise effects on the error vector magnitude. IEEE Trans. Veh. Technol. 53(2), 443–449 (2004) 10. Lin, F., Chen, S., Chen, I., Chung, H.: Computer simulation and adjacent channel power ratio (ACPR) for amplifier. In: Automotive Technology Conference, pp. 2024–2028 (1999) 11. Qizheng, G.: RF System of Transceivers for Wireless Communications. Springer, US (2005) 12. Wang, A.K., Ligmanowski, R.: EVM simulation and analysis techniques. In: IEEE Conference on Military Communications, pp. 3043–3049 (2006) 13. Olgaard, C.: Using WLAN transmitter degradations. RF Design 27, 28–36 (2004)

Chapter 5

Threshold Effect Indicator Analysis for Template-Based Processing in Microwave Imaging Aleksey S. Gvozdarev and Tatyana K. Artyomova

Abstract A set of methods indicating the threshold signal-to-noise ratio in signal parameter estimation is analyzed. The approaches rely on the behavior of the obtained estimates’ variances lower bounds specifically the Cramer-Rao bound, finite point Barankin and Abel bounds, and the proposed “hybrid” indicator. Validation of proposed methods is illustrated with an example of cumulative phase difference estimation in template phase matching method for microwave imaging applications. Keywords Threshold · Estimation · Variance · Signal-to-noise ratio · Lower bounds · Cramer-Rao bound · Baranking bound · Abel bound

5.1 Introduction The problem of objects’ and signals’ parameters inference is one of the key problems in modern wireless and radio engineering in general and microwave imaging in particular. Two most widely used quality indicators for point estimators are bias and variance [1, 2], which being tested against theoretical lower bounds can possibly show a potential for an improvement of the inference procedure. Among others one of the most widespread is Cramer-Rao bound, which is well known (from classical estimation theory) to be the lowest amidst all other lower bounds and in many practical cases can be evaluated analytically [3]. Unfortunately, it is often unattainable or can be attained only asymptotically, at the same time the support of the probability density function of the estimated variable should not depend on the variable itself and the probability density function should be differentiable. All these restrictions degrade its practical usefulness. In addition to the aforementioned, one of the key practical limitations of Cramer-Rao Lower Bound (CRLB) is its total insensitivity to the A. S. Gvozdarev (B) · T. K. Artyomova Department of Infocommunications and Radiophysics, P.G. Demidov Yaroslavl State University, 14, Sovetskaya Str., Yaroslavl 150003, Russian Federation e-mail: [email protected] T. K. Artyomova e-mail: [email protected] © Springer Nature Switzerland AG 2020 M. Favorskaya and L. C. Jain (eds.), Advances in Signal Processing, Intelligent Systems Reference Library 184, https://doi.org/10.1007/978-3-030-40312-6_5

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threshold effect [4–9], whose indication beforehand (before the estimation procedure is performed) can help to predict possible discrepancy between the expected and actual estimation quality. The problems of CRLB practical application are well-studied and nowadays there exists a wide range of more suitable models for lower bound of estimate’s variance. The research concerns the threshold effect predictive capabilities comparison of the CRLB, finite Barankin [4, 5, 8], and Abel [7] bounds, as well as, the proposed hybrid bound. The chapter is organized as follows. Parameter estimation quality bounds and threshold effect in classical inference are reported in Sect. 5.2. Section 5.3 provides approaches to the problem of threshold SNR indication. Threshold indication for microwave imaging problem is discussed in Sect. 5.4. Summary and conclusions are given in Sect. 5.5.

5.2 Parameter Estimation Quality Bounds and Threshold Effect in Classical Inference It is well known [1–4] that CRLB for the variance of any unbiased estimate ψˆ of scalar parameter ψ, obtained in terms of N stat independent identically distributed samples is given by Eq. 5.1, where CR(ψ, q) is the CRLB for parameter ψ, E[ · ] is the expectation operator, w(x|ψ, q) is joint Probability Density Function (PDF) of the sample, q is some accessory deterministic parameter or vector of deterministic parameters in general (in our case it would denote Signal-to-Noise Ratio (SNR) at system input) and the integration is performed over all possible values of ψ.     2  ∂ ln w(x|ψ, q) 2 ≥ CR(ψ, q) = w(x|ψ, q)d x E ψˆ − ψ ∂ψ

(5.1)

It is worth noticing that the parameter under estimation can have deterministic nature or random with some prior probability distribution. The latter case is usually referred as Bayesian and the former one as non-Bayesian. Hence, all the possible inference procedures and statistical bounds are described as Bayesian or nonBayesian. Throughout the subsequent sections only deterministic (non-Bayesian) case will be assumed. At the same time, it is a common knowledge that most of the real life estimators at some extent suffer from the so called “threshold effect”, which can be described by the dramatic loss of estimation quality (in terms of its variance) as some of the guiding parameters (signal-to-noise ratio, analysis time, sample size etc.) being decreased. To illustrate this (see Fig. 5.1), one can split the whole parameter estimators’ plot into three regions [4]:

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Fig. 5.1 Threshold effect demonstration

• Asymptotic region, where practically any “good” (in some statistical and analytical sense) estimator almost coincides with the theoretical lower bound for estimators’ variance. • Non-information region, where the dominating outliers almost diminish the effect of informative signal presence. • Threshold region, where the variance of some given practical estimator begin to deviate from the possible lower bound. The nature of the threshold effect [4] is in the dramatic loss of information about the signal (with parameters under estimation) as SNR or analysis time are decreased. In such a situation, the inference procedure (because of more frequent outliers) tends to underestimate or overestimate thus introducing some estimation bias. Hence, the detection of the threshold effect essentially relies on the form of the lower bound that is chosen for analysis. Among the existing forms of estimation variance lower bounds one should give prominence to Barankin Lower Bound (BLB) [8], which is theoretically the tightest (hence, most accurate) among all the lower bounds. At the same time, it is evaluated it terms of multidimensional (in most general case infinite-dimensional) constrained optimization procedure relative to some set (generally, countable set) of test points, which makes it unrealistic for analytical closed form derivation and very impractical even for numeric evaluation. It is important to note that even single-point BLB (utilizing only one test point), though being the least accurate among Barankin bounds, is sensitive to threshold effect. Hence, even finite approximations of BLB with a small number of test points being used as an indicator can yield some practical effect. Amidst the side benefits of BLB, one should mention the absence of PDF regularity requirement and its asymptotic (in SNR or sample time/amount of sample) convergence to CRLB. In the most general case, BLB with M test-points is defined as:

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   →  q = N T (h)B  ψ, q −   −1 h, N (h), B M ψ, h,

(5.2)

 − → where h = (h 1 , . . . , h M )T is the column vector of test-points, N ψ, h =    ψ, q is the matrix, inverse to the Barankin (1, h 1 − ψ, . . . , h M − ψ)T and B−1 h, information matrix: ⎛

⎞ CR(ψ, q) J (ψ, h 1 , q) . . . J (ψ, h M , q)   ⎜ J (ψ, h 1 , q) G(ψ, h 1 , h 1 , q) · · · G(ψ, h 1 , h M , q) ⎟ ⎟  q =⎜ B M ψ, h, ⎜ ⎟, .. .. .. .. ⎝ ⎠ . . . . J (ψ, h M , q) G(ψ, h 1 , h M , q) · · · G(ψ, h M , h M , q)

(5.3)

with elements 

 2 J (ψ, h, q) = w(x|ψ+h,q) dx − 1 w(x|ψ,q) .   w(x|ψ+h i ,q)w(x|ψ+h j ,q )  dx G ψ, h i , h j , q = w(x|ψ,q)

(5.4)

It should be noted that the zero test-point BLB (i.e. M = 0), as it was mentioned earlier, coincides with CRLB. At the same time, the magnitude of BLB itself does not depend on the choice of test-points [8], but essentially varies with their amount. Hence, hereafter BLB with the number of test-points from 1 to 3 (equally distributed within the range of the parameter under estimation) will be used. As a compromise between BLB and CRLB that combines “proper” behavior (possibly admitting non-differentiability of PDF, sensitivity to threshold effect, etc.) with relative simplicity of computation, one can assume Abel Lower Bound (ALB) [7], which is defined in a similar way in terms of the solution of the optimization problem over some set of test-points. Similar to BLB, when M = 0, ALB degenerates into CRLB. In case of equal number of test-points, interrelation of magnitudes of those bounds is still an open problem. In the simplest non-trivial case (assumed in the research), single test-point ALB can be represented in the following manner: A1 (ψ, q) = sup A1 (ψ, h, q) h ⎧ ⎫ ⎪ ⎨ CR−1 (ψ, q) + J (ψ,h,q) − 2 (ψ,h,q) ⎪ ⎬ h2 h2 = sup ,   2 ⎪ (ψ,h,q) h ⎪ ⎩ CR−1 (ψ, q) · J (ψ,h,q) ⎭ − h2 h2

(5.5)

where J (ψ, h, q) was defined by Eq. 5.4 and (ψ, h, q) can be expressed by Eq. 5.6.  (ψ, h, q) =

∂ ln w(x|ψ, q) w(x|ψ + h, q)d x. ∂ψ

(5.6)

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65

5.3 Approaches to the Problem of Threshold SNR Indication The problem of the threshold region beginning detection (in terms of its signalto-noise ratio) requires some sort of indicating function pointing on the sufficient discrepancy between the variance of the estimator used and one of the lower bounds. Let us assume several possible variations: 1. Let qB be set as a point defining the divergence between the BLB and CRLB (as the limiting case of BLB) on some arbitrary small value δ:       q B − CR(ψ, q B ) ≤ δ. q B : B M ψ, h,

(5.7)

2. Let, as it was stated in [7], qA be set as a point defining in terms of ALB:    ∂h op (q)  , q A = arg sup ∂q  q

(5.8)

where h op (q) = arg suph A1 (h, q). 3. Let us introduce a hybrid indicator qAB of the threshold region that assumes the following strategy:    ∂ H (q)  , q AB = arg sup ∂q  q where H (q) =

(5.9)

CR(ψ,q) . arg suph A1 (h,q)

It is reasonable to use as a reference guide for all of the assumed indicators the discrepancy of the variance of some practically used estimator D(ψ, qth ) and the lowest of all possible lower bounds (that is CRLB): qth :

|D(ψ, qth ) − CR(ψ, qth )| ≤ ε,

(5.10)

where ε is some arbitrary small positive value. Equations 5.7–5.10 in most general form (without specifying some probabilistic model of the observation process) are, unfortunately, impossible. Hence further analysis will be performed for the problem of cumulative phase difference estimation in the case of microwave imaging of objects hidden behind opaque obstacles [9–12].

5.4 Threshold Indication for Microwave Imaging Problem Following [9–12], let us assume a template matching microwave imaging objects parameters estimation method. Assume that for a given analysis time interval (for

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which the parameters of the microwave imaging system setup are being constant: type and size of aperture, object and reference signals, etc.) the object, whose parameters are under investigation, is being illuminated by an exploring signal with known characteristics (time structure, bandwidth, carrier wavelength, polarization properties, etc.). There is no loss of generality in supposing that the object signal is monochromatic (if not so, then the spectral decomposition should be performed and following analysis can be applied to each practically sufficient harmonic). In such a case for a fixed frequency, an object with some parameter R that is being estimated for a microwave image or complex scattered field u(α, ˙ R) (depending on the absence/presence or reference signal and its type) at some discreet sample point on the aperture is described with the impinging angle α. Hence, the main problem is to find the joint estimates of all possible unknown objects parameters: its geometry, position, electromagnetic properties, etc. For further analysis only a single unknown parameter will be assumed. For definiteness, assume R is a constant real-valued positive parameter (for instance, one of the geometric parameters). Hence u(α, ˙ R) is a complex-valued function of real-value arguments α, R. Let’s suppose that u(α, ˙ R) is at least twice differentiable in both arguments (which is a valid suggestion since its physical meaning is an angular spread of an electric or magnetic field). Assuming that for a fixed value R = Rob there is N stat sample points being   Nα the realizations of u(α, ˙ R) at each discreet sample point α j j=1 within the given aperture range [α1 , α2 ] (which can be regarded as an antenna array snapshot). Let us  Net  form beforehand a set of for a set of objects’ estimated parameter values Reti i=1   Net    Net reference (etalon or template) microwave images u˙ eti = u˙ et Reti . i=1 i=1 Function ξ˙i (Rob ) describing the connection between the observed sample and ith reference object can be represented in the following form: ξ˙i (Rob ) =

Nα 

    u˙ eti α j · u˙ ∗ Rob , α j .

(5.11)

j=1

Let us define a cumulative phase difference using Eq. 5.12.  ⎧ ⎫   Nα     ∗ ⎬   ⎨ ϕi (Rob ) = arg ξ˙i (Rob )  = arg u˙ eti α j · u˙ Rob , α j  ⎭  ⎩ j=1

(5.12)

As an estimate of the object parameter, let us assume the corresponding value of the reference object Ret minimizing ϕi (Rob ): Rˆ ob = arg min{ϕi (Rob )}. i

(5.13)

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67

Following the outline of the method given in [9, 10] basing on N stat samples, we will be interested in the estimate of the cumulative phase difference ϕ(Rob ) formed on the output of the microwave imaging system. For the case of observation, a process being performed in presence of complex circular symmetric white Gaussian noise with zero mean and equal variance of inphase and quadrature components statistical characteristics of ϕ(Rob ) were derived in [9–12]. For instance, the probability density function is given by: √ √   π π 2 − ρ2 −v12 −v22 2 1+ v1 e erf(v1 ) + v2 e erf(v2 ) , wϕ (x, ρ, ψ) = e π 2 2

(5.14)

where v1 = √12 ρ cos(x − ψ), v2 = √12 ρ cos(x + ψ), ψ is the true value of the cumulative phase difference without noise, erf(·) is the error function, and ρ equals to the input signal-to-noise ratio q multiplied by the coefficient η defining amplitude discrepancy of the reference and analyzed objects, which will be set to 1 for further convenience. Following [12], method of moments will be used for the cumulative phase difference inference, i.e. in Eq. 5.10 qP = qMM and D = DMM . It was shown that although it exhibits the same threshold effect (which is being of the main importance for the proposed research) as the maximum likelihood method it is computational less intensive. Based on Eq. 5.12, a signal-to-noise ratio value q that indicates the beginning of the threshold effect (or the transition between the asymptotic and non-information regions) will be searched in accordance with the assumed criteria (Eqs. 5.7–5.10) with the help of the expressions for lower bounds (Eqs. 5.1–5.6). To exemplify it a microwave imaging system was simulated with a following setup: the carrier wavelength λ = 0.008 m, arc antenna array with an angular aperture varying form 1° to 20° and arc radius equals 21.2 m, the distance from the object under investigation equals 21.2 m. Theoretical resolution of an imaging system in such a configuration is 15 λ. The signal-to-noise ratio was varied from 0 dB to 20 dB with a step of 1 dB. The sample size used for averaging at each point of the antenna aperture N stat was 1000 samples. The threshold values indicating the spread of the bounds δ and ε were set on the level of 0.5 dB (that is half of the signalto-noise ratio step). Electromagnetic fields of the test objects under investigation were recorded in presence of circular symmetric zero mean complex additive white Gaussian noise with equal variances of inphase and quadrature components that was chosen in accordance with the required signal-to-noise ratio, and the fields of the reference (matching templates) objects were recorded without noise. The obtained samples were used to produce the values of hop (q) and H(q), that enter into Eqs. 5.8–5.9. It should be noted (see, for example, Fig. 5.2) that they exhibit multiextremal behavior essentially depending on the value of the estimated cumulative phase difference. This in its turn imposes additional stringent requirements to the numerical methods used for the solution of constrained optimization problems (Eqs. 5.8–5.9) while searching for indicators qA and qAB . Hence, a practical

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Fig. 5.2 The dependence of the optimal test point choice (in case of ALB) from SNR for of ψ = 3° (black curve) and ψ = 10° (red curve)

comparison of the computational time or complexity of all the assumed bounds shall heavily rely upon the utilized computer libraries used for numerical optimization, which in its turn depend on the platform being used. At the same time, for a fixed platform it can be noticed that the computational complexity of qA is quiet close to, but a little bit less than that of qAB and both of them are less computationally intensive comparing to qB . In [12], thorough analysis of cumulative phase difference estimation variance dependence on various factors was performed (meaning that classical method of moments estimation procedure is used), where it was shown that the decrease of the true value of the estimated variable ψ expands the non-information region, and the functional relation of the threshold SNR (see Fig. 5.3) calculated in accordance with Eq. 5.10 decreases monotonically approaching the saturation level for ψ > 35°. This essentially means that the prior uncertainty region for cumulative phase difference estimation cannot be compressed to more than 5–6 dB for any true value of ψ. The performed analysis of the single-point BLB demonstrated that for all assumed research values of the parameters its divergence from CRLB is negligible. As an illustration Fig. 5.4 depicts in logarithmic scale normalized to the true value of ψ, square roots of variances for single-point BLB ( Dˆ B1 ) and CRB ( Dˆ C R ) for two values of cumulative phase difference (ψ = 1° and 10°) were calculated. As one can see, the resultant obtained divergence was less than 0.5 dB that is less than the assumed values for δ and ε. In accordance with the first approach (see Eq. 5.7) the threshold SNR is defined in terms of the absolute divergence of those bounds. Hence, it is evident that the single-point BLB will not be helpful in threshold detection and we should resort to a higher dimensional BLB. A similar behavior is demonstrated by the single-point ALB (see Fig. 5.5 where Dˆ A1 corresponds to single-point ALB). At the same time due to the fact that in the second (Eq. 5.8) and third approaches (Eq. 5.9) it is nonlinearly transformed. Hence,

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Fig. 5.3 The dependence of qMM from the true value of the estimated cumulative phase difference

Fig. 5.4 Comparison of the normalized variances of single-point BLB and CRLB for cases ψ = 1° (upper curve) and ψ = 10° (lower curve)

even small discrepancies in the input arguments can possibly yield large deviations in output. Further analysis demonstrated that the difference between CRLB and single-point ALB increases for smaller ψ and for ψ > 10° is of the order of magnitude of the chosen δ and ε. Hence, for further illustrative convenience in Figs. 5.5–5.6, hereafter, it will not be presented as a separate graph, but rather will be understood to closely fit CRLB. Figures 5.6 and 5.7 present in logarithmic scale plots for normalized square roots of CRLB ( Dˆ C R ), two- and three-point BLB ( Dˆ B2 , Dˆ B3 ), and the square root of variance for moment estimator of the cumulative phase difference ( Dˆ M M ). The dot-dashed

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Fig. 5.5 Comparison of the normalized variances of single-point ALB and CRLB for cases ψ = 1° (upper curve) and ψ = 10° (lower curve)

Fig. 5.6 Comparison of the threshold SNR obtained by Eqs. 5.7–5.10 for ψ = 2°

vertical lines indicate the threshold SNR obtained by applying rules: Eq. 5.7—qB , Eq. 5.8—qA , Eq. 5.9—qAB , and Eq. 5.10—qMM . It should be noted that after the analysis of the obtained data it was found out that for an assumed SNR simulation step the reference indicator qMM and the proposed indicator qAB coincide for all the given range of ψ,. Hence, on the graphs only qMM is plotted. In the context of its predictive properties in indicating the beginning of the threshold region, the three-point BLB outperformed the two-point. Hence, the indicator qB (in Figs. 5.6–5.7 and Table 5.1) uses it. It could be argued that the four-point bound would yield even greater effect, but, as it was shown, the gain in overall effect by increasing the number of test-points is not growing even linearly. At the same time,

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Fig. 5.7 Comparison of the threshold SNR, obtained with help of Eqs. 5.7–5.10 for ψ = 2°

Table 5.1 Comparison of the threshold SNR obtained with various indicators ψ, °

qMM , dB

qA , dB

qBB , dB

qAB , dB

2

19

1

9

19

3

17

2

8

17

4

16

2

8

16

5

15

2

8

15

10

12

6

8

12

15

10

4

6

10

the computational burden and, hence, the computational time grows almost exponentially. This limits the application of such indicators, thus it is rational to resort to smaller number of test points. The performed simulation demonstrated that the increase of the true value of the estimated cumulative phase difference lessens the discrepancy between the indicators, which can be explained by the contraction of the non-information region. The obtained results are summarized in Table 5.1. It can be seen the proposed hybrid indicator of the beginning of the threshold region demonstrates the best correspondence with the practically observed discrepancy between the variance of a moment estimator and CRLB. The worst results are obtained with an indicator based on ALB dominated by the indicator and based on three-point BLB, with a loss decreasing with the increase in the true value of the estimated cumulative phase difference, reaching the spread of 2 dB for ψ > 10°.

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5.5 Conclusions The performed research allows to derive following conclusions. The behavior of the Cramer-Rao, Barankin, and Abel lower bounds for variances of microwave imaging objects’ parameters estimates demonstrates their usefulness for beforehand divergence detection between the predicted and actual quality of the estimation procedure in case of realistic observation. The decay rate of the threshold SNR decreases with the increase of true value of ψ that is being estimated with coming to a saturation of around 6 dB for ψ > 35°. This means that in the case of cumulative phase difference estimation the non-information region cannot be compressed to less than 5–6 dB. The divergence of the threshold SNR derived with the help of single-point Abel lower bound decreases with the increase of the estimated true value ψ and reaches up to 17 dB in the case of small values of ψ. A new proposed hybrid estimator enables the threshold region detection with accuracy of 1 dB (which is being SNR step in simulation process). For the cumulative phase difference varying in the range of 2°–55°, the proposed method based on the divergence between Cramer-Rao lower bound and Barankin lower bound outperforms the existing method based on Abel lower bound, by up to 7 dB.

References 1. de Ridder, D., Tax, D.M.J., Lei, B., Xu, G., Feng, M., Zou, Y., van der Heijden, F.: Classification, Parameter Estimation and State Estimation: An Engineering Approach Using MATLAB, 2nd edn. Wiley, Hoboken, NJ (2017) 2. Aster, R.C., Borchers, B., Thurber, C.H.: Parameter Estimation and Inverse Problems. 3th edn. Academic Press (2012) 3. Kay, S.M.: Fundamentals of Statistical Signal Processing, vol. 1. Prentice Hall, NY (1993) 4. Harry, L., Van Trees, K.L.: Bell Bayesian Bounds for Parameter Estimation and Nonlinear Filtering and Tracking. Wiley, IEEE Press (2007) 5. Chaumette, E., Renaux, A., Larzabal, P.: New trends in deterministic lower bounds and SNR threshold estimation: from derivable bounds to conjectural bounds. In: IEEE Sensor Array Multichannel Workshop, Jerusalem, Israel, pp. 121–124 (2010) 6. Chaumette, E., Renaux, A., Larzabal, P.: Lower bounds on mean square error derived from mixture of linear and non-linear transformations of the unbiasedness definition. In: IEEE International Conference on Acoustics, Speech, Signal Process. Taipei, Taiwan, pp. 3045–3048 (2009) 7. Renaux, A., Najjar-Atallah, L., Forster, P., Larzabal, P.: A useful form of the Abel bound and its application to estimator threshold prediction. IEEE Trans. Sig. Process. 55(5), 1001–1007 (2007) 8. McAulay, R.J., Seidman, L.P.: A useful form of the Barankin lower bound and its application to PPM threshold analysis. IEEE Trans. Inform. Theory, IT 15(2), 273–279 (1969) 9. Gvozdaryev, A.S.: Threshold signal-to-noise ratio analysis in application for integral-phase difference estimation problem. In: 23rd International Crimean Conference on Microwave & Telecommunication Technology, Sevastopol, pp. 951–952 (2013) 10. Artyomova, T.K., Gvozdaryev, A.S.: Minimum-phase method for the reference estimation of object size in problems of RF holography. Izv. Vyssh. Uchebn. Zaved. Radioelektron. 54(4), 22–30 (2011)

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11. Gvozdaryev, A.S.: Statistical measures of moment estimator for integral phase difference for application of reference phase method. In: 23rd International Crimean Conference Microwave & Telecommunication Technology. Sevastopol, pp. 1224–1225 (2013) 12. Gvozdaryev, A.S., Artyomova, T.K, Artyomov, K.S.: The comparative study of integral phase estimation with method of moments and maximum likelihood procedures. J. Radio Electron. 12 (in Russian) (2013)

Chapter 6

The Development of the Algorithm for Estimating the Spectral Correlation Function Based on Two-Dimensional Fast Fourier Transform Timofey Shevgunov, Evgeniy Efimov, Vladimir Kirdyashkin and Tatiana Kravchenko Abstract This chapter presents the algorithm for estimating spectral correlation function of a random process that is a valid bi-frequency description of the probabilistic properties of any wide-sense cyclostationary process and relates to its cyclic autocorrelation function via Fourier transform. The key point of the algorithm is that it is based on the two-dimensional discrete Fourier transform of the sample dyadic correlation function weighted by the two-dimensional windowing function, which is rectangular in the direction orthogonal to the current-time axis shape. The dedicated mathematical software libraries implementing fast Fourier transform, which is typically used for image processing, achieve higher performance in comparison with other algorithms involving spectra accumulation. The signal containing a pulse sequence with random amplitudes masked by the additive stationary white Gaussian noise is used in numerical simulation to provide an example of the spectral correlation function estimation procedure and obtain results demonstrating the effectiveness of the proposed algorithm. Keywords Cyclostationarity · Cyclic frequency · Spectral correlation function · Spectral correlation density · Fast fourier transform · Two-dimensional FFT · Pseudo-power

T. Shevgunov (B) · E. Efimov · V. Kirdyashkin Moscow Aviation Institute (National Research University), 4 Volokolamskoe shosse, Moscow 125993, Russian Federation e-mail: [email protected] E. Efimov e-mail: [email protected] V. Kirdyashkin e-mail: [email protected] T. Shevgunov · T. Kravchenko National Research University Higher School of Economics, 20 Myasnitskaya ulitsa, Moscow 101000, Russian Federation e-mail: [email protected] © Springer Nature Switzerland AG 2020 M. Favorskaya and L. C. Jain (eds.), Advances in Signal Processing, Intelligent Systems Reference Library 184, https://doi.org/10.1007/978-3-030-40312-6_6

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6.1 Introduction The non-parametric estimation of the cyclic characteristics of a random process involving long digital samples is one of the important problems to be solved by means of signal processing techniques based on the models representing signals as realizations of cyclostationary random processes [1]. The most acknowledged algorithms performing cyclic spectral estimation, such as based on Wigner-Ville sample transformation, fast Fourier transform accumulation method [2], and spectral strip correlation analyzer [3], were developed in the early 1990s. Despite their relatively high performance that allowed them to be run on that time personal computers, they have a significant drawback. It consists in the fact that estimators provided by them do not cover the bispectral plane (cycles— frequency) completely with such a grid of nodes that can guarantee the absence of missing components of the cyclic spectra. The considerable increase in computer performance having happened over last decades has renewed interest to the problem. Within methods developed recently, it is reasonable to notice the following algorithms based on cyclic periodograms or cyclograms: the double-length Fast Fourier Transform (FFT) estimator (2 N-FFT) [4] and averaged absolute spectral correlation density estimator [5, 6]. Although both algorithms rule out a chance of missing any components of spectral correlation functions during the analysis, their requirements of computational resources, the performance of central processor unit, and, especially, the capacity of random-access memory are rather large even for modern computers. The further search for a simple yet computationally effective digital signalprocessing algorithm conducting spectral correlation function estimation for the finite-length observations of random processes stands to be actual so far. Such an algorithm that could be a conceptual analogue of the well-known FFT used for the classical non-parametric spectral analysis [7] could be very profitable for both scientific and engineering usages. From engineering point of view, the algorithm must be easy to implement, and the standard well-known building blocks should be used as much as possible. The algorithm for spectral correlation estimation described in the chapter is based on two-dimensional fast Fourier transform, the technique intensively used in image processing [8] rather than in the classical signal processing. The key point is the application of this transform to the windowed correlation matrix of the observed signal, where the purpose of the windowing is to increase the robustness. The rest of the chapter is organized as follows. Section 6.2 gives a brief overview of the cyclostationary description of non-stationary random signals focusing more on the dyadic forms. Section 6.3 describes the proposed algorithm for the estimation of the spectral correlation function. The results of numerical simulation are presented in Sect. 6.4, where the comparison to the known analytic solution is given. The chapter ends with the conclusions.

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6.2 Dyadic Cyclostationary Description The probabilistic properties of a wide-sense cyclostationary random process x(t) with zero mean E{x(t)} = 0 can be described in time domain by the dyadic autocorrelation function depending on two time instants:   Rs (t1 , t2 ) = E x(t1 )x ∗ (t2 ) ,

(6.1)

where E{·} stands for the probabilistic expectation and the superscript * does for the complex conjugation. The linear transformation of the arguments 

2 t = t1 +t 2 τ = t1 − t2

(6.2)

allows one to build the symmetric form of the two-Dimensional AutoCorrelation Function (2D-ACF):     Rs (t, τ ) = Rs t + τ 2, t − τ 2 ,

(6.3)

where t means the current time and τ does the relative time or the time shift between two instants, where the correlation is evaluated. Equation 6.3 can be expanded into the generalized Fourier series of the current time t: Rx (t, τ ) =



Rαx (τ ) exp( j2π αt),

(6.4)

α∈A2

where α is the cyclic frequency taking values out of the countable set A2 (alpha), Rαx (τ ) are the components of Cyclic AutoCorrelation Function (CACF), which is a function of the relative time τ = t 1 – t 2 . CACF itself is a set of its components {Rαx (τ )} and it is equivalent to 2D-ACF in the same sense as the set of Fourier series coefficients could replace the periodic signal. The component R0x (τ ) at α = 0 is of particular interest. Firstly, it always exists; secondly, it corresponds to the one-dimensional time-invariant autocorrelation function that can fully characterize as a wide-sense stationary zero-mean random process: ∀t : Rs (t, τ ) = R0x (τ ).

(6.5)

Other than that, if the suggestion (Eq. 6.5) does not hold, it will mean that the random process x(t) is non-stationary. This automatically makes the problem of its formal investigation extremely difficult unless Eq. 6.4 remains valid leading to the cyclostationary case. It is important to notice that stationarity can be considered a particular case of cyclostationarity. In the case of a strictly periodic behavior of 2D-ACF (Eq. 6.3), when it reproduces exactly with the period T, the set A2 consist of such cyclic frequencies that are

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multiples of the fundamental frequency reciprocal to the period T:  k A2 = α|α = , k ∈ Z . T

(6.6)

The strictly periodic case results in the ordinary Fourier series Rx (t, τ ) =



k Rkx/ T (τ ) exp j2π t , T k=−∞ +∞ 

(6.7)

k T where its components Rx/ (τ ) is being evaluated via the classical Fourier series analyzer [9] for T-periodic functions:

Rkx/ T (τ )

1 = T

T

 k Rx (t, τ ) exp − j2π t dt. T

(6.8)

0

Each component of CACF Rαx (τ ) provides one with the function revealing the correlation in time domain in terms of the time shift τ . This can be replaced by the correlation description in frequency domain by means of Fourier transform with respect to τ performed over Rαx (τ ) component-wisely: Sx(α) ( f ) =

+∞ R(α) x (τ ) exp(− j2π f τ )dτ ,

(6.9)

−∞

where the components Sx(α) ( f ) build up the Spectral Correlation Function (SCF) of the random process. In practice, one usually deals with finite-time observations of random processes, leading to changing SCF for Spectral Correlation Density (SCD), which is a function of continuous cyclic frequency rather than discrete. The formal expression of SCD for the theoretical model of the cyclostationary random process, which is infinite time and possesses SCF (Eq. 6.4), is also possible by means of generalized Dirac delta functions:  Sx(ν) ( f )δ(α − ν), (6.10) Sx (α, f ) = ν∈A

where the auxiliary variable ν is used for indexing since α already stands for the cyclic frequency. Using the transformation in the frequency domain 

f = f1 +2 f2 α = f1 − f2

(6.11)

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that is similar to the one applied to ACF in the time domain for Eq. 6.2, SCD can be rewritten in the functional form depending on two frequencies f 1 and f 2 :

Sx ( f 1 , f 2 ) = S x

 f1 + f2 f1 − f2 , . 2

(6.12)

This form reveals the meaning of spectral correlation as the probabilistic expectation:   Sx ( f 1 , f 2 ) = E X ( f 1 )X ∗ ( f 2 ) ,

(6.13)

where X(f ) is a random complex measure [10] that can be considered as the spectral density of the random process x(t) via Crámer’s representation [11]: exp( j2π f t)X ( f )d f .

x(t) =

(6.14)

f ∈R

Then, the one-to-one relationship between ACF Rx (t 1 , t 2 ) (Eq. 6.1) and the density S x (f 1 , f 2 ) is given by means of two-dimensional Fourier transform: +∞ +∞ Sx ( f 1 , f 2 ) =

Rx (t1 , t2 ) exp[− j2π( f 1 t1 − f 2 t2 )]dt2 dt1 ,

(6.15)

−∞ −∞

where the integral is considered in the generalized or distributional sense [12] that leads to an arbitrary mixture of regular and generalized functions. The possible change of ACF for its estimation in Eq. 6.15 opens the way to the design of relatively simple and fast algorithm for the estimation of SCF (Eq. 6.9).

6.3 Two-Dimensional FFT Algorithm Let us consider a digital signal being processed as a time sequence of N samples, which are acquired by uniform sampling of the realization x(t) of the continuous-time random process observed over the finite-time interval [0, T x ): x[n] = x(nTs ),

(6.16)

where T s is the sampling period. Suppose that the length of the observation is much greater than the maximal correlation time to be seen in the 2D-ACF (Eq. 6.3) of the process: Tx  τcor . Starting with the sample sequence x[n], one can simply construct two-dimensional sample function:

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R x [n 1 , n 2 ] = x[n 1 ]x ∗ [n 2 ],

(6.17)

which expresses, in discrete time, the estimation of ACF, which is a function of two time instants



R x [n 1 , n 2 ] = R x (n 1 Ts , n 2 Ts )

(6.18)

defined on the compact support (t 1 , t 2 )∈[0, T x ] × [0, T x ]. However, it is shown in [13] that the estimator R x [n 1 , n 2 ] itself is not a consistent estimator for the true SCF (Eq. 6.9), and for obtaining a consistent one, an appropriate two-dimensional windowing has to be applied to Eq. 6.18 before it makes possible to use the transformation (Eq. 6.15). The simplest choice for the window is a rectangular one defined as



n1 − n2 w[n 1 , n 2 ] = rect d





t1 − t2 = rect w

 = w(t1 , t2 ),

(6.19)

where d = w /T s , and  defines the width of the window in the direction parallel to the line t 1 = –t 2 , the function rect(v) is defined by Eq. 6.20.  rect(v) =

 1 |v| < 1 2 0 |v| > 1 2

(6.20)

On the one hand, the width  of the windows w(t 1 , t 2 ) does not have to be chosen less than twice as short as the expected length of the maximal correlation time of the processes τk : w ≥ 2τcor . On the other hand, the choice of excessively wide window inevitably leads to the higher variation of the estimated SCD (Eq. 6.15). The product of the sample ACF (Eq. 6.18) and the window (Eq. 6.19) yields

x [n 1 , n 2 ] = R x [n 1 , n 2 ]w[n 1 , n 2 ], R

(6.21)

which is to be transformed by two-dimensional discrete Fourier transform:  Sx [m 1 , m 2 ] =

  2π  (n 1 m 1 − n 2 m 2 ) . Rx [n 1 , n 2 ] exp − j N =0

N −1  N −1  n 1 =0 n 2

(6.22)

This transformation can be calculated by means of FFT algorithm carried out x [n 1 , n 2 ]. The majority of modern mathematical platover the matrix containing R forms, such as Matlab or Octave, have got the built-in function fft2. In the case of some other packages, where this function could not be included, the transformation (Eq. 6.12) can be carried out via 2 N one-dimensional FFT. At first, N transformations are to be made along the matrix rows. Then N inverse transformations are to

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be made along the matrix columns or in reverse. The resultant matrix contains the values of  Sx [m 1 , m 2 ] and represents, up to the scaling factor Ts2 , the estimation of SCD S x (f 1 , f 2 ) defined by Eq. 6.22:

S ( f 1 , f 2 ) = Ts2  Sx [m 1 , m 2 ].

(6.23)

The transfer from the dyadic form of SCD (Eq. 6.13) to the symmetric form (Eq. 6.12), where the cyclic frequency is explicitly brought out, can be done via Eq. 6.24.     S x (α, f ) = S x f + α 2, f − α 2



(6.24)

Actually, the particular value of the cyclic frequency α defines the straight line in the bifrequency plane (f 1 , f 2 ), according to the equation f 1 = f 2 + α, whereas the frequency axis f is defined alongside this line: f = (f 1 + f 2 )/2. However, if one wishes to get the estimation of SCF using Eq. 6.9 rather than SCD (Eq. 6.10), they have to use scaling factor as follows: α

Sx ( f ) =

1 S x (α, f ). Tx

(6.25)

Here, it is crucial to notice that units of measurement for SCF and SCD differ. Thus, provided the process is measured in V and the time unit is measured in s, the unit of SCF is V2 s, which is same as for Power Spectral Density (PSD), whereas SCD is measured in V2 s2 .

6.4 Simulation Results The numerical simulation, where the estimation of SCF is carried out for the random process [14] exhibiting strong cyclostationary properties, can be a clear demonstration revealing the performance of the proposed algorithm. Let us consider the regular periodic pulse train, where each pulse has the same waveform but the amplitudes are random. This signal is also known as Pulse-Amplitude Modulated (PAM) signal:  t − qT , Abq rect x(t) =  q=−∞ +∞ 



(6.26)

where  is the width of the pulse, T is the period, A is the amplitude, bq are independent identically distributed (i.i.d.) Rademacher random variables with zero mean E{Aq } = 0 and a finite variance is provided by Eq. 6.27.

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Fig. 6.1 A typical realization of PAM random process

 E{bq b p } =

1, p = q 0, p = q

(6.27)

The typical realization of the random process (Eq. 6.26) is shown in Fig. 6.1. The form of each pulse in Eq. 6.26 was deliberately chosen rectangular to provide the closed-form analytical expression [14] that can be explicitly used for further comparison: Sxα ( f ) =

    α  α  A 2 2 sinc π f +  sinc π f −  , T 2 2

(6.28)

where sinc(v) = sin(v)/v with 1 at v = 0. For the simulation purpose, let us choice the following numerical values for the parameters of the random process (Eq. 6.25): A = 1 V, τ = 5 μs, T = 2τ ; for the observation parameters: T s = τ /16, N = 4096, T x = NT s . Thus, there are T x /T = 128 totally observed within the observation time T x . The width w of the window (Eq. 6.19) is chosen a bit more than double the correlation time:  = 2.1τcor , since the one-side correlation time τ cor is equal to the pulse width τ for this kind of processes. A stationary Gaussian noise z[n] with uniform PSD was added, so SNR is as low as 0 dB. The intensity two-dimensional diagram of the absolute value of the estimated spectral correlation density is shown in Fig. 6.2. This plot gives one the overall description of the correlation in frequency domain. However, the values of the cyclic frequencies characterizing the random process are not evident from the plot in Fig. 6.2. A possible approach to find them consists in applying the integral characteristic [5]: F max

|Sx (α, f )|d f ,

I (α) =

(6.29)

−Fmax

which allows one to estimate the pseudo-power concentrating on each cyclic frequency, where F max is a frequency determined by the borders of the principal domain [15]:

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Fig. 6.2 The color intensity plot for the estimation of the spectral correlation function of PAM random process

 1 1 Fmax = − |α| . 2 Ts

(6.30)

The plot shown in Fig. 6.3 helps one to determine the cyclic frequencies, at which the strong cyclostationarity presents. Observing the plot, one can easily notice the outstanding peaks showing that the process under investigation exhibits those strong cyclostationary properties at cyclic frequencies that are multiples of 1/T = 0.1 MHz. The absolute value of several initial components of the estimated SCF are shown in Figs. 6.4, 6.5, 6.6, and 6.7 together with the analytic curves drawn according to Eq. 6.25. In Fig. 6.4, the component at zero cyclic frequency is depicted. Actually, one can see the power spectrum density of the random process there. Moreover, one

Fig. 6.3 The integral characteristic: pseudo-power concentrating at cycles

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Fig. 6.4 The component of the spectral correlation function at α = 0

Fig. 6.5 The component of the spectral correlation function at α = 1/T

Fig. 6.6 The component of the spectral correlation function at α = 2/T

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Fig. 6.7 The component of the spectral correlation function at α = 3/T

can easily notice that the estimated component goes higher than the analytic one by some constant. This constant has been brought out by the additive stationary noise with the constant PSD. The component at the first cyclic frequency α = 1/T is shown in Fig. 6.5. Since the additive noise does not exhibit cyclostationary properties, one can see how well one curve matches another. However, the estimation error tends to increase as the component number becomes greater. A small difference between the estimated and analytical curves can be easily observed for the second component, at cyclic frequency α = 2/T, shown in Fig. 6.6. Furthermore, Fig. 6.7 demonstrates that the difference is even greater for the third component at α = 3/T, especially at frequencies over than 0.5 MHz, where the values of the component itself is relatively small.

6.5 Conclusions The proposed algorithm for estimating the spectral correlation function of widesense cyclostationary processes is the effective tool for revealing the cyclic frequencies exhibited by the process in case of a long finite-time observation of the process is available. The algorithm is a simultaneous wideband estimator providing such coverage of the bifrequency plane that is dense enough to avoid missing any SCF components. The numerical simulation carried out for the process with strong cyclostationary behavior has proved the effectiveness of the algorithm due to the comparison made between the estimated components of SCF and the curves drawn for their analytical expressions. The first component of the estimated SCF matches the analytic curve very well; that is not so for the component with greater number because they are more sensitive to the influence of the noise.

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The algorithm is based on the techniques traditionally used in image processing such as spatial windowing and two-dimensional fast Fourier transform. The application of the window is a crucial step, otherwise the estimation will not be sufficient due to the variance of the noise increasing significantly. The rectangular window appears to suit well for the estimation task although the formal search for the best shape of the window might be a problem for further investigation as well as seeking the optimal window width for the bias-variance trade-off. Acknowledgements This work was supported by state assignment of the Ministry of Education and Science of the Russian Federation (project 8.8502.2017/BP).

References 1. Gardner, W.A.: Cyclostationarity in Communications and Signal Processing. IEEE Press, New York (1994) 2. Roberts, R.S., Brown, W.A., Loomis, H.H.: Computationally efficient algorithms for cyclic spectral analysis. IEEE Signal Process. Mag. 8(2), 38–49 (1991) 3. Brown, W.A., Loomis, H.H.: Digital implementations of spectral correlation analyzers. IEEE Trans. Signal Process. 41(2), 703–720 (1993) 4. Shevgunov, T., Efimov, E., Zhukov, D.: Algorithm 2 N-FFT for estimation cyclic spectral density. Electrosvyaz 2017(6), 50–57 (2017) 5. Shevgunov, T., Efimov, E., Zhukov, D.: Averaged absolute spectral correlation density estimator. In: Moscow Workshop on Electronic and Networking Technologies, pp. 1–4 (2018) 6. Efimov, E., Shevgunov, T., Kuznetsov, Y.: Cyclic spectrum power density estimation of infocommunication signals. Trudy MAI 97, 14 (2017) 7. Marple Jr., S.L.: Digital Spectral Analysis: With Applications. Prentice Hall, N.Y. (1987) 8. Gonzalez, R.C., Woods R.E.: Digital Image Processing, 4th edn. Pearson (2018) 9. Kammler, D.W.: A First Course in Fourier Analysis, 2nd edn. Cambridge University Press (2007) 10. Samorodnitsky, G., Taqqu M.S.: Stable Non-gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman and Hall/CRC (1994) 11. Napolitano, A.: Cyclostationary Processes and Time Series: Theory, Applications, and Generalizations. Academic Press (2019) 12. Lighthill M.J.: An Introduction to Fourier Analysis and Generalised Functions. Cambridge University Press (1958) 13. Lenart, L.: Asymptotic distributions and subsampling in spectral analysis for almost periodically correlated time series. Bernoulli 17(1), 290–319 (2011) 14. Shevgunov, T.: A comparative example of cyclostationary description of a non-stationary random process. J. Phys.: Conf. Ser. 1163, 012037 (2019) 15. Efimov, E., Shevgunov, T., Kuznetsov, Y.: Time Delay Estimation of Cyclostationary Signals on PCB Using Spectral Correlation Function, pp. 184–187. Baltic URSI Symposium, Poznan (2018)

Chapter 7

Systems Analysis of Discrete Two-Dimensional Signal Processing in Fourier Bases Alexey V. Ponomarev

Abstract The problems of two-dimensional signals processing on the base of twoDimensional Discrete Fourier Transform (2D DFT) are considered. A general definition and mathematical description of two-dimensional discrete signal is given. The algebraic form of 2D DFT is presented, the basic properties of 2D DFTs are briefly considered. The system analysis of the application of two-dimensional signal processing methods based on 2D DPF is carried out. The advantages and disadvantages of digital methods for discrete two-dimensional processing based on 2D DFT are considered. It is shown that the two-dimensional version of the discrete canonical decomposition of random signals proposed by Pugachev implies (by default) a modification of the standard cyclic two-dimensional correlation function of the original signal. A working hypothesis is proposed for solving the problem of discrete two-dimensional signal processing in the spatial-frequency domain. Keywords Two-dimensional signal · Two-dimensional discrete Fourier transform · System analysis · Canonical decomposition of random signals · Cyclic two-dimensional correlation function · Aperiodic two-dimensional correlation function

7.1 Introduction In various subject areas, there are many sources of formation of discrete 2D signals, which, on the one hand, are informational (since they contain information about the states, properties, and characteristics of certain complex applied objects) and, on the other hand, are fundamentally two-dimensional, which requires processing 2D methods. From a mathematical point of view, a discrete 2D signal is a twodimensional sequence that is a set of real (or, in the general case, complex) numbers defined for ordered pairs of integers m and n, where −∞ < m, n < +∞. A. V. Ponomarev (B) Kalashnikov Izhevsk State Technical University, Studencheskay str., 7, Izhevsk, Udmurt Republic 426069, Russian Federation e-mail: [email protected] © Springer Nature Switzerland AG 2020 M. Favorskaya and L. C. Jain (eds.), Advances in Signal Processing, Intelligent Systems Reference Library 184, https://doi.org/10.1007/978-3-030-40312-6_7

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This chapter addresses the discrete two-dimensional signals of finite length (in other words, two-dimensional sequences of finite length). There are two types of descriptions of a discrete two-dimensional signal x(m, n) in a rectangular reference plane, i.e. with 0 ≤ m ≤ N1 − 1 and 0 ≤ n ≤ N2 − 1 [1, 2]: • Matrix representation of a two-dimensional signal x(m, n) has a view of Eq. 7.1.

X N1 ×N 2

0 1 . ⎡ 0 x(0, 0) x(0, 1) . ⎢ 1 x(1, 0) x(1, 1) . ⎢ ⎢ = . . . . ⎢ ⎢ ⎣ . . . . (N1 − 1) x(N1 − 1, 0) x(N1 − 1, 1) . m

. (N2 − 1) . x(0, N2 − 1) . x(1, N2 − 1) . . . . . x(N1 − 1, N2 − 1)

⎤n ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(7.1) • Vector representation of a two-dimensional signal x(m, n), the essence of which is to form a single vector from a matrix description of a two-dimensional signal x(m, n) by “reading” the matrix X N1 ×N2 in columns (or rows). The practice of applying methods and algorithms of signal processing based in one way or another on DFT revealed, in addition to the advantages, their disadvantages, which significantly reduce the effectiveness of solving signal processing problems. The chapter is organized as follows. Problems of discrete two-dimensional signal processing in Fourier bases are discussed in Sect. 7.2. Two-dimensional discrete Fourier transform and its basis are represented in Sect. 7.3. Section 7.4 concludes the chapter.

7.2 Problems of Discrete Two-Dimensional Signal Processing in Fourier Bases Problems of discrete two-dimensional signal processing in Fourier bases can be considered as a set of problems that are in relations and connections with the main problem being solved and whose solution essentially depends on the solution of the main problem. From a formal point of view, a problem is understood as a set [3]: H = {Q, F, V}, where Q = {Qi } is the set of goals, the achievement of which solves the problem, F = {F j } is the set of properties of a problem, V = {Vk } is the set of problem solving hypotheses.

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Problem-solving based on the system analysis implies the development of a sequence of operations (steps) aimed at eliminating the discrepancy (difference) between the necessary (desired) and real (existing) choosing the preferred alternative (hypothesis) to solve the problem. Let us briefly discuss the principal differences and coincidences of the digital spectral processing of one-dimensional and two-dimensional signals. Both in the first and second cases, there are operations of switching from analog to digital waveforms (sampling and quantization operations), as well as, signal conversion operations using unitary transforms. At the same time, when generalizing digital spectral methods for processing one-dimensional signals to a two-dimensional case, not only computational difficulties arise, but also serious theoretical problems (for example, the inability to decompose a two-dimensional polynomial into lower-dimensional polynomials). Theoretical and applied research in this scientific direction has shown that the transition from digital one-dimensional signal processing to two-dimensional processing is a far from trivial task and is not only quantitative, but also qualitative in nature [4–6]. Section 7.2.1 presents a system analysis of the nature of origin, sources, and subject areas of applications of discrete two-dimensional signals, while the main directions of scientific and applied research in the field of digital spectral processing of one-dimensional and two-dimensional signals are discussed in Sect. 7.2.2.

7.2.1 System Analysis of the Nature of Origin, Sources, and Subject Areas of Applications of Discrete Two-Dimensional Signals System analysis of the nature of the origin and sources of discrete two-dimensional signals in literature [7–9] makes possible to identify the following main types of sources of their formation, as well as, the subject areas of their applications. Electromagnetic waves: • Gamma radiation (wavelength 10−11 m). Subject areas of application are the medical radiology, astronomical research, and control of equipment in the nuclear industry. • X-rays (wavelength 10−10 –10−9 m). Subject areas of application are the computed tomography, angiography, control and technical diagnostics in industry, and astronomical research. • Ultraviolet radiation (wavelength 10−8 –10−7 m). Subject areas of application are the technical diagnostics and control in industry, fluorescence microscopy, lithography, laser technology, biological, and astronomical research.

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• Visible light and infrared radiation (wavelength 10−6 –10−3 m). Subject areas of application are the microscopy, astronomical observations, monitoring and technical diagnostics of products, traffic control, remote sensing of the Earth’s surface, environmental monitoring, meteorology, and criminology. • Microwave radiation (wavelength 10−2 –101 m). Subject areas of the application are the radar monitoring and study of remote areas of the Earth. • Radio waves (wavelength 10−2 –101 m). Subject areas of application are the medical diagnostics based on the method of nuclear magnetic resonance and astronomical research. Acoustic waves: • Infrasonic oscillations (frequency of oscillations 1–100 Hz). Subject areas of application are the geological exploration of minerals, passive sonar, marine geology, and seismology. • Sound vibrations (frequency of oscillations 100–20,000 Hz). Subject areas of the application are the acoustics, psychoacoustics, passive sonar, and musical acoustics. • Ultrasonic vibrations (oscillation frequency 20 kHz–30 MHz). Subject areas of application are the medical diagnostics, control and technical diagnostics in industry. Multidimensional time series. Subject areas of application are the acoustics, psychoacoustics, passive sonar, musical acoustics, measurements, medical diagnostics, geophysics, industrial control and technical diagnostics, economics, environmental monitoring, meteorology, biological and astronomical research. Focused electron beams. Subject areas of application are the transmission and scanning electron microscopy and high-temperature plasma physics. Computer-generated two-dimensional signals. Subject areas of application are the construction of 3D models of complex applied objects and computer graphics.

7.2.2 The Main Directions of Scientific and Applied Research in the Field of Digital Spectral Processing of One-Dimensional and Two-Dimensional Signals The following main directions of scientific and applied research in the field of digital spectral processing of one-dimensional and two-dimensional signals can be distinguished [10–16]: • Conventional methods of digital spectral signal processing (vector and spectral analysis, linear and homomorphic filtering, correlation analysis, and sliding Fourier analysis).

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• Non-conventional methods of digital spectral signal processing (methods based on autoregressive and other signal models, Prony method, the minimum dispersion method, and the eigenvalue method). • Discrete multi-resolved signal analysis. Conventional methods of digital spectral processing. The conventional methods of digital spectral processing of both one-dimensional and two-dimensional signals retain their leading role and effectiveness of their applications in almost all subject areas. They are used to almost all classes of stationary and mixed random signals. The spectral estimates obtained by classical methods are robust (the most structurally stable) spectral estimates. Obtaining spectral estimates using conventional methods is the most effective from a computational point of view due to the use of Fast Fourier Transform (FFT) algorithms. However, the practice of applying methods and signal processing algorithms based in one way or another on DFT, revealed, in addition to the advantages, their disadvantages, which significantly reduce the effectiveness of solving signal processing problems using conventional methods. The disadvantages of conventional methods of digital spectral processing of informational signal derive directly from the properties of DFT and manifest themselves in the form of well-known effects (overlap effects, paling, leakage, and amplitude modulation effects). Non-conventional methods of digital spectral processing. Non-conventional methods of spectral signal processing were created as an alternative to the conventional methods of spectral signal processing in order to overcome the shortcomings of the latter. However, as shown in practice applying the developed non-conventional methods of spectral processing, the existing problem was solved only partially. In practice, a set of non-conventional methods of digital spectral processing of stationary random and mixed signals is increased using the autoregressive models and other models of signals. In such methods, in contrast to the conventional digital spectral processing methods, a certain parametric signal model is selected (autoregressive model, moving average model, or combined autoregression model considering a moving average model) or another model described by a certain set of parameters, whose values are determined during processing. The analysis of these methods and signal processing algorithms show that their significant shortcomings are the following. First, the requirements include larger assumptions than in the case of conventional methods. Second, the obtained spectral estimates are structurally stable (robust) only for a limited class of stationary random signals. Third, these methods possess the considerable subjectivity. Discrete multiresolution signal analysis. The theory of multiresolution (wavelet) analysis provides a complex, but quite effective tool for solving some practical problems, where the intuition and experience of a researcher play a significant role. Unlike conventional signal processing, this theory is not a fundamental physical theory. Wavelet analysis, which has greatly expanded the information technology for processing non-stationary signals, is not a universal method for solving signal processing and analysis problems.

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Multiresolution analysis should not be considered as a substitute for classical methods of spectral processing and analysis of information, just because of the fact that polyharmonic and stationary discrete signals are widespread, both in nature and technical systems. Additionally, we note that the results obtained in the framework of the conventional directions of spectral processing of both one-dimensional and twodimensional signals are significant and tested by time. Existing attempts to estimate the absolute superiority of one or another method of spectral digital spectral signal processing over other methods are initially counterproductive. The system analysis of two-dimensional methods of spectral signal processing and subject areas of their applications led to the conclusion that in almost all subject areas, two-dimensional discrete Fourier transform plays a leading role (Fig. 7.1).

Fig. 7.1 Subject areas of application of two-dimensional discrete Fourier transform

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7.3 Two-Dimensional Discrete Fourier Transform and Its Basis In the case of two-dimensional signals when choosing a system of lattice functions, a system of two-dimensional Discrete Exponential Functions (2D DEF), which is based on discrete two-dimensional unitary Fourier transform, can be represented as:   2π 2π (k1 n 1 ) · exp − j (k2 n 2 ) de f N1 ,N2 (k1 , n 1 , k2 , n 2 ) = exp − j N1 N2 = W Nk11n 1 · W Nk22n 2   

 2π 2π 2π = cos k1 n 1 − j sin k1 n 1 · cos k2 n 2 N1 N1 N2   2π (7.2) − j sin k2 n 2 , N2 where k1 , k2 are the spatial frequencies; k1 , n 1 = 0, (N1 − 1), k2 , n 2 = 0, (N2 − 1). We come to the algebraic form of direct 2D DFT of two-dimensional signal x(n 1 , n 2 ): S N1 ,N2 (k1 , k2 ) =

N 1 −1 N 2 −1 1 x(n 1 , n 2 )W Nk11n 1 · W Nk22n 2 , N1 · N2 n =0 n 1

(7.3)

2

where k1 = 0, (N1 − 1), k2 = 0, (N2 − 1) are the spatial frequencies; x(n 1 , n 2 ) is the two-dimensional signal, n 1 = 0, N1 − 1, n 2 = 0, N2 − 1,S N1 ,N2 (k1 , k2 ) are coefficients of 2D DFT (two-dimensional vector spatial-frequency spectrum of signal x(n 1 , n 2 ). Figure 7.2 shows 2D DEF basis functions with k1 = {0, 1, 2}, k2 = {1, 2}. It can be shown that the energy spectrum of the signal x(n 1 , n 2 ):

2 G(k1 , k2 ) = S N1 ,N2 (k1 , k2 )

(7.4)

is only a thinned version of the “true” energy spectrum of the signal x(n 1 , n 2 ) obtained by the two-dimensional version of Wiener–Khinchin theorem. A similar statement holds with respect to Pugachev canonical decomposition. Systems analysis of two-dimensional signal processing methods and their application domains showed that: • The physical nature of the sources of formation of discrete two-dimensional signals is diverse, and the subject areas of their applications are extensive and numerous.

94 Fig. 7.2 The basic functions of two-dimensional discrete Fourier transform for different values k 1 , k 2 : a k 1 = 0, k 2 = 1; b k 1 = 1, k 2 = 1; c k 1 = 3, k 2 = 2

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• Various unitary transformations, such as Fourier transform, cosine transform, Haar transform, Walsh transform, Karhunen–Loew transform, and wavelet analysis, have found wide application in discrete two-dimensional signal processing methods. • In almost all subject areas, two-dimensional discrete Fourier transform plays a leading role due to a number of inherent advantages. The main advantage of two-dimensional discrete Fourier transform consists in the fact that the transform kernels are symmetric and separable, the transform can be implemented by fast algorithms, and the estimates obtained with its help are structurally stable. • The practice of applying two-dimensional discrete Fourier transform revealed a number of negative effects inherent in this transform. The main disadvantages of two-dimensional discrete Fourier transform are the manifestation of the effects: leakage effect, the effect of parasitic amplitude modulation, paling, and the overlapping effect in the spatial, spatial-frequency, and spatial-correlation regions.

7.4 Conclusions Two-dimensional signals are a universal mathematical apparatus for describing information intended both for human perception and storing, transmitting, and presenting information. In order to increase the efficiency of discrete two-dimensional spectral signal processing, it is important and relevant to carry out theoretical and applied research on the development of new and improve existing methods and algorithms for two-dimensional Fourier processing. As a working hypothesis of solving a set of problems of discrete two-dimensional signal processing, it is proposed to consider the development of new two-dimensional basic systems combining the advantages of two-dimensional discrete Fourier basis and eliminating its negative effects.

References 1. Ponomareva, O.V.: Development of the theory and development of methods and algorithms for digital processing of information signals in parametric Fourier bases. Dissertation of the Dr. Sci., Izhevsk (in Russian) (2016) 2. Pratt, W.K.: Digital Image Processing, 4th edn. Wiley-Interscience publication (2007) 3. Druzhinin, V.V., Kantor, D.S.: Problems of systemology: problems of the theory of complex systems. Moscow, Sov. Radio (in Russian) (1976) 4. Dudgeon, D.E.: Multidimensional Digital Signal Processing. Prentice Hall (1995) 5. Marpl, S.L.: Digital Spectral Analysis: With Applications. Prentice-hall, New Jersey (1986) 6. Yaroslavsky, L.P.: Compression, restoration, resampling, “compressive sensing”: Fast transforms in digital imaging. J. Optics 17(7), 073001 (2015) 7. Favorskaya, M.N., Jain, L.C.: Development of mathematical theory in computer vision. In: Favorskay, M.N., Jain, L.C. (eds.) Computer Vision in Control Systems-1: Mathematical Theory, ISRL, vol. 73, pp. 1–8. Springer International Publishing, Switzerland (2015) 8. Gonzalez, R.C., Woods, R.E.: Digital Image Processing, 4th edn. Published by Pearson (2018)

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9. Rabiner, L., Gold, B.: Theory and Application of Digital Signal Processing. Prentice-hall, New Jersey (1975) 10. John, W.: Multidimensional Signal, Image, and Video Processing and Coding. Academic Press is imprint of Elsevier (2006) 11. Ponomarev, V.A., Ponomareva, O.V., Ponomarev, A.V.: Method for effective measurement of a sliding parametric Fourier spectrum. Optoelectron. Instrum. Data Process. 50(2), 1–7 (2014) 12. Ponomareva, O., Ponomarev, A., Ponomarev, V.: Evolution of forward and inverse discrete Fourier transform. In: IEEE East-West Design & Test Symposium, pp. 313–318 (2018) 13. Ponomareva, O., Ponomarev, A., Ponomareva, N.: Window—presume parametric discrete Fourier transform. IEEE East-West Design & Test Symposium, pp. 364–368 (2018) 14. Favorskaya, M.N., Buryachenko, V.V., Zotin, A.G., Pahirka, A.I.: Video completion in digital stabilization task using pseudo-panoramic technique. Int. Arch. Photogramm. Remote Sens. Spatial Inf. Sci. XLII-2/W4, 83–90 (2017) 15. Favorskaya, M.N., Jain, L.C., Savchina, E.I.: Perceptually tuned watermarking using nonsubsampled shearlet transform. In: Favorskay, M.N., Jain, L.C. (eds.) Computer Vision in Control Systems-3: Aerial and Satellite Image Processing, ISRL, vol. 135, pp. 41–69. Springer International Publishing AG (2018) 16. Petrovsky, N.A., Rybenkov, E.V., Petrovsky, A.A.: Two-dimensional non-separable quaternionic paraunitary filter banks. In: IEEE International Conference on Signal Processing: Algorithms, Architectures, Arrangements, and Applications, Poznan, Poland, pp. 120–125 (2018)

Chapter 8

Sliding Spatial Frequency Processing of Discrete Signals Olga V. Ponomareva, Alexey V. Ponomarev and Natalya V. Smirnova

Abstract The definition of sliding spatial-frequency processing of discrete signal is given. Fast methods for analyzing two-dimensional discrete signals in the spatial-frequency domain are proposed. The mathematical apparatus of direct twodimensional discrete Fourier transform in the algebraic and matrix forms is considered. A step by step implementation of two-dimensional discrete Fourier transform based on one-dimensional fast Fourier transform is considered. Effective methods and algorithms for horizontally sliding two-dimensional discrete Fourier transform have been developed that allow us to calculate the coefficients of this transform in time. The developed algorithms efficiency (in terms of computational costs) of real horizontally sliding two-dimensional discrete Fourier transform is evaluated in comparison with the known algorithms. As a result of experimental studies on model two-dimensional discrete signals, the validity, efficiency, and reliability of the proposed methods and algorithms for horizontally sliding two-dimensional discrete Fourier transform have been proved. The relative saving of computations in the developed fast algorithms of horizontal sliding two-dimensional discrete Fourier transform was compared with the standard algorithm. Keywords Sliding spatial frequency processing · Discrete two-dimensional signal · Two-dimensional discrete Fourier transform · Horizontally sliding two-dimensional discrete Fourier transform · Spatial-frequency domain · Efficiency

O. V. Ponomareva · A. V. Ponomarev (B) · N. V. Smirnova Kalashnikov Izhevsk State Technical University, Studencheskay str., 7, Izhevsk, Udmurt Republic 426069, Russian Federation e-mail: [email protected] O. V. Ponomareva e-mail: [email protected] © Springer Nature Switzerland AG 2020 M. Favorskaya and L. C. Jain (eds.), Advances in Signal Processing, Intelligent Systems Reference Library 184, https://doi.org/10.1007/978-3-030-40312-6_8

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8.1 Introduction It is difficult to overestimate the role and place of digital processing methods of discrete two-dimensional signals, which they occupy in various fields of scientific research. These methods have the widest application in such subject areas as medical radiology, biological and astronomical research, computed tomography, angiography, industrial control and technical diagnostics, laser technology, traffic control, remote sensing of the Earth’s surface, environmental monitoring, meteorology, forensics, geological prospecting of minerals, passive and active sonar, marine geology, seismology, acoustics, psychoacoustics, musical acoustics, transmission and scanning electron microscopy, high-temperature plasma physics, build 3D models of complex application objects, and computer graphics. Note that this type of signals, firstly, belongs to informational ones, because it contains information about the properties, states, and characteristics of complex technical systems under study and, secondly, due to its specificity, it requires the development of new and improving existing methods of their two-dimensional spectral processing. The classical method of spatial-frequency processing of two-dimensional discrete signals is two-dimensional discrete Fourier transform, which allows to obtain a two-dimensional spatial-frequency spectrum [1–13]. At the same time, there are a number of applications [6–10, 13], where it is necessary to find the values of the twodimensional spatial-frequency spectrum not at every spatial frequencies, but at their subset only. In this case, the use of the full version of two-dimensional discrete Fourier transform, even on the basis of fast Fourier transform becomes ineffective, since most of the obtained coefficients of the two-dimensional discrete Fourier transform are not used. In this chapter for the decision of the specified problem, a concept of sliding spatial-frequency processing of two-dimensional discrete signals on the basis of two-dimensional discrete Fourier transform is introduced. The horizontal spatialfrequency processing of two-dimensional discrete signals is specifically considered. The chapter is organized as follows. Direct two-dimensional discrete Fourier transform is suggested in Sect. 8.2. Sliding spatial-frequency processing of discrete signals is discussed in Sect. 8.3. Section 8.4 concludes the chapter.

8.2 Direct Two-Dimensional Discrete Fourier Transform Suppose we are given a discrete two-dimensional signal x(n 1 , n 2 ) in the form of a two-dimensional sequence of finite length 0 ≤ n 1 ≤ (N1 −1) and 0 ≤ n 2 ≤ (N2 −1) or with a matrix of size n 1 × n 2 in a rectangular reference zone (specifically on a plane). Direct two-Dimensional Discrete Fourier Transform (2D DFT) of a twodimensional signal x(n 1 , n 2 ) is a special case of direct two-dimensional z-transform:

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S N1 ,N2 (k1 , k2 ) = X (z 1 , z 2 )|z1 =W k1 , z2 =W k2 N1

(8.1)

N2

and can be specified both in the algebraic and matrix form. Algebraic form S N1 ,N2 (k1 , k2 ) is provided by Eq. 8.2, where k1 = 0, (N1 − 1) and k2 = 0, (N2 − 1) are the spatial frequencies, x(n 1, n 2 ) is the two-dimensional  (k n ) and W kN22n 2 = signal, n 1 = 0, N1 − 1, n 2 = 0, N2 − 1, W kN11n 1 = exp − j 2π N1 1 1   exp − j 2π (k n ) are the coefficients (bins) of 2D DFT (two-dimensional vector N2 2 2 spatial-frequency spectrum). S N1 ,N2 (k1 , k2 ) =

   N 1 −1 N 2 −1   1 k1 n 1 k2 n 2 x(n 1 , n 2 ) exp − j2π + N1 · N2 n =0 n =0 N1 N2 1

=

1 N1 · N2

2

N 1 −1 N 2 −1  

x(n 1 , n 2 ) · W Nk11n 1 · W Nk22n 2

(8.2)

1 F (2) · X N 1 ×N2 · FN(1)2 ×N 2 , N1 · N2 N1 ×N1

(8.3)

n 1 =0 n 2 =0

Matrix form has a view: S N1 ×N2 = where

X N1 ×N 2

0 1 ⎡ 0 x(0, 0) x(0, 1) ⎢ 1 x(1, 0) x(1, 1) ⎢ ⎢ = . . . ⎢ ⎢ ⎣ . . . (N1 − 1) x(N1 − 1, 0) x(N1 − 1, 1) n1

.. (N2 − 1) .. x(0, N2 − 1) .. x(1, N2 − 1) .. . .. . . . x(N1 − 1, N2 − 1)

⎤ n2 ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(8.4) ⎡

FN(1)2 ×N2

0 W N0·02 W N1·02 . .

1 W N0·12 W N1·12 . .

0 ⎢ 1 ⎢ ⎢ = . ⎢ ⎢ ⎣ . (N2 − 1) W N(N2 2 −1)·0 W N(N2 2 −1)·1 n2

. . . . . .

. (N2 − 1) 2 −1) . W N0·(N 2 2 −1) . W N1·(N 2 . . . . . W N(N2 2 −1)·(N 2 −1)

⎤ k2 ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(8.5)

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⎡

FN(2)1 ×N1

0 W N0·01 W N1·01 . .

1 W N0·11 W N1·11 . .

0 ⎢ 1 ⎢ ⎢ = . ⎢ ⎢ ⎣ . (N1 − 1) W N(N1 1 −1)·0 W N(N1 1 −1)·1 k1

. . (N1 − 1) 1 −1) . . W N0·(N 1 1 −1) . . W N1·(N 1 .. . .. . (N1 −1)·(N 1 −1) . . W N1

⎤ n1 ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(8.6)

In the future, without loss of generality in Eq. 8.3, we omit the multiplier 1/(N1 · N2 ). Due to the fact that for the product of the matrices from Eq. 8.3 holds the associative property:     S N1 ×N2 = FN(2)1 ×N1 · X N 1 ×N2 · FN(1)2 ×N 2 = FN(2)1 ×N1 · X N 1 ×N2 · FN(1)2 ×N 2

(8.7)

and 2D DFT core is separable, then according to Eq. 8.7 you can get 2D DFT bins S N1 ×N2 , k1 = 0, (N1 − 1), k2 = 0, (N2 − 1), in two ways, each one consists of two stages. It is easy to find out that for obtaining S N1 ,N2 (k1 , k2 ) k1 = 0, (N1 − 1), k2 = 0, (N2 − 1) it is necessary to perform N1 · N2 one-dimensional representation of 2D DFT, for calculating which the algorithms of fast Fourier transform can be effectively applied. Let us consider options for building fast methods and algorithms for horizontal spatial-frequency analysis of two-dimensional discrete signals with horizontal single shift of the spatial window to the right (HS+ ) (Fig. 8.1c). Suppose we need to find the coefficient (bin) of two-dimensional discrete transform S N1 , N2 (k 1 k2 ) at the spatial frequency (k1 , k 2 ) from the samples of the input signal x(n 1 , n 2 ). In this case, Eq. 8.3 is converted into the form:

S(k1 , k2 ) =

[W Nk11·0 , W Nk11·1 , . . . W Nk11·(N1 −1) ]

·

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

⎡ X N 1 ×N2

⎢ ⎢ ⎢ ·⎢ ⎢ ⎣

⎤⎫ ⎪ ⎪ ⎪ ⎥⎪ ⎬ ⎥⎪ ⎥ ⎥ . ⎥⎪ ⎪ ⎦⎪ ⎪ ⎭ (N2 −1)·k2 ⎪

W N0·k2 2 W N1·k2 2 . .

(8.8)

W N2

At the first stage, according to Eq. 8.8, we multiply the basis function of frequency k2 and duration N2 by a matrix of a discrete two-dimensional signal x(n 1 , n 2 ). As a result, we obtain a columned matrix S N2 (n 1 , k2 ) of size N2 , having spent on this procedure N2 · N1 complex multiplications and (N2 − 1) · N1 complex additions. Further, at the second stage, we multiply the basis function of frequency k1 and duration N1 by a columned matrix of size obtained at the first stage, having spent on this procedure N1 complex multiplications and (N1 − 1) complex additions.

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Fig. 8.1 A star chart and examples of sliding of discrete spatial analysis window of 3 × 3 on a two-dimensional discrete signal: a star chart illustrating four types of sliding of a spatial window, b zero shift, c shift HS+ one transition, d VS– one transition; e LDS– one transition

Thus, it is necessary to expend N2 · (N1 + 1) complex multiplications and (N2 − 1) · (N1 + 1) complex additions to obtain one coefficient of two-dimensional discrete transform S N1 , N2 (k 1 k2 ) at spatial frequency (k1 , k 2 ). Considering that performing one complex multiplication requires four real multiplications and two real additions, and one complex addition of two real additions, it is necessary to spend 4 · N1 · (N2 + 1) real multiplications and 4 · N1 N2 real additions to obtain the value of one coefficient of two-dimensional discrete transform S N1 , N2 (k 1 k2 ). Note that this amount of computation needs to be performed at each shift of a two-dimensional spatial analysis window using a two-dimensional signal (Fig. 8.1c). At the same time, from Fig. 8.1 it is easy to see that for any kind of shift of a twodimensional signal a large number of values X N1 ×N2 of the complex matrix in the spatial analysis window remains unchanged.

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Note that the shift of the spatial window along a two-dimensional discrete signal can be considered as a shift of a two-dimensional discrete signal in the spatial analysis window in the opposite direction to the movement of the spatial window.

8.3 Sliding Spatial-Frequency Processing Let us consider the spatial-frequency processing of two-dimensional discrete signals in a sliding spatial analysis window. In contrast to the one-dimensional case [1] for the two-dimensional case, there are four possible ways of sliding of the spatial analysis window on the original two-dimensional discrete signal: • First way consists of horizontal right (HS+ ) and horizontal left (HS– ) shifts, respectively. • Second way consists of vertical up (VS+ ) and vertical down (VS– ) shifts, respectively. • Third way consists of right diagonal up (RDS+ ) and right diagonal down (RDS– ) shifts, respectively. • Forth way consists of left diagonal up (LDS+ ) and left diagonal down (LDS– ) shifts, respectively. Figure 8.1a shows a star diagram illustrating four types of sliding of a spatial analysis window using a discrete two-dimensional signal and examples of its shift. The first horizontal right shift HS+ method and fast Algorithm 1 of sliding processing of two-dimensional discrete signals are discussed in Sect. 8.3.1. Horizontal left shift HS– method of sliding processing of two-dimensional discrete signals is proposed in Sect. 8.3.2. The second horizontal right shift method and fast Algorithm 2 of sliding processing of two-dimensional discrete signals are suggested in Sect. 8.3.3.

8.3.1 First Method and Fast Algorithm 1 of HS+ Sliding Processing Consider the method of HS+ sliding processing of two-dimensional discrete signals in the spatial-frequency domain, which allows to eliminate the above redundancy in finding the coefficient (bin) S N1, N2 (k 1 , k2 ) of two-dimensional discrete transform at the spatial frequency (k1 , k 2 ). ,k2 ) (r ) for a bin S N1, N2 (k 1 , k2 ) of two-dimensional Let us introduce the notation S N(k11,N 2 discrete transform obtained by shifting HS+ spatial analysis window by r samples to the right on a two-dimensional discrete signal x(n 1 , n 2 ):

8 Sliding Spatial Frequency Processing of Discrete Signals ,k2 ) S N(k11,N (r ) = 2

N 1 −1 N 2 −1   1 x[(n 1 ), (n 2 + r )] · W Nk11n 1 · W Nk22n 2 , N1 · N2 n =0 n =0 1

103

(8.9)

2

where r = 0, 1, 2, . . . From Eq. 8.9, it immediately follows that for r = 0: ,k2 ) (0) = S N(k11,N 2

N 1 −1 N 2 −1   1 x(n 1 , n 2 ) · W Nk11n 1 · W Nk22n 2 N1 · N2 n =0 n =0 1

(8.10)

2

and when r = 1: ,k2 ) (1) = S N(k11,N 2

N 1 −1 N 2 −1   1 x[(n 1 ), (n 2 + 1)] · W Nk11n 1 · W Nk22n 2 . N1 · N2 n =0 n =0 1

(8.11)

2

Entering the variable m 2 = n 2 + 1, we convert Eq. 8.11 to the form: ,k2 ) (1) = S N(k11,N 2

N N2 1 −1   1 x(n 1 , m 2 ) · W Nk11·n 1 · W Nk22·(m 2 −1) . N1 · N2 n =0 m =1 1

(8.12)

2

Using the separability property of the 2D DFT core, Eq. 8.12 can be represented by Eq. 8.13. ,k2 ) (1) S N(k11,N 2

 N −1   N2 1 1  1  k 1 ·n 1 k2 ·m 2 −k2 = W x(n 1 , m 2 ) · W N2 · W N2 N1 n =0 N1 N2 m =1 1

(8.13)

2

Let us change the limits of summation in the braces (curly brackets) of Eq. 8.13 by adding and subtracting the corresponding terms:  N −1  N2 2  W N−k2 2 1  −k2 k2 ·m 2 k2 ·m 2 x(n 1 , m 2 ) · W N2 · W N2 = x(n 1 , m 2 ) · W N2 N2 m =1 N2 m 2 =0 2   − x(n 1 , 0) · W Nk22·0 − x(n 1 , N 2 ) · W Nk22·N 2   1 (8.14) = W N−k2 2 · S N(n21 , k2 ) (0) − (x(n 1 , 0) − x(n 1 , N2 )) , N2 where S N(n21 ,k 2 ) (0) is the result of one-dimensional DFT at the frequency k2 . Equation 8.13, taking into account Eq. 8.14, can be rewritten as follows: ,k2 ) (1) S N(k11,N 2

1 = N1

 N −1 1  n 1 =0

W Nk 11·n 1

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  1 ×W N−k2 2 · S N(n21 ,k 2 ) (0) − (x(n 1 , 0) − x(n 1 , N2 )) . N2

(8.15)

Equation 8.15 defines a sliding processing method for shifting HS+ in the spatialfrequency domain, which allows recurrently calculating the coefficient of two,k2 ) (r ) at the spatial frequency (k 1 , k 2 ) with input dimensional discrete transform S N(k11,N 2 samples x[n 1 , (n 2 + r )] n 1 = 0, N1 − 1, n 2 = 0, N2 − 1, r = 1, 2, 3, . . .. We present an algorithm that implements the developed sliding processing method for HS+ shift in the spatial-frequency domain, which allows using the recurrent procedure to find the coefficient of two-dimensional discrete transform S N(n21 ,k 2 ) (r ) at a step using a result of the previous step—S N(n21 ,k 2 ) (r − 1) (let us call it Algorithm 1). Fast Algorithm 1 sliding processing in the spatial-frequency domain with HS+ shift of two-dimensional discrete signals includes the following steps: Step 1. Find the columned matrix S N2 (n 1 , k2 ) of size n 1 by multiplying the base function of duration n 2 and frequency k2 by the matrix of discrete two-dimensional signal x(n 1 , n 2 ). Step 2. Store the columned matrix S N2 (n 1 , k2 ) as a columned matrix S N(n21 ,k 2 ) (0). Step 3. Calculate the value of the coefficient of two-dimensional discrete transform S N1, N2 (k 1 k2 ) by multiplying the columned matrix S N2 (n 1 , k2 ) by the base function of duration n 1 and frequency k1 . Step 4. Do HS+ shift of the discrete spatial window one sample to the right on the two-dimensional signal x(n 1 , n 2 ) and get the matrix of discrete two-dimensional signal x[n 1 , (n 2 + 1)]. Step 5. Form a columned matrix S N(n21 ,k 2 ) (1) in accordance with Eq. 8.15. Step 6. Calculate the value of the coefficient of two-dimensional discrete transform S N1, N2 (k 1 k2 ) by multiplying the columned matrix S N(n21 ,k 2 ) (1) by the base function of duration and frequency k1 . Step 7. Go to the implementation of step 4. Note that the pass of the sliding processing Algorithm 1 in the spatial-frequency domain with HS+ shift of two-dimensional discrete signals to the sliding mode (the first three stages) can be performed using fast Fourier transform algorithms. Consider the effectiveness of the proposed sliding processing algorithm for twodimensional discrete signals in the spatial-frequency domain in comparison with the standard method for obtaining the two-dimensional discrete transform S N1, N2 (k 1 k2 ) coefficient. After simple calculations, you can see that: • The standard sliding processing algorithm for two-dimensional discrete signals in the spatial-frequency domain (Algorithm A) requires 4 · N1 · (N2 + 1) real multiplications and 4 · N1 · N2 real additions to obtain a coefficient S N1, N2 (k 1 k2 ) of two-dimensional discrete transform. • The proposed algorithm for sliding processing of two-dimensional discrete signals in the spatial-frequency domain (Algorithm B) requires 8 · N1 real multiplications (10 · N1 − 2) and real additions to obtain the coefficient S N1 ,N2 (k1 , k2 ) of twodimensional discrete transform.

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Fig. 8.2 The relative savings γ in multiplication as a percentage compared with the standard algorithm

Let us introduce the efficiency criterion, for which, similarly to [1], we use the relative savings of calculations when applying the compared Algorithms A and B: λ=

NOA − NOB · 100%, NOA

(8.16)

where NOA is the number of operations in Algorithm A, NOB is the number of operations in Algorithm B. By the number of operations in Eq. 8.16, we mean the number of real multiplications or the number of real multiplications and additions (in the case of using highspeed multipliers) in the compared algorithms. Figure 8.2 shows the relative savings of multiplication operations in percent, when applying the proposed algorithm in comparison with the standard algorithm for obtaining the coefficient S N1, N2 (k 1 k2 ) of two-dimensional discrete transform. Figure 8.3 shows the relative savings in multiplication and addition operations as a percentage, when applying the proposed algorithm in comparison with the standard algorithm for obtaining the coefficient S N1, N2 (k 1 k2 ) of two-dimensional discrete transform.

8.3.2 Method of HS– Sliding Processing Consider the method of HS– sliding processing of two-dimensional discrete signals in the spatial-frequency domain, which allows you to eliminate the above-mentioned redundancy in finding the coefficient (bin) S N1, N2 (k 1 k2 ) of two-dimensional discrete transform at spatial frequency (k1 , k 2 ). ,k2 ) (−r ) for a bin S N1, N2 (k 1 k2 ) of a twoLet us introduce the notation S N(k11,N 2 dimensional discrete transform obtained by shifting HS– spatial analysis window

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Fig. 8.3 The relative savings γ of multiplication and addition in percent in comparison with the standard algorithm

by r samples to the left on a two-dimensional discrete signal x(n 1 , n 2 ): ,k2 ) (−r ) = S N(k11,N 2

N 1 −1 N 2 −1   1 x[(n 1 ), (n 2 − r )] · W Nk11n 1 · W Nk22n 2 , N1 · N2 n =0 n =0 1

(8.17)

2

where r = 0, 1, 2, . . . From Eq. 8.9, it immediately follows that for r = 0: ,k2 ) (0) = S N(k11,N 2

N 1 −1 N 2 −1   1 x(n 1 , n 2 ) · W Nk11n 1 · W Nk22n 2 N1 · N2 n =0 n =0 1

(8.18)

2

and when r = −1: ,k2 ) S N(k11,N (−1) = 2

N 1 −1 N 2 −1   1 x[(n 1 ), (n 2 − 1)] · W Nk11n 1 · W Nk22n 2 . N1 · N2 n =0 n =0 1

(8.19)

2

Entering the variable m 2 = n 2 − 1, we convert Eq. 8.19 to the form of Eq. 8.20. ,k2 ) (−1) = S N(k11,N 2

N 1 −1 N 2 −2   1 x(n 1 , m 2 ) · W Nk11·n 1 · W Nk22·(m 2 +1) N1 · N2 n =0 m =−1 1

(8.20)

2

Using the separability property of 2D DFT kernel, Eq. 8.20 can be represented as follows: ,k2 ) (−1) = S N(k11,N 2

N 1 −1 N 2 −2   1 x(n 1 , m 2 ) · W Nk11·n 1 · W Nk22·(m 2 +1) . N1 · N2 n =0 m =−1 1

2

(8.21)

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Let us change the limits of summation in the braces (curly brackets) of Eq. 8.21 by adding and subtracting the corresponding terms:  N −1  N2 −2 2  W N+k2 2 1  k2 ·m 2 +k2 k2 ·m 2 x(n 1 , m 2 ) · W N2 · W N2 = x(n 1 , m 2 ) · W N2 N2 m =−1 N2 m 2 =0 2   + x(n 1 , −1) · W Nk22·0 − x(n 1 , N 2 − 1) · W Nk22·N 2   1 (8.22) = W N+k2 2 · S N(n21 , k2 ) (0) + (x(n 1 , −1) − x(n 1 , N2 − 1)) , N2 where S N(n21 ,k 2 ) (0) is the result of one-dimensional DFT at frequency (first stage, Eq. 8.9). Equation 8.21 with regard to Eq. 8.22 can be rewritten by Eq. 8.23. ,k2 ) (−1) S N(k11,N 2

 N −1 1 1  = W k 1 ·n 1 × W N+k2 2 N1 n =0 N1 1   1 (n 1 ,k 2 ) · S N2 (0) + (x(n 1 , −1) − x(n 1 , N2 − 1)) N2

(8.23)

Equation 8.23 defines a sliding processing method for HS– shift in the spatialfrequency domain, which allows recurrently calculating the two-dimensional discrete ,k2 ) (−r ) at the spatial frequency (k 1 , k 2 ) with input signal transform coefficient S N(k11,N 2 samples x[n 1 , (n 2 − r )] n 1 = 0, N1 − 1, n 2 = 0, N2 − 1, r = 1, 2, 3, . . .. The sliding processing algorithm in the spatial-frequency domain during shifting HS– two-dimensional discrete signals conceptually is similar to the above HS+ processing algorithm that directly follows from the comparison Eqs. 8.15 and 8.23. Particular attention should be paid to the positive/negative signs in Eqs. 8.15 and 8.23 in programming fast algorithms. As for the efficiency of the sliding processing algorithm in the spatial-frequency domain during shifting HS– two-dimensional discrete signals, it is exactly the same as the efficiency of the sliding processing algorithm in the spatial-frequency domain when shifting HS+ .

8.3.3 Second Method and Fast Algorithm 2 of HS+ Sliding Processing In Sects. 8.3.1 and 8.3.2, we considered the methods of horizontally sliding processing of two-dimensional discrete signals in the spatial-frequency domain at HS+ and HS– shifts. Referring to the matrix Eq. 8.8, it is easy to see that another method of HS+ horizontally sliding processing of two-dimensional discrete signals in the

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spatial-frequency domain is possible, which differs from those described in the order of operations in the matrix Eq. 8.8: S(k1 , k2 ) =



   T W Nk11·0 , W Nk11·1 , . . . , W Nk11·(N1 −1) · X N 1 ×N2 · W 0·k2 , W N1·k2 2 , . . . , W N(N2 2 −1)·k2 .

(8.24)

Consider a fast horizontal sliding processing algorithm for two-dimensional discrete signals in the spatial-frequency domain that implements Eq. 8.24 (let us call it Algorithm 2). Fast Algorithm 2 sliding processing in the spatial-frequency domain with HS+ shift of two-dimensional discrete signals involves the steps: Step 1. Find the rowed matrix S N1 (n 1 , m 1 ) with the size N1 by multiplying the base function with the duration N1 and frequency m 1 by the matrix of the discrete two-dimensional signal x(n 1 , n 2 ). Step 2. Remember the rowed matrix S N1 (n 1 , m 1 ) as a rowed matrix S N(n11 ,m 1 ) (0). Step 3. Transform S N1 ,N2 (m 1 , m 2 ) by multiplying the rowed matrix S N1 (n 1 , m 1 ) by the base function of duration N1 and frequency m 1 . This procedure ends the output of the algorithm on the operating mode. Next, a recurrent procedure is performed to obtain the coefficient (bin) S N1, N2 (m 1 , m 2 ), when the window is shifted by one sample to the right by a two-dimensional signal x(n 1 , n 2 ). Step 4. Do HS+ shift of the discrete spatial window one sample to the right on the two-dimensional signal x(n 1 , n 2 ) and get the matrix of discrete two-dimensional signal x[n 1 , (n 2 + 1)]. Step 5. Find according to Eq. 8.23 S N(n11 ,m 1 ) (−1) = S N((n1 1 +1),m 1 ) (0), 1 −1),m 1 ) (−1) must be found additionally by multiplyn 1 = 0, N1 − 2. The value S N((N 1 ing the row m 2 of the matrix FN(2)2 ×N 2 by the column (N1 − 1) of the matrix S N(n11 ,m 1 ) (1) after HS+ shift. Step 6. Calculate the value of the coefficient of two-dimensional discrete transform S N1, N2 (m 1 , m 2 ) by multiplying the rowed matrix S N(n11 ,m 1 ) (1) by the base function of duration N1 and frequency m 2 . Step 7. Go to the implementation of Step 4. Thus, fast Algorithm 2 for horizontal sliding processing of two-dimensional discrete signals in the spatial-frequency domain requires 4·(N1 +N2 ) real multiplications and 4·(N1 + N2 −1) real additions to obtain two-dimensional discrete transformation S N1, N2 (k 1 , k2 ).

8.4 Conclusions As was shown above, fast Algorithm 1 for horizontal sliding processing of twodimensional discrete signals in the spatial-frequency domain in order to obtain twodimensional discrete transform S N1, N2 (k 1 , k2 ) requires 8 · N1 real multiplications and (10· N1 −2) real additions. A distinctive feature of Algorithm 1 is the independence of

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the computational cost (when the algorithm goes to the operating mode) on the size of the sliding window in a variable n 2 . Fast Algorithm 2 for horizontal sliding processing of two-dimensional discrete signals in the spatial-frequency domain requires 4 · (N1 + N2 ) real multiplications and 4 · (N1 + N2 − 1) real additions to obtain a twodimensional discrete transform S N1, N2 (k 1 , k2 ). A distinctive feature of this algorithm is the dependence of the computational cost (when the algorithm goes to the operating mode) on the size of the sliding window in both variables n 1 and n 2 . These properties of the algorithms and the required computational costs for their implementation allow the researcher to choose either Algorithm 1 or Algorithm 2 depending on the specific situation. The considered fast methods of HS+ and HS– sliding processing of twodimensional discrete signals in the spatial-frequency domain allow by eliminating redundancy to provide a real-time scale for obtaining 2D DFT coefficients (bins).

References 1. 2. 3. 4. 5.

6. 7. 8. 9. 10. 11.

12.

13.

Pratt, W.K.: Digital Image Processing. 4th edn. Wiley (2007) Dudgeon, D.E.: Multidimensional Digital Signal Processing. Prentice Hall (1995) Marpl, S.L.: Digital Spectral Analysis: With Applications. Prentice-Hall, New Jersey (1986) Yaroslavsky, L.P.: Compression, restoration, resampling, “compressive sensing”: fast transforms in digital imaging. J. Opt. 17(7), 073001 (2015) Favorskaya, M.N., Jain, L.C.: Development of mathematical theory in computer vision. In: Favorskay, M.N., Jain, L.C. (eds.) Computer Vision in Control Systems-1: Mathematical Theory, ISRL, vol. 73, pp. 1–8. Springer, Switzerland (2015) Gonzalez RC, Woods, R.E.: Digital Image Processing. 4th edn. Published by Pearson John, W.: Multidimensional Signal, Image, and Video Processing and Coding. Academic Press is imprint of Elsevier (2006) Ponomarev, V.A., Ponomareva, O.V., Ponomarev, A.V.: Method for effective measurement of a sliding parametric Fourier spectrum. Optoelectron. Instrum. Data Process 50(2), 1–7 (2014) Ponomareva, O., Ponomarev, A., Ponomarev, V.: Evolution of forward and inverse discrete Fourier transform. In: IEEE East-West Design & Test Symposium, pp. 313–318 (2018) Ponomareva, O., Ponomarev, A., Ponomareva, N.: Window-presume parametric discrete Fourier transform. IEEE East-West Design & Test Symposium, pp. 364–368 (2018) Favorskaya, M.N., Buryachenko, V.V., Zotin, A.G., Pahirka, A.I.: Video completion in digital stabilization task using pseudo-panoramic technique. In: The International Archives of Photogrammetry Remote Sensing and Spatial Information Sciences, XLII-2/W4, pp. 83–90 (2017) Favorskaya, M.N., Jain, L.C., Savchina, E.I.: Perceptually tuned watermarking using nonsubsampled shearlet transform. In: Favorskay M.N., Jain L.C. (eds.) Computer Vision in Control Systems-3: Aerial and Satellite Image Processing, ISRL, vol. 135, pp. 41–69. Springer (2018) Petrovsky, N.A., Rybenkov, E.V., Petrovsky, A.A.: Two-dimensional non-separable quaternionic paraunitary filter banks. In: IEEE International Conference on Signal Processing: Algorithms, Architectures, Arrangements, and Applications, Poznan, Poland, pp. 120–125 (2018)

Chapter 9

Interpolation of Real and Complex Discrete Signals in the Spatial Domain Olga V. Ponomareva, Alexey V. Ponomarev and Natalya V. Smirnova

Abstract The problem solution of two-dimensional signal interpolation on the reference plane in the spatial domain is considered. The difference between the interpolation and approximation problems of two-dimensional discrete signals is shown. The mathematical apparatus of matrix and algebraic descriptions of two-dimensional discrete signals in the spatial and spatial-frequency domain is given. The methods of interpolation of real and complex discrete two-dimensional signals in the spatial domain are developed. The validity, accuracy, and efficiency of the proposed interpolation algorithms are illustrated. The results of experimental study using two-dimensional discrete signal model are presented. Keywords Two-dimensional signal · Two-dimensional discrete Fourier transform · Systems analysis · Interpolation · Approximation · Spatial domain

9.1 Introduction The following definition of a one-dimensional signal and purpose of its processing is given in [1]: “A signal is a material carrier of information of various physical nature about processes, phenomena, states or physical quantities characterizing an object of the material world. A signal is a means of transferring information in space and time. Signal processing is extraction of information contained in signals about states, system connections and patterns of objects functioning, processes and phenomena under investigation”. This definition of signal and the purposes of its processing are not valid only in relation to one-dimensional, but also in relation to two-dimensional (in the general case, multidimensional) signals. O. V. Ponomareva · A. V. Ponomarev (B) · N. V. Smirnova Kalashnikov Izhevsk State Technical University, Studencheskay str., 7, Izhevsk, Udmurt Republic 426069, Russian Federation e-mail: [email protected] O. V. Ponomareva e-mail: [email protected] © Springer Nature Switzerland AG 2020 M. Favorskaya and L. C. Jain (eds.), Advances in Signal Processing, Intelligent Systems Reference Library 184, https://doi.org/10.1007/978-3-030-40312-6_9

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Discrete two-dimensional signals are informational signals because they contain information about the states, properties, and characteristics of complex application objects under study [1–13]. Note that the generalization of one-dimensional processing methods to the two-dimensional case is far from trivial task. For example, in the general case, it is often not possible to generalize the one-dimensional concept of isolated poles, zeros, and roots to a two-dimensional case. In addition, an important role in the limited applications of promising one-dimensional methods of signal processing, which in principle allow a generalization to the two-dimensional case, plays the so-called “curse of dimension”. The aim of this work is to develop the interpolation methods for real and complex discrete one-dimensional and two-dimensional signals in the spatial domain based on fast 2D Fourier transform. The tasks of interpolation of one-dimensional and two-dimensional signals, researchers face in many subject areas. Among them, for example, digital filtering of finite impulse response and infinite impulse response filters and spectral analysis of one-dimensional and two-dimensional signals in the spatial domain, improvement and reconstruction, compression and segmentation, visualization and encoding of images in medicine, seismology, and hydroacoustics. Consider a relationship between the two concepts of digital signal processing given a finite number of samples: the interpolation of discrete signals and approximation of discrete signals. The relationship question of the aforementioned concepts for discrete one-dimensional and two-dimensional signals (signals discrete variables) has its own specific character in force the following propositions [1]: • Dimension (a number of degrees of freedom) of a set of discrete signals is determined by a number of samples, with which one can choose any discrete signal from this set. • Number of degrees of freedom of a set of discrete signals does not depend on the chosen basic system. • If a set of discrete signals is specified by dimension N, then the complete basic system ought to be defined in N points and contains N functions. Based on these propositions, the interpolation of discrete signals consists in finding the values of a discrete signal between already known values, which in Fig. 9.1 are highlighted with bold lines. In the case of approximation of discrete signals, there is no requirement of maintaining the known values of the signal x(n). The chapter is organized as follows. Direct and inverse two-dimensional discrete Fourier transforms and their fast algorithms are discussed in Sect. 9.2. Interpolation method of two-dimensional discrete real and complex signals in the spatial domain is considered in Sect. 9.3. Section 9.4 concludes the chapter.

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Fig. 9.1 Interpolation of a discrete signal x(n), n = 0, 7

9.2 Direct and Inverse Two-Dimensional Discrete Fourier Transform From a mathematical point of view, a discrete two-dimensional signal of finite length is a two-dimensional sequence of finite length, which is a set of real (or in the general case complex) numbers defined for ordered pairs of integers n 1 and n 2 with 0 ≤ n 1 ≤ N1 − 1, 0 ≤ n 2 ≤ N2 − 1. A discrete two-dimensional signal x(n 1 , n 2 ) on a rectangular spatial reference plane with 0 ≤ n 1 ≤ N1 − 1 and 0 ≤ n 2 ≤ N2 − 1 can be represented as a matrix (Eq. 9.1).

X N1 ×N 2

0 1 ⎡ 0 x(0, 0) x(0, 1) ⎢ 1 x(1, 0) x(1, 1) ⎢ ⎢ = . . . ⎢ ⎢ ⎣ . . . (N1 − 1) x(N1 − 1, 0) x(N1 − 1, 1) n2

.. (N2 − 1) .. x(0, N2 − 1) .. x(1, N2 − 1) .. . .. . . . x(N1 − 1, N2 − 1)

⎤ n2 ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(9.1) When choosing for the vector and spectral analysis of discrete two-dimensional signals of unitary two-dimensional Fourier transform, we use the system of twoDimensional Discrete Exponential Functions (2D DEF) (the system is the basis of two-dimensional discrete Fourier transform):

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 2π 2π de f N1 ,N2 (k1 , n 1 , k2 , n 2 ) = exp − j (k1 n 1 ) · exp − j (k2 n 2 ) N1 N2 = [W Nk11n 1 ] · [W Nk22n 2 ] 



 2π 2π k1 n 1 − j sin k1 n 1 = cos N1 N1 



 2π 2π · cos k2 n 2 − j sin k2 n 2 , N2 N2

(9.2)

where k1 , k2 are the spatial frequencies, k1 , n 1 = 0, (N1 − 1), k2 , n 2 = 0, (N2 − 1). Section 9.2.1 presents the algebraic forms of direct and inverse two-dimensional discrete Fourier transform. Matrix forms of direct and inverse two-dimensional discrete Fourier transforms are discussed in Sect. 9.2.2. Section 9.2.3 presents fast algorithms of direct and inverse two-dimensional discrete Fourier transforms.

9.2.1 Algebraic Form of Direct and Inverse Two-Dimensional Discrete Fourier Transforms Algebraic form of direct two-Dimensional Discrete Fourier Transform (2D DFT) of a two-dimensional signal has a view: S N1 , N2 (k1 , k2 ) =

N 1 −1 N 2 −1 1 x(n 1 , n 2 )W Nk11n 1 · W Nk22n 2 , N1 · N2 n =0 n =0 1

(9.3)

2

where k1 = 0, (N1 − 1), k2 = 0, (N2 − 1) are the spatial frequencies; x(n 1 , n 2 ) is a two-dimensional signal, n 1 = 0, N1 − 1, n 2 = 0, N2 − 1, S N1 , N2 (k1 , k2 ) are 2D DFT coefficients (the two-dimensional vector spatial-frequency spectrum of a signal x(n 1 , n 2 ). The algebraic form of Inverse 2D DFT (2D IDFT) of two-dimensional vector spatial-frequency spectrum S N1 , N2 (k1 , k2 ) can be represented as: x(n 1 , n 2 ) =

N 1 −1 N 2 −1

S N1 , N2 (k1 , k2 ) W N−k1 1 n 1 · W N−k2 2 n 2 ,

(9.4)

k1 =0 k2 =0

where n 1 = 0, (N1 − 1), n 2 = 0, (N2 − 1), S N1 , N2 (k1 , k2 ) is the two-dimensional vector spatial-frequency spectrum of a signal x(n 1 , n 2 ).

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9.2.2 Matrix Form of Direct and Inverse Two-Dimensional Discrete Fourier Transform Two-dimensional vector spatial frequency spectrum S N1 , N2 (k1 , k2 ) on a rectangular frequency-spatial reference plane with 0 ≤ k1 ≤ N1 − 1 and 0 ≤ k2 ≤ N2 − 1 can be represented as a matrix (Eq. 9.5).

S N1 ×N 2

0 1 ⎡ 0 S(0, 0) S(0, 1) ⎢ 1 S(1, 0) S(1, 1) ⎢ ⎢ = . . . ⎢ ⎢ ⎣ . . . (N1 − 1) S(N1 − 1, 0) S(N1 − 1, 1) k1

.. (N2 − 1) .. S(0, N2 − 1) .. S(1, N2 − 1) .. . .. . . . S(N1 − 1, N2 − 1)

⎤ k2 ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(9.5) We introduce matrices of basis systems of 2D DFT provided by Eqs. 9.6 and 9.7. ⎡

FN(1)2 ×N2

FN(2)1 ×N1

0 W N0·02 W N1·02 . .

1 W N0·12 W N1·12 . .

0 ⎢ 1 ⎢ ⎢ = . ⎢ ⎢ ⎣ . (N2 − 1) W N(N2 2 −1)·0 W N(N2 2 −1)·1 n2 0 1 ⎡ W N0·11 W N0·01 0 ⎢ 1 W N1·01 W N1·11 ⎢ ⎢ = . ⎢ . . ⎢ ⎣ . . . (N1 − 1) W N(N1 1 −1)·0 W N(N1 1 −1)·1 k1

. . . . . .

. (N2 − 1) 2 −1) . W N0·(N 2 2 −1) . W N1·(N 2 . . . . . W N(N2 2 −1)·(N 2 −1)

⎤ k2

. . (N1 − 1) 2 −1) . . W N0·(N 1 1·(N2 −1) . . W N1 .. . .. . (N1 −1)·(N 1 −1) . . W N1

⎤ n1

⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(9.6)

(9.7)

Using Eqs. 9.1, 9.5–9.7, we get the matrix form of the direct 2D DFT: S N1 ×N2 =

1 F (2) · X N 1 ×N2 · FN(1)2 ×N 2 N1 · N2 N1 ×N1

(9.8)

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and matrix form of inverse 2D DFT: X N1 ×N2 = [FN(2)1 ×N1 ]∗ · S N 1 ×N2 · [FN(1)2 ×N 2 ]∗ ,

(9.9)

where the symbol * denotes the complex conjugation.

9.2.3 Fast Algorithms of Direct and Inverse Two-Dimensional Discrete Fourier Transform Due to the fact that for the product matrix (Eqs. 9.8–9.9), the associative properties are held:  

S N1 ×N2 = FN(2)1 ×N1 · X N 1 ×N2 · FN(1)2 ×N 2 = FN(2)1 ×N1 · X N 1 ×N2 · FN(1)2 ×N 2 ,  

X N1 ×N2 = [FN(2)1 ×N1 ]∗ · S N 1 ×N2 · [FN(1)2 ×N 2 ]∗ = [FN(2)1 ×N1 ]∗ · S N 1 ×N2 · [FN(1)2 ×N 2 ]∗ and the core direct 2D DFT and inverse 2D DFT are separable (getting bins direct 2D DFT S N1 ×N2 ,k1 = 0, (N1 − 1), k2 = 0, (N2 − 1) and sample values X N1 ×N2 n 1 = 0, (N1 − 1), n 2 = 0, (N2 − 1)), two ways can be done, each of which consists of two stages. 

Direct 2D DFT, 1st way. Get the product X N 1 ×N2 · FN(1)2 ×N 2 and then multiply the matrix FN(2)1 ×N1 by the result of the product obtained in the previous step.

 Inverse 2D DFT, 1st way. Get the product S N 1 ×N2 · [FN(1)2 ×N 2 ]∗ and then multiply the matrix [FN(2)1 ×N1 ]∗ by the result of the product obtained in the previous step.

 Direct 2D DFT, 2nd way. Get in the first stage the product FN(2)1 ×N1 · X N 1 ×N2 , the result of which is then multiplied by the matrix FN(1)2 ×N 2 .

 Inverse 2D DFT, 2nd way. Get in the first stage the product [FN(2)1 ×N1 ]∗ · S N 1 ×N2 ,

the result of which is then multiplied by the matrix [FN(1)2 ×N 2 ]∗ . It is easy to establish that to obtain S N1 , N2 (k1 , k2 ) k1 = 0, (N1 − 1), k2 = 0, (N2 − 1) or X N1 ×N2 n 1 = 0, (N1 − 1), n 2 = 0, (N2 − 1) it is necessary to perform N1 · N2 one-dimensional DFT or IDFT, respectively. Fast Fourier Transform (FFT) algorithms (the direct and inverse FFT algorithms) can be effectively used to calculate one-dimensional DFTs or IDFTs.

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9.3 Interpolation Method of Two-Dimensional Discrete Real and Complex Signals in Spatial Domain The interpolation method of two-dimensional discrete real and complex signals in the spatial domain is based on three main propositions: • Two-dimensional version of Shannon–Kotelnikov theorem. • Two-dimensional z-transform. • Interrelations of sampling and periodization operations in the spatial and spatialfrequency domains in the case of applying two-dimensional discrete Fourier transform. According to Kotelnikov (Shannon, Nyquist, sampling) theorem as a result of sampling a two-dimensional signal, the two-dimensional spectrum is limited by spatial frequencies f 1 and f 2 , occurs, firstly, periodization in the spatial-frequency domain of its two-dimensional spectrum, secondly, the original continuous signal can be theoretically accurately reconstructed at the sampling rates of the original two-dimensional continuous signal 2 f 1 and 2 f 2 , respectively. Practically impossible to do so, since for this the original signal must be known at the reference plane in infinite intervals. In the case when the original signal x(n 1 , n 2 ) is discrete and casual (x(n 1 , n 2 ) ≡ 0 for n 1 ≥ N1 , n 2 ≥ N2 ), the direct and inverse two-dimensional z-transformations are true for it: Direct two-dimensional z-transform: X (z 1 , z 2 ) =

∞

∞

x(m, n) · z 1−m z 2−n .

(9.10)

m=−∞ n=−∞

Inverse two-dimensional z-transform:   1 X (z 1m−1 z 2n−1 ) dz 1 dz 2 . x(m, n) = 4π

(9.11)

B1 B2

It is easy to see that 2D DFT is a special case of a direct two-dimensional ztransform:  | S N1 , N2 (k1 , k2 ) = X (z 1 , z 2 )z1 =W k1 n1 , z2 =W k2 n2 . N1

(9.12)

N2

From Eqs. 9.3–9.4, 9.9–9.12, the method and algorithm for the interpolation of two-dimensional discrete signals (both real and complex) in the spatial domain directly follow: • Perform 2D DFT discrete signal x(n 1 , n 2 ). • Select interpolation coefficients m 1 , m 2 for variables k 1 , k2 .

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Fig. 9.2 Two-dimensional discrete exponential function de f N1 ,N2 (k1 , n 1 , k2 , n 2 ) defined on the 4 × 8 reference plane, 0 ≤ n 1 ≤ 3, 0 ≤ n 2 ≤ 7: a real part, b imaginary part

• Insert zeroed rows and zeroed columns in the number (m 1 −1)·N1 and (m 2 −1)·N2 , respectively, in the middle of the two-dimensional spectrum S N1 , N2 (k1 , k2 ). • Perform 2D IDFT obtained at the previous step of the spectrum. As a result, we obtain an interpolated two-dimensional discrete signal xint (n 1 , n 2 ), 0 ≤ n 1 ≤ m 1 · N1 , 0 ≤ n 2 ≤ m 2 · N2 . In Figs. 9.2, 9.3, 9.4, 9.5, 9.6, and 9.7, the efficiency of the considered method and algorithm is illustrated by the example of interpolation of two-Dimensional Discrete Exponential Functions (2D DEFs) at various values of k1 and k2 . Figures 9.2, 9.3, 9.4, 9.5, 9.6, and 9.7 show the three-dimensional surface envelopes of the original and interpolated two-dimensional discrete signals. Let us remark here, when the frequencies of a two-dimensional signal match with the frequencies of 2D DFT, a leakage effect (leakage) is not observed, and interpolation error equals zero. When the frequencies of two-dimensional signal do not match with the frequencies of 2D DFT, it is observed and shows itself in error of interpolation at the beginning and at the end of the original two-dimensional signal. The authors have not yet found the closed-form mathematical expression enabling us to predict these errors.

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Fig. 9.3 Interpolated two-dimensional discrete exponential function de f N1 ,N2 (k1 , n 1 , k2 , n 2 ), 0 ≤ n 1 ≤ 31, 0 ≤ n 2 ≤ 63 with the interpolation coefficient m 1 = m 2 = 8, defined on the 4 × 8 reference plane: a real part, b imaginary part

Fig. 9.4 Two-dimensional discrete exponential function de f N1 ,N2 (k1 , n 1 , k2 , n 2 ) defined on the 4 × 8 reference plane, 0 ≤ n 1 ≤ 3, 0 ≤ n 2 ≤ 7: a real part, b imaginary part

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Fig. 9.5 Interpolated two-dimensional discrete exponential function de f N1 ,N2 (k1 , n 1 , k2 , n 2 ), 0 ≤ n 1 ≤ 31, 0 ≤ n 2 ≤ 63 with the interpolation coefficient m 1 = m 2 = 8, defined on the 4 × 8 reference plane: a real part, b imaginary part

Fig. 9.6 Two-dimensional discrete exponential function de f N1 ,N2 (k1 , n 1 , k2 , n 2 ) defined on the 4 × 8 reference plane, 0 ≤ n 1 ≤ 3, 0 ≤ n 2 ≤ 7: a real part, b imaginary part

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Fig. 9.7 Interpolated two-dimensional discrete exponential function de f N1 ,N2 (k1 , n 1 , k2 , n 2 ), 0 ≤ n 1 ≤ 31, 0 ≤ n 2 ≤ 63 with the interpolation coefficient m 1 = m 2 = 8, defined on the 4 × 8 reference plane: a real part, b imaginary part

9.4 Conclusions When implemented, the reviewed the interpolation method of two-dimensional discrete signals (both real and complex) in the spatial domain occurs three problems that need to be addressed: • Need for large amounts of memory to store zero samples of the signal. • Significant non-productive computational costs (even taking into account the separability of 2D DFT and 2D IDFT cores and use of the corresponding FFT algorithms), since a significant number of samples in the spatial frequency domain are zero. We note that the “cropping” method of the signal graph 2D IDFT (not performing operations with zero spectral samples) yields only a small computing cost savings. • Appearance of noticeable relative interpolation errors of initial and final values of signal samples, as a result of two-dimensional leakage, which occurs, when the initial signal contains the harmonic components that do not coincide with 2D DFT bins. To solve the mentioned above problems, a synthesis (active search) of a basis is necessary, which would take into account the specifics of this interpolation method in the spatial domain of two-dimensional discrete signals (both real and complex).

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References 1. Ponomareva, O.V.: Development of the theory and development of methods and algorithms for digital processing of information signals in parametric Fourier bases. Dissertation of the Dr. Sci., Izhevsk (in Russian) (2016) 2. Marpl, S.L.: Digital Spectral Analysis: With applications. Prentice-Hall, New Jersey (1986) 3. Yaroslavsky, L.P.: Compression, restoration, resampling, “compressive sensing”: fast transforms in digital imaging. J. Opt. 17(7), 073001 (2015) 4. Favorskaya, M.N., Jain, L.C.: Development of mathematical theory in computer vision. In: Favorskay, M.N., Jain, L.C. (eds.) Computer Vision in Control Systems-1: Mathematical Theory, ISRL, vol. 73, pp. 1–8. Springer, Switzerland (2015) 5. Gonzalez, R.C., Woods, R.E.: Digital Image Processing. 4th edn. Published by Pearson (2018) 6. Rabiner, L., Gold, B.: Theory and Application of Digital Signal Processing. Prentice-Hall, New Jersey (1975) 7. John, W.: Multidimensional Signal, Image, and Video Processing and Coding. Academic Press is imprint of Elsevier (2006) 8. Ponomarev, V.A., Ponomareva, O.V., Ponomarev, A.V.: Method for effective measurement of a sliding parametric Fourier spectrum. Optoelectron. Instrum. Data Process 50(2), 1–7 (2014) 9. Ponomareva, O., Ponomarev, A., Ponomarev, V.: Evolution of forward and inverse discrete Fourier transform. In: IEEE East-West Design & Test Symposium, pp. 313–318 (2018) 10. Ponomareva, O., Ponomarev, A., Ponomareva, N.: Window—presume parametric discrete Fourier transform. In: IEEE East-West Design & Test Symposium, pp. 364–368 (2018) 11. Favorskaya, M.N., Buryachenko, V.V., Zotin, A.G., Pahirka, A.I.: Video completion in digital stabilization task using pseudo-panoramic technique. In: The International Archives of Photogrammetry Remote Sensing and Spatial Information Sciences, XLII-2/W4, pp. 83–90 (2017) 12. Favorskaya, M.N., Jain, L.C., Savchina, E.I.: Perceptually tuned watermarking using nonsubsampled shearlet transform. In: Favorskay, M.N., Jain, L.C. (eds.) Computer Vision in Control Systems-3: Aerial and Satellite Image Processing, ISRL, vol. 135, pp. 41–69. Springer (2018) 13. Petrovsky, N.A., Rybenkov, E.V., Petrovsky, A.A.: Two-dimensional non-separable quaternionic paraunitary filter banks. In: IEEE International Conference on Signal Processing: Algorithms, Architectures, Arrangements, and Applications, Poznan, Poland, pp. 120–125 (2018)

Chapter 10

Topography of the z-Plane Discretized by Quantizing the Coefficients of the Canonical Form of Recursive Digital Filter Vladislav A. Lesnikov, Tatiana V. Naumovich, Alexander V. Chastikov and Alexander P. Metelyov Abstract It is established that the zeros and poles of recursive digital filters with finite word length are elements of the set of algebraic numbers. Therefore, not every point of the unit circle of the z-plane can be the zero and/or pole of such digital filters. The position of admissible positions for zeros and poles depends on the degree of algebraic numbers and the length of the fractional part of the coefficients of the equivalent canonical structure of the corresponding order. The formation of the corresponding configurations is considered as a z-plane discretization due to quantization of the filter coefficients. The z-plane discretization for second-order filters has been well studied. The geometric locus of the corresponding algebraic numbers is a system of concentric circles, that is, plane algebraic curves of the second degree. This chapter confirms the hypothesis that for digital filters of the higher orders, zeros and poles are points of algebraic numbers of the higher degrees. Keywords Infinite impulse response · Possible pole-zero locations · Grid of allowable pole and zero positions · Variation curve of the poles and zeros · Quantization of pole locations · Plane algebraic curves

V. A. Lesnikov (B) · T. V. Naumovich · A. V. Chastikov · A. P. Metelyov Institute of Mathematics and Information Systems, Vyatka State University, 36, Moscowskaja str., Kirov 610000, Russian Federation e-mail: [email protected] T. V. Naumovich e-mail: [email protected] A. V. Chastikov e-mail: [email protected] A. P. Metelyov e-mail: [email protected] © Springer Nature Switzerland AG 2020 M. Favorskaya and L. C. Jain (eds.), Advances in Signal Processing, Intelligent Systems Reference Library 184, https://doi.org/10.1007/978-3-030-40312-6_10

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10.1 Introduction The classical technique of synthesis of Infinite Impulse Response (IIR) digital filters includes two stages, which can be called functional and structural synthesis [1–3]. At the stage of functional synthesis, a digital filter is considered as a “black box”, for which the relationship between input and output signals is determined. Effects related to the finite word length are not considered at this stage. The filter is represented as a linear discrete system. The result of functional synthesis is to obtain a transfer function described by the ratio of two polynomials of the corresponding degree from the complex variable z. At the stage of structural synthesis, one of the possible structural schemes is selected, and the coefficients of the structural scheme are calculated. After that, the quantization of the filter structure coefficients is performed, and the consequences of this quantization are analyzed. Quantization of the coefficients leads to errors in their representation, distortion of the coefficients of the transfer function, a shift in the position of zeros and poles and, as a result, distortion of the filter characteristics and violation of the specification of requirements. In short, at the stage of structural synthesis, the results of functional synthesis are distorted. To remedy the situation, the researchers resorted to various actions. An increase in the digit capacity could lead to an unacceptable increase in the need for computing resources. The choice of a different filter structure from among the classical ones is limited by their small nomenclature. Many publications are known, in which structures with low sensitivity of filter characteristics to the accuracy of the coefficient representation are proposed. However, these structures are characterized by specific unique calculation algorithms that are not implemented in development systems. There are known attempts to apply discrete optimization methods to meet the specification of requirements. The state-of-the-art in the field of synthesis of IIR filters with a finite length of a word with stringent characteristics requirements can be described as follows. Despite the long history the solving the problem of the synthesis of such filters is far from a satisfactory solution. In a series of publications [4–10], the authors proposed an alternative approach to the synthesis of IIR digital filters, believing that it would allow overcoming the existing difficulties. The use of this approach assumes that finite word length is taken into account already at the stage of functional synthesis. At the stage of structural synthesis, the structures are generated that correspond to the results of the previous stage. Structural synthesis with this approach does not distort the results of functional synthesis. This is possible when using the algebraic properties of matrix models of structures studied by the authors. In this chapter, we limit ourselves to problems related to the functional stage of the synthesis. The possibility to solve the above tasks of this stage is based on the results of research of the number-theoretical nature of the zeros and poles of IIR filters with a finite word length, made by the authors [6–10]. These studies have shown that the considered zeros and poles are the elements of the set of algebraic numbers of the

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corresponding degree. Therefore, not every point of the z-plane can be a zero or a pole. The allowed values form the topography of zeros and poles in the z-plane. For filters of the second order, this fact was established a long time ago [11–18]. In the works of the authors [7], this topography was studied in detail. Get a similar solution for the filters of higher orders could not get up until recently. Therefore, it was necessary to resort to an indirect approach based on the transition from the representation of filters in the z-plane to the representation in the transfer function coefficient space. In this chapter, the study of the discrete structure of localization of zeros and poles in the z-plane is extended to complex algebraic numbers of a higher degree. The remainder of the chapter is organized as follows. Section 10.2 describes the applicability of the theory of algebraic numbers to the description of the nature of zeros and poles of IIR digital filters with a finite word length. Section 10.3 describes the discrete z-plane structure for the localization of complex algebraic numbers of the second degree. Section 10.4 is devoted to a brief description of the recently obtained results for the geometric place of complex algebraic numbers of the third degree [10]. In Sect. 10.5, equations of plane algebraic curves are provided, which describe the allowed position in the z-plane of fourth-degree complex algebraic numbers. Section 10.6 concludes the chapter.

10.2 The Number-Theoretic Nature of the Zeros and Poles of IIR Digital Filters with Finite Word Length It is known [19] that all possible roots zi of a polynomial of degree n are calculated using Eq. 10.1 Pn (z) =

n 

ci z n−i ,

(10.1)

i=0

whose coefficients ci ∈ Q

(10.2)

belong to the set of real rational numbers, are algebraic numbers of degree n z i ∈ An ⊂ A.

(10.3)

Without considering the finite word length, the coefficients of the digital filter transfer function are defined by Eq. 10.4:

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n H(z) =

i=0 bi z n 

zn +

n−i

,

(10.4)

ai z n−i

i=1

in the general case bi , ai are real numbers: bi , ai ∈ R.

(10.5)

Real numbers are also real and imaginary parts of zeros zzi and poles zpi . Any point of the z-plane in this case can be a zero or a pole of some digital filter. All practically implemented digital filters have finite coefficient bitness. Therefore, the digit capacity of the transfer function coefficients will also be finite, and these coefficients will be elements of the set of real rational numbers Q. In this case, only those points of the z-plane, which correspond to algebraic numbers of degree n, can be zeros and poles. The allowed positions of the roots of the polynomials of the transfer function numerator (denominator) for fixed values of the degree of algebraic numbers and the bit depth of the fractional part of the coefficients of the polynomials form the z-plane topography. This process will be called the z-plane discretization due to quantization of the polynomial coefficients. As noted above, the discrete nature of the z-plane topography for second-order filters was known for a long time and was studied in detail in the works of the authors. It was found that the geometrical place of the roots of a second-degree polynomial with quantized coefficients is a system of concentric circles. The center of this system can be any rational point on the (Re z)-axis. The radii of the circles are determined by their center and the digit capacity of the fractional part of the polynomial coefficients. Circles are plane second-degree algebraic curves (conics). For algebraic numbers of a higher degree, it was hypothesized that the geometric locus of algebraic numbers of nth degree are flat algebraic curves of nth degree. However, this hypothesis was not directly confirmed in this formulation. In [10], the problem of determining the zplane topography for algebraic numbers of the third degree was solved. This chapter presents results for fourth-degree algebraic numbers.

10.3 The Topography of the Second-Degree Algebraic Numbers in a Discretized Complex z-Plane Here we consider only the complex conjugate roots of the polynomial P2 (z) that are of practical interest. Obviously, the roots of the polynomial P2 (z) are such numbers: ⎧  ⎪ ⎨ z 1 = Re z 1 + j Im z 1 = −0.5c1 + j c2 − 0.25c12 ,  ⎪ ⎩ z 2 = Re z 1 − j Im z 1 = −0.5c1 − j c2 − 0.25c2 . 1

(10.6)

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Equations 10.6 convert to this form: ⎧  ⎪ ⎨ Re z 1 − xc + j Im z 1 = −0.5c1 − xc + j c2 − 0.25c12 ,  ⎪ ⎩ Re z 1 − xc − j Im z 1 = −0.5c1 − xc − j c2 − 0.25c2 , 1

(10.7)

where x c is a real number. Introducing for convenience the notations

x = Re z 1 y = Im z 1

(10.8)

and multiplying Eqs. 10.7, we get (x − xc )2 + y 2 = c2 + xc2 + c1 xc .

(10.9)

Equation 10.9 is the equation of a circle with center x c , and a square of radius equals c2 + xc2 + c1 xc . Looking through all possible quantized values of the coefficients c1 and c2 , we obtain a system of concentric circles, which is the geometric locus of the corresponding algebraic numbers of the second degree (Fig. 10.1). Figure 10.1 shows only the first quadrant of the z-plane, on which the green points represent possible roots of a polynomial P2 . The absolute value of these roots does not exceed 1. The bitness of 1

0.75

0.5

0.25

0.25

0.5

0.75

1

Fig. 10.1 Systems of concentric circles with different centers for algebraic numbers of the second degree

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the polynomial coefficients fractional part is equal to 3. Black circles have a center equal to x c = 0. The center of the red circles is x c = 0.5, while x c = 1 is the center of the blue circles.

10.4 The Topography of the Third-Degree Algebraic Numbers in a Discretized Complex z-Plane Here we consider the locus of complex algebraic numbers of the third degree (cubic numbers). The variant, in which all three roots of a polynomial P3 (z) are real numbers, is of no interest. For definiteness, we will assume that z1 and z2 are complex conjugate numbers, and z3 is a real number. Under these conditions, the polynomial P3 (z) can be represented b y Eq. 10.10. P3 (z) = z 3 + c1 z 2 + c2 z + c3 = (z − z 1 )(z − z 2 )(z − z 3 ) = (z − Re z 1 − j Im z 1 )(z − Re z 2 + j Im z 2 )(z − z 3 )

(10.10)

From Eq. 10.6, it follows that ⎧ ⎪ ⎨ c1 = −2 Re z 1 − z 3 , c2 = Re2 z 1 + Im2 z 1 + 2z 3 Re z 1 , ⎪ ⎩ c3 = −z 3 Re2 z 1 − z 3 Im2 z 1 .

(10.11)

Applying the notation of Eq. 10.8, we get: ⎧ ⎪ ⎨ c1 = −2x − z 3 , c2 = x 2 + y 2 + 2z 3 x, ⎪ ⎩ c3 = −z 3 x 2 − z 3 y 2 .

(10.12)

There are three variants of the solution of Eqs. 10.12 to derive the equations of plane curves, points which are the roots of the polynomial P3 (z): 1. We exclude the variables y2 , c1 , and z3 , solving the system Eq. 10.12 relative to them. After the elementary transformations of the resulting expression for y2 , one can obtain the equation:

x 2 + y2

2

 − c2 x 2 + y 2 − 2c3 x = 0.

(10.13)

As you can see, this equation describes a plane fourth-order algebraic curve (quartics).

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1

c2=1

0.8

c2=0.5

0.6

c2=-1 c2=-1.5 c2=-2 c2=-2.5

-0.2

c2=-0.5

c2=0

0

c2=0

Im z

0.2

c2=-1 c2=-0.5

0.4

c2=3

-0.4 -0.6

c2=0.5

-0.8

c2=1

-1 -1

-0.8

-0.6

-0.4

-0.2

0

Re z

0.2

0.4

0.6

0.8

1

Fig. 10.2 Planar algebraic curves described by Eq. 10.15 with c1 = –0.75

2. We exclude the variables y2 , c2 , and z3 , solving the system Eq. 10.12 relative to them. After the elementary transformations of the resulting expression for y2 , one can obtain the equation:

 x 2 + y 2 (2x − c1 ) + c3 = 0.

(10.14)

This equation describes plane algebraic curves of the third degree (cubics). 3. We exclude the variables y2 , c3 , and z3 , solving the system Eq. 10.12 relative to them. After the elementary transformations of the resulting expression for y2 , one can obtain the equation: y 2 = 3x 2 − 2c1 x − c2 .

(10.15)

This equation describes plane algebraic curves of the second degree (conics). To describe the position of the roots of the polynomial P3 (z), you can choose any variant. As the third variant is the simplest, natural to focus on it. There are two ways to obtain a graphic image of the geometric place of the thirddegree algebraic numbers described by Eq. 10.15. The first way is to obtain a family of curves for different values of c2 with a fixed value of c1 .

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-0.2

c1=2

c1=-1.5

c1=-1.5

c1=3 c1=-3

c1=1.5

c1=1.5

Im z

0.2 0

c1=-1 c1=-2.5

c1=1 c1=2.5

0.4

c1=-2

c1=1

-0.4

c1=-1

-0.6 -0.8 -1 -1

-0.8

-0.6

-0.4

-0.2

0

Re z

0.2

0.4

0.6

0.8

1

Fig. 10.3 Planar algebraic curves described by Eq. 10.15 with c2 = –0.5

When implementing the second way, it is necessary to fix c2 and obtain a family of curves for different values of c1 . Examples based on the use of the first and second methods are presented in Figs. 10.2 and 10.3, respectively.

10.5 The Topography of the Fourth-Degree Algebraic Numbers in a Discretized Complex z-Plane Here we consider the locus of complex algebraic numbers of the fourth degree (quartic numbers). The variant, in which all three roots of a polynomial P3 (z) are real numbers, is of no interest. For definiteness, we will assume that z1 and z2 are complex conjugate numbers, and z3 is a real number. Under these conditions, the polynomial P4 (z) can be represented by Eq. 10.16. P4 (z) = z 4 + c1 z 3 + c2 z 2 + c3 z + c4 = (z − z 1 )(z − z 2 )(z − z 3 )(z − z 4 ) = (z − Re z 1 − j Im z 1 )(z − Re z 2 + j Im z 2 )(z − z 3 )(z − z 4 ) From Eq. 10.6, it follows that

(10.16)

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⎧ c1 ⎪ ⎪ ⎨ c2 ⎪ c ⎪ ⎩ 3 c4

= −2 Re z 1 − (z 3 + z 4 ), = Re 2 z 1 + Im2 z 1 + 2 Re z 1 (z 3 + z 4 ) + z 3 z 4 , 2 =− z 1 + Im2 z1 (z 3 + z 4 ) − 2 Re z 1 (z 3 z 4 ),

Re 2 = Re z 1 + Im2 z 1 z 3 z 4 .

131

(10.17)

Applying the notation Eq. 10.8, we get ⎧ c1 ⎪ ⎪ ⎨ c2 ⎪ c ⎪ ⎩ 3 c4

= −2x − u,

= x 2 + y 2 +  2ux + v, 2 2 =− x + y  u − 2vx,

= x 2 + y 2 v,

(10.18)

where

u = z3 + z4, v = z3 z4.

(10.19)

It is possible to construct families of curves that are a geometric place of algebraic numbers of the fourth degree in four ways:

Fig. 10.4 Planar algebraic curves described by Eq. 10.23 with c2 = 1, c3 = 2

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Fig. 10.5 Planar algebraic curves described by Eq. 10.23 with c1 = 1, c3 = 2

1. Exclusion of variables y2 , c1 , u, and v from Eqs. 10.18 leads to the equation for y2 :



 x 2 + y 2 4x 2 y 6 + (2c3 x − c4 )y 4 + c2 c4 y 2 − c42 − c4 = 0.

(10.20)

This equation describes a tenth-order plane algebraic curve. 2. Exclusion of variables y2 , c2 , u, and v from Eqs. 10.18 leads to the equation for y2 :

 (c1 + 2x)y 4 + 2x 2 c1 + 4x 3 − c3 − 1 y 2 + c1 x 4 − c3 x 2 − 2c4 x = 0. (10.21) This equation describes a fifth-order plane algebraic curve. 3. Exclusion of variables y2 , c3 , u, and v from Eqs. 10.18 leads to the equation for y2 :

 y 4 − 4x 2 + 2c1 x + c2 − 1 y 2 − 3x 2 − 2c1 x + c4 = 0. This equation describes a fourth-order plane algebraic curve.

(10.22)

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4. Exclusion of variables y2 , c4 , u, and v from Eqs. 10.18 leads to the equation for y2 : (c1 + 4x)y 2 − 4x 3 − 3c1 x 2 − 2c2 x − c3 = 0.

(10.23)

This equation describes a third-order plane algebraic curve. This option is the simplest, so it will be used to describe the z-plane topography. If we fix the values of the coefficients c2 and c3 , then we obtain a family of curves for different values of c1 . Figure 10.4 shows the corresponding example. Fixing the coefficients c1 and c3 allows us to calculate a family of curves for different values of the coefficient c2 . An example is shown in Fig. 10.5. And, finally, a family of curves for varying values of c3 will be obtained by fixing c1 and c2 . An example of this option is illustrated in Fig. 10.6.

Fig. 10.6 Planar algebraic curves described by Eq. 10.23 with c1 = 2, c2 = 3

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10.6 Conclusions This work continues the cycle of publications devoted to the new approach developed by the authors to the design of IIR digital filters. In this approach, the finite word length is taken into account when calculating zeros and poles even before the stage of structural synthesis. In other words, structural synthesis does not distort the results of functional synthesis. The possibility of this is based on considering the numbertheoretic nature of the zeros and poles of IIR filters with a finite bitness of the coefficients. It is established that in this case the zeros and poles are elements of the set of algebraic numbers. Therefore, all possible positions of zeros and poles in the z-plane form a certain discrete structure. The discrete nature of the allowed location of algebraic numbers of the second degree in the z-plane has long been known. The authors in their works studied in detail the corresponding topography. In 2019, the authors finally managed to obtain equations describing the topography of the third-degree algebraic numbers. Here, these results are extended to the fourth-degree algebraic numbers. Future work in this domain will consist in determining the topography of algebraic numbers of arbitrary degree in the z-plane. The results will be used in the development of software that implements the currently being elaborated approach to the design of IIR digital filters with a finite word length. Acknowledgements The work was supported by a grant from the Russian Foundation for Basic Research 18-07-00986.

References 1. 2. 3. 4.

5.

6.

7.

8.

Losada, R.A.: Digital Filters with MATLAB® . The MathWorks, Inc. (2008) Signal Processing Toolbox™: User’s Guide. The MathWorks, Inc. (2017) DSP System Toolbox™: Getting Started Guide. The MathWorks, Inc. (2017) Lesnikov, V., Chastikov, A., Naumovich, T., Armishev, S.: A new paradigm in design of IIR digital filters. In: 8th IEEE East-West Design and Test Symposium, St. Petersburg, Russia, pp. 282–285 (2010) Lesnikov, V., Chastikov, A., Naumovich, T., Armishev, S.: Implementation of a new paradigm in design of IIR digital filters. In: 8th IEEE East-West Design and Test Symposium, St. Petersburg, pp. 156–159 (2010) Lesnikov, V., Naumovich T.: Number-theoretic and algebraic aspects of structural synthesis of digital filters. In: International Embedded Solutions Event (The Embedded Signal Processing Conference), Santa Clara, USA, pp. 1374.1–1374.4 (2004) Lesnikov, V., Naumovich, T., Chastikov, A.: Topography of z-plane which is discretized due to quantization of coefficients of digital biquad filters. In: 12th International Siberian Conference on Control and Communications, Moscow, Russia, pp. 1–4 (2016) Lesnikov, V., Naumovich, T., Chastikov, A.: The sampling of the z-plane due to the quantization of the digital filter coefficients. In: 7th Mediterranean Conference on Embedded Computing. Budva, Montenegro, 10–14 June 2018, 4p. (2018)

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9. Lesnikov, V., Naumovich, T., Chastikov, A.: Number-theoretical analysis of the structures of classical IIR digital filters. In: 7th Mediterranean Conference on Embedded Computing. Budva, Montenegro, 4p. (2018) 10. Lesnikov, V., Naumovich, T., Chastikov, A.: The topography of a third order IIR digital filter zeros and poles in the z-plane discretized due to the quantization of the direct form coefficients. In: 8th Mediterranean Conference on Embedded Computing. Budva, Montenegro, 4p. (2019) 11. Schlichthärle, D.: Digital Filters: Basics and Design. Springer, Berlin, Heidelberg (2011) 12. Weinstein C.J.: Quantization effects in digital filters. Technical Report 468, Lincoln Laboratory, Massachusetts Institute of Technology, Lexington, Massachusetts (1969) 13. Oppenheim, A.V., Weinstein, C.J.: Effects of finite register length in digital filtering and the fast Fourier transform. Proc. IEEE 60(8), 957–976 (1972) 14. Rahman, M.H., Fahmy, M.M.: On pole distribution in digital filters. Int. J. Circuit Theory Appl. 3(1), 95–100 (1975) 15. Yli-Kaakinen, J., Saramaki, T.: An efficient algorithm for the design of lattice wave digital filters with short coefficient wordlength. In: 32nd IEEE International Symposium on Circuits and Systems VLSI, Orlando, FL, USA, vol. 3, pp. 443–448 (1999) 16. Yli-Kaakinen, J., Saramaki, T.: An algorithm for the design of multiplierless approximately linear-phase lattice-wave digital filters. In: 33rd IEEE International Symposium on Circuits and Systems, Geneva, Switzerland, vol. 2, pp. 77–80 (2000) 17. Saramäki, T., Yli-Kaakinen, J.: Design of digital filters and filter banks by optimization: Applications. In: 10th European Signal Processing Conference, Tampere, Finland, pp. 1–32 (2000) 18. Yli-Kaakinen, J., Saramaki, T.: A systematic algorithm for the design of lattice wave digital filters with short-coefficient wordlength. IEEE Trans. Circ. Syst. I: Regular Pap. 54(8), 1838– 1851 (2007) 19. Hilbert, D.: The Theory of Algebraic Number Fields. Springer, Berlin, Heidelberg, New York (1998)

Chapter 11

Method for Adaptive Control of Technical States of Radio-Electronic Systems Pavel A. Budko, Alexey M. Vinogradenko, Alexey V. Mezhenov and Nina G. Zhuravlyova Abstract At present, monitoring of technical state of complicated technical objects under different attacks and destabilizing factors, aging, and dispersion of technological parameters is a crucial problem. Requirements to the quality, security, and reliability of complicated technical systems are consistently increased. In this chapter, we propose new method for adaptive control of technical states of the radio-electronic systems. This approach is based on the interval complex estimation of parameters, use of knowledge base of the critical and regular states, and also inner connections between the controlled parameters considering the false negative and false positive ratios. Multidimensional presentation of technical state of the controlled systems is possible using the accurate monitoring of technical states of the radioelectronic systems, increased accuracy and reliability of state identification, and extended possibilities of control and diagnostic equipment. Keywords Parameter estimation · Control · Radio-electronic system · Technical state · Knowledge base · Complex parameter evaluation

P. A. Budko (B) · A. M. Vinogradenko · A. V. Mezhenov Public Joint Stock Company Information and Telecommunication Technologies, 8, Kantemirovskaya ave., Sankt-Petersburg 197342, Russian Federation e-mail: [email protected] A. M. Vinogradenko e-mail: [email protected] A. V. Mezhenov e-mail: [email protected] N. G. Zhuravlyova V. A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences (ICS RAS), 65, Profsoyuznaya str, Moscow 117997, Russian Federation e-mail: [email protected] © Springer Nature Switzerland AG 2020 M. Favorskaya and L. C. Jain (eds.), Advances in Signal Processing, Intelligent Systems Reference Library 184, https://doi.org/10.1007/978-3-030-40312-6_11

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11.1 Introduction Radio-Electronic Equipment (REE) is a part of modern technological systems in different industries of manufacturing electronics and electrical equipment. The importance and relative volume of REE in technical systems increases continuously that demands a creation of effective control and diagnostic equipment applied at the test and exploitation stages. Nondestructive control is especially important for uninhabited objects in remote territories, where execution of the control and diagnostic functions of REE by staff is impossible. At the same time, a volume of increased Measuring Information (MI) during the control of the remote and territorial distributed objects requires the reduction and reliability of the transmitted messages. New approaches of contactless diagnostics (for example, computer vision) are successfully developed. These approaches are applied in information and measuring systems, telemetry systems, and autonomous automated control systems. They are implemented on the basis of contactless ways of MI exchange, digital signal processing, etc. Functions of telemetry systems are the following [1, 2]. First, MI entering from control objects is collected and processed by one or several peripheral control elements. Second, after MI delivering through communication channels, its full processing is implemented, and resulting outputs are provided to the staff managing Radio-Electronic Systems (RES). Transitions from pre-emergency Technical State (TS) of REE (TS-REE) to the accidents are not allowed. This requires the on-line collection of diagnostic information of control objects. At the same time, the tasks of TS estimation and identification of failure location in REE [3–5] are solved. The special properties are as follows: • Volume of processed MI is increased. • It is required to process several MI streams under the restricted resources of standard control elements. • Necessary control and measuring information, as well as, specialists for the analysis of REE technical condition may by unavailable sometimes in control centers. This requires a development of new approaches for MI assessment in information and measuring systems, telemetry systems, and autonomous automated control systems. In order to eliminate the redundancy of measuring information under completeness of object control through communication channels, it is reasonable to transmit not all measuring results but only messages about an exit of object parameters outside of the limits of the set admissions. The systems realizing such method of collecting telemetric data are called the adaptive systems of prestart control [6]. Achievement by the controlled parameter of threshold level in random time instant is an event initializing emergency signal [7]. In this case, the outlier value of parameter over threshold level is also a random value. Conventional telemetry systems are divided into the Remote Signaling Systems (RSS) and Telemetry Measurement Systems (TMS). However, we propose a complex

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model of emergency signal processing. It includes the integration of the existing classes of systems as follows: the emergency signal is formed only in the case of exceeding by controlled parameter x the predefined threshold level x T h (as in RSS) with the subsequent measurement (as in TMS) of the emission value above the threshold [6]. In the integrated system, random variables are the time instances ti of emergency signals and levels of these signals u i . For control of TS-REE, the approach based on integration of measuring information entering from a group of polytypic sensors is offered. Such approach promotes to increase an informational content of emergency monitoring, as well as, the reliability and accuracy in terms of False Positive Ratio (FPR) and False Negative Ratio (FNR). The modelled representation of REE failures consists in the following: 1. Complex use of the polytypic sensors (for example, sensors of temperature, tension of magnetic field, and tension and humidity of air). 2. Processing of emergency signals in control systems integrating the properties of RSS and TMS. 3. Mistakes check of FPR and FNR. In this research, the new adaptive control method of autonomous REE, which is based on estimation of their TS, is presented. It occurs by means of integration of critical results of measurements of parameters (emissions) by various sensors, ellipsoidal approximation of area of working capacity of Controlled Object (CO) and formation of multidimensional failure. At the same time, FPR and FNR values are considered that allows to eliminate a redundancy of MI and also to increase reliability and accuracy of control. Chapter has the following structure. Section 11.2 includes the analysis of works in the field of monitoring and estimation (recognition) of TS-REE types. Problem definition and a method of the decision are given in Sect. 11.3. Section 11.4 considers the mechanisms of functioning of the offered adaptive control system and results of researches. Conclusions and further research directions are given in Sect. 11.5.

11.2 Related Work Analysis of the works [1, 2, 5, 8] shows that in order to ensure the effective functioning of REE while reducing the cost of their life cycle, it is necessary to introduce tools and methods for automated monitoring and diagnostics of TS. It is also necessary to use effective methods and means of ensuring the safety and reliability of the operation of REE. At the same time, the specifics of functioning of TS-REE with accounting of their operating modes rely on to use of nondestructive ways of control and diagnostics. These methods are used in various branches of industrial electronics and electrical equipment [1–5]. Their disadvantage is the high probability of failure during exploitation. This is due to the fact that the control thresholds are assigned without

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taking into account the general state of the communication system and the load level of the buffer devices in the switching nodes. These conditions cause the locks in the loaded network, rather low productivity and high coefficient of downtime. It is caused by the fact that for control of complex technical systems and identification of their state it is necessary to perform the measurement, transformation, and processing of a large number of parameters during an equipment downtime. Among the existing ways of monitoring, the control of TS focused on the reliability of CO providing adjusting and forecasting is the most optimum strategy, in other words the preventive and forecasting control methods [9]. Thus, in [10] TS of the controlled objects is characterized by comparison of an emergency with the reference table of conditions of emergence of malfunction. Along with the known methods of estimation [3–5, 11–16], the predetermine REE models are offered in [17]. They have properties of nonlinearity and multiconnectivity and also allow to build the adaptive control algorithms with identification or reference model. Predetermination of the assessment of TS-REE in [2] is carried out due to the comparison of the measured value and values of the predetermined weight coefficients characteristic of the controlled equipment. However, in many cases the prior information is not enough for the implementation of this or that assessment of TS-REE, and selection of a posteriori data is small for statistical conclusions. In these conditions, the methods of statistical classification [13], theory of neural networks [4], and intelligent agent systems [14] provide enough reliable results. These methods have the merits and demerits, which are used for TS assessment and forecasting of changes of REE controllable parameters. Thus, in [15] for control a robotic hand the neural network with back-propagation is used. In [16], it is offered to use the positioning ontology, which models the spatial and temporal relationship between the observations from different sensors for assessment of a condition of Internet of robotic things’ elements. This says about the efficiency of the adaptive methods for REE technical estimation. Monitoring of autonomous objects is, as a rule, characterized by the automated wireless exchange of MI [18] that allows to reduce considerably a time resource and participation of the person. At the same time, a configuration of the system of continuous monitoring allows to carry out adjustment of the parameters, by response to sudden fluctuations in a state of CO or sharp changes in resource requirements [19]. An alternative to the aforementioned methods is the collection and processing of MI implemented in multi-level monitoring systems as TS of RES (TS-RES), in which the collection and processing of MI is based on its comprehensive assessment. And MI collection stage is presented in the form of transmission and processing of data on the output of object parameters beyond the specified tolerances This stage promotes to eliminate a redundancy of MI due to integration of RSS and TMS. Such integration in a complex with classification of emergencies and errors’ check as FPR and FNR values promotes to increase the productivity (efficiency) of control system. In general, the conducted researches in the field of TS-RES control and recognition of failures’ types with their forecasting are characterized by wide range of approaches in the subject domain.

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11.3 Mathematical Backgrounds: Representation of Control Results For implementation of TS-REE assessment, it is offered to use a working capacity, which dimension is defined by the number of output parameters. TS-REE is defined by parameters and characteristic of concrete type of the equipment within admissions. Detailed representation of TS-REE information requires the complex accounting of characteristic signals and errors during transform, receiving, and processing of MI. Hereinafter, a complex nature of the transmitted signals is discussed in Sect. 11.3.1. Creation of three-dimensional MI model at the touch level is considered in Sect. 11.3.2. Section 11.3.3 provides a formation of an ellipsoid of working capacity considering FPR and FNR errors.

11.3.1 Complex Nature of the Transmitted Signals During TS-REE monitoring at the stage of MI collecting for controlled parameters’ estimation, it is offered to use a way of complex statistical control of TS-REE, which has to meet the following conditions: 1. Integration of sensors’ indications so that the signals characterizing TS of CO by one parameter, “invisible” to one type of sensors, will be identified by sensors of other type. 2. Detection of emergency signals using multidimensional statistical control (at various levels of a system). 3. Scopes’ extension of technical means for control and diagnostics. The complex nature of control consists in MI obtaining about TS-REE. It is based on universal parameters: temperature, electromagnetic response, humidity of air, tension, etc. MI obtaining is carried out from the sensors of temperature, the tension of magnetic field, humidity of air, the voltmeter, etc. Considering that MI transfer from sensors of the controlled equipment is carried out constantly during the normal operation of the equipment, its volume will be superfluous. For elimination of redundancy, it is necessary to involve one type of the sensors removing information at the time instant on the most critical for a certain type of the equipment and its operating mode to the parameters. In these time instants, when the controlled parameters go beyond allowable limits, information entering from sensors will confirm an emergency (pre-fault) condition of the equipment. In this case, it is important that the data entering from the each sensor supplemented each other, giving a full picture.

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On the basis of the statistical analysis of the measured parameters X i of several REE samples (for example, radio-electronic modules), we can establish a reliability range as an area of operating states DP , which represents an interval of dispersion values of the parameters (low X l and upper X u limits) corresponding to the operating state of RES in general.

11.3.2 Creation of Three-Dimensional MI Model at the Touch Level For combination of the indications of various types of sensors, a method based on a grid of emissions and Bayesian conclusion, modified for creation of threedimensional model of TS-REE on the basis of a surface of points is used. The measurements received from each sensor are presented in a form of a surface of points in three-dimensional space (Fig. 11.1), at the same time, each point r of a surface is presented by the following parameters: 1. Mathematical expectation of a point position in three-dimensional space r x , r y , rz . 2. Matrix of the covariance rδ setting dispersion of three-dimensional normal distribution of a point position. 3. Probability of a parametrical failure rr e f .

Fig. 11.1 Example of three-dimensional model of TS-RES assessment: a hypersurface of TS of control object, b quality ellipse, c entered ellipsoid, d described ellipsoid

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4. Probabilities of received measuring signals in the pre-critical conditions N of REE elements rdi , where i ∈ 1, . . . , N . MI received from the magnetic field sensor is presented in a form of several points with situation r x yz = Mr−1 Mn−1 v, where Mr is the matrix setting position of the sensor concerning the REE controllable element, Mn−1 is the matrix of TS dimension H × H, where H is the length of v of vectors (with coordinate z) the digital sequence, which represents h measurements of instant values obtained from the magnetic field sensor, h = 1,2,…, H. The results of measurements in a form of thermograms of REE controllable element are received from a temperature sensor. During analysis of control parameters of REE element, the results of measurements add to a surface of points in threedimensional space (Fig. 11.1). At the same time, the identification of possible violations of its TS stability on the basis of existence of non-random structures and use of borders of reliability range is implemented [3, 4]. The results of the voltage measurements are registered in a form of a separate point, for which the point interfaced by it is found with a sufficient measure of trust. Formulas of calculation of a point measurement in three-dimensional space are similar to the points of the magnetic field sensor. Information obtained from the sensor of air humidity can be presented in a form of color range of points of a surface (Fig. 11.1). It occurs at corresponding change of humidity in the field of a controlled element, which is characterized by an exit of the parameter (humidity) out of reliability range limits. Coordinates of points can be expressed as r x yz = Mr−1 Ml−1 R X H R X B , where R X H ,Y is the matrix of deviations of the controlled parameter to the lower bound of admission concerning axis Y, R X B ,Z is the matrix of deviations of controlled parameter to the upper bound of admission concerning axis Z of area of operating states DP . The covariance matrix in the global system of coordinates is set by expression rδ = Mr−1 Ml−1 R X H ,Y R X B ,Z Mδ . At the same time, a value of diagonal of matrix Mδ corresponding to axes X and Y in the local system of coordinates of all sensors are set proceeding from the chosen approximation step, and the value corresponding to axis Z is set proceeding from the accuracy of specific sensor. This model can be expressed mathematically as follows. Let employment of a point of r be a variable of a state Cell, which accepts one or the other values: Cell(r ) = {it is busy, free}. Emission is a signal of an exit of controlled parameters for reliability range borders (an accident signal) arriving from of the magnetic field sensors, tension, air humidity, and tension in some direction ϕ, is defined by the Cond variable: Cond(r, ϕ) = {accident signal, norm signal}. All possible directions ϕ (from 0° to 360°) breaks into q of the sites designated ϕi . Variable conditions of Cell and Cond unite through logical operation of following. Let’s consider the following assumptions:

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O : Cell(r ) = {it is busy), Ri : Cond(r, ϕ) = {emission}. Then O is expressed through Ri as follows: R1 ∨ R2 ∨ . . . ∨ Rn−1 ∨ Rn → O.

(11.1)

For probability definition that the point is occupied, Bayes method is used. For each point r, a validity of offer O is defined. As O treats R through logical following, the probability can be determined by Eq. 11.2. P(O) = P(R1 ∨ R2 ∨ . . . ∨ Rn )

(11.2)

Let’s find the probabilities of offers O and Ri : P(O) = 1 −



(1 − P(Ri )).

(11.3)

Equation 11.3 can be used for calculation of the probability that a point is occupied (an exit of controlled parameter for reliability range borders) if the probabilities of emissions P(Ri ) are known. For practical application, Eq. 11.3 can be written down on the basis of a formula of total probability concerning P(r ). Let’s apply the rule of Bayes for probability definition P(Ri /r ) on new measurement of controlled parameter r provided by Eq. 11.4, where P(Ri ) is the initial probability of the received signal of emission. P(Ri |r ) =

P(r |Ri )P(Ri ) P(r )

(11.4)

In Bayesian rule, it is usually accepted equal 1/2 as it is impossible to define initially if a point is busy or free. Complex model of a sensor. Probability P(r |Ri ) describes a sensor model. The sensor model determines a probability of receiving measurement of r if it is known that offer Ri is true. Sensor model of magnetic field sensor PN (r |Ri ) can be presented as follows. parameter Ustm f (tension Let X i be the random variable characterizing controlled  fixed by of the magnetic field sensor) and f Ustm f be the function of density of probability of stay X i in the reliability range. Then a sensor model for of the magnetic field sensor is defined by Eq. 11.5.  PN (r |Ri ) =

  0 if f Ustm f  < 0 1 if f Ustm f > 0

(11.5)

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At the registration of the controlled parameter r in this range, emission is formed. Taking into account fixing of violation of reliability range of the sensor at the exit of the controlled parameter from the range of admissions of two and more REE elements on the decisive device simultaneously, several signals of emissions can arrive at the same time, and several points in three-dimensional space are received, respectively. Generally, a number of such points is in the proportion to the measurements by the magnetic field sensor. Thus, a probability that a point displays a parameter exit in this direction of range of reliability is inversely proportional to the magnetic field. Sensor model of temperature sensor PT (r |Ri ) can be presented as follows. Let X¯ t = (x¯t1 . . . x¯tm )T be the vector of averages in the tth instant selections of measurements of temperature (t = 1, . . . , m), xi j be the average value in the tth instant selection in the parameter j. The main criterion of the violation of process stability is an exit of controlled parameter for the threshold level (for reliability range borders). Then a sensor model for the temperature sensor PT (r |Ri ) has a view of Eq. 11.6.  PT (r |Ri ) =

1 if Tt < Tthr eshold 0 if Tt > Tthr eshold

(11.6)

The sensor model of tension PU (r |Ri ) can be described by the expression similar to of the magnetic field sensor model provided by Eq. 11.7, where Umeasur ed is the measured value tension sensor.  0 if f (Umeasur ed ) < 0 PU (r |Ri ) = (11.7) 1 if f (Umeasur ed ) > 0 Sensor model PV (r |Ri ) for the air humidity sensor to within values of probability can be presented as follows. Let s be the color range corresponding to the range of change of the humidity of air in the field of the controlled element (Fig. 11.1a), which is in various states, and f δ (s) be a function of density of probability of a range of s. Then a model of air humidity sensor has a view of Eq. 11.8. ⎧ if f δ (s) < ε ∧ s < r ⎨ 0.3 PV (r |Ri ) = 0.5 if f δ (s) < ε ∧ s ≤ r ⎩ 0.95 f δ (s) if f δ (s) ≥ ε

(11.8)

Formation of a complex signal of technical MI. As a rule, TS-REE is characterized not only by one parameter, but a whole group of parameters. The existing method, which assigns the independent intervals to each parameter separately, does not allow to consider the correlation of REE parameters. For accounting of correlation of parameters and, as a result, increase of control reliability, the area in parameters’ space, in which the values of controlled parameters are defined with a set probability, is offered to consider. Therefore, at assessment of TS-REE is offered to use an area

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of working capacity DP , which dimension is defined by a number of the output parameters characterizing TS-REE in general [4, 5]. Influence of the external indignations, which exact properties are unknown and cannot be described by simple ratios, leaves a mark on area of working capacity DP , washing away its borders. In this regard, values of REE controllable parameters are always defined with a final error. Taking into account the normal distribution of real measurements in the presence of FPR and FNR errors, in which values of the measured output parameters of CO are defined with a set probability, is expedient to approximate an area of working capacity as an ellipsoid [13, 20, 21] (Fig. 11.1c, d).

11.3.3 Formation of an Ellipsoid of Working Capacity Considering FPR and FNR Errors Control of REE separate parameters without their interrelations does not provide the required size of control reliability or excessively overestimates the operational indicators. At the same time, numerous false signals of an accident are possible [6–8, 22]. For control of CO parameters using several correlated indicators, the multidimensional methods of the statistical analysis are used [3–5]. It assumes a correction of CO parameters values during its operation by selective control. This procedure is necessary for maintenance of statistically operated and stable process of CO work. However, at the same time there is no accounting of FPR and FNR errors. In the modelled monitoring system, at a deviation of the controlled parameters of REE elements, a comparison of parameters with threshold values within DP is made. By the results of comparison, a normal state of REE is defined using a probability p1 , and its abnormal state is described by a probability p2 . And TS-REE recognition (critical condition) is carried out taking into account FPR (α) and FNR (β) errors. Minimization of a probability α comes down to creation of the entered rectangle Bv of the maximum area (Fig. 11.1b) using a method of diagonals. According to this method, the top of the entered parallelepiped (Fig. 11.1d) are in a point of intersection of diagonals of the described parallelepiped with an ellipsoid (Figs. 11.1c and 11.2). For this purpose, the values of the entered parallelepiped (Fig. 11.1d) corresponding to the admissions on parameters of RES elements are used under zero probability of error β and minimum possible of error α. Ensuring the required correlation α/β between FPR and FNR errors for approximating the region DP of admissible parameter values (the hatched described ellipsoid in Fig. 11.1c, d) determines the tolerances for the parameters. With these tolerances, a certain probability of an undetected failure of REE elements is preliminarily set or the cost of the control and monitoring system is minimized, when the established requirements for TS indicator of CO (quality of operation) are implemented. Thus, a set of the controlled parameters, which are in limits of admissions at ellipsoidal approximation in a complex case, will represent a characteristic form

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Fig. 11.2 Ellipsoid of working capacity (informational content)

of an ellipsoid (Fig. 11.2). Its dimension (parameters) will also be written down in memory elements. If a deviation of parameters is outline from the norm values, the proportion of a figure will change, and the dynamics of the changing parameter will be characterized by a color range from violet (normal state) to red (critical condition). Complex use of MI received from the diverse sources is presented in threedimensional space as the ellipsoidal approximation. It is applicable for the solution of problems of TS control in the conditions of uncertainty. Such approach promotes to increase MI reliability under a condition of the observed objects in the systems of monitoring, expansion a scope of technical means of control and diagnostics, and also eliminate MI redundancy at the stages of its transfer. In general, it promotes increase the efficiency of control process.

11.4 Implementation of TS-RES Adaptive Control Method For definition of the stages of TS-RES control, we define the following basic data: – List of the controlled parameters (temperature, tension, of the electromagnetic field strength, humidity of air, etc.). – Frequency of the poll of RES sensors. – Greatest possible dynamics (frequency, speed) of the controlled parameter change. MI RES is received when functioning is necessary to transfer information at remote dispatch center for the analysis and final definition of TS-RES. The solution of a problem of TS-RES control is presented in the form of the sequence of the following stages (procedures). Main stages of control of TS-RES are the following: I. Preprocessing of MI on the remote terminal: Stage 1. Scan poll controllers (servers) of sensors on the RES elements.

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Stage 2. Creation of knowledge base (statistical data) using the controlled parameters. Stage 3. Primary estimation of the values of the received group of signals (remote signaling) from the RES controllable element: appraisal evaluating degradations of the controlled parameter (application). II. Processing of MI at remote dispatch center: Stage 4. Secondary complex estimation of the received group of signals of RSSTMS: formation of a multidimensional complex image of TS-RES (in the most critical parameters) and comparison of the received image with reference values of images of signals from the knowledge base; Stage 5. Identification of TS-RES taking into the account of FPR and FNR errors. In the case of exit of different controlled parameters of objects out of the allowable limits, the critical signal in the sensor located directly on elements of objects is formed. In the existing systems of the telemetry, each parameter of an object is controlled with the period T0 irrespective of its speed of change. However, if the speed of change of the separate parameters is increased, they can reach the permissible values in a time smaller than the fixed period T0 . In this case, the control system will not be able to react in due time to inadmissible changes of parameter that will lead to failure of a controlled object. For efficiency of TS control and subsequent assessment of parameter, a frequency of proportional speed of change is calculated. Depending on exit speed (time t1 , t2 achievement of permissible value), the controlled parameter U out of the allowable limits the priority of a signal is defined. This is set due to a multi-level tolerance system (the higher the tolerance level, the higher the priority of the service request). An autonomous electrical installation (power plant) was used as the control object, and the proposed approach was used for TS monitoring using the internal parameters (output voltage, temperature of heating of the generator (anchor), air humidity, and tension of the electromagnetic field). Fixed deviations of the controlled parameters are displayed in the form of an ellipsoid of various color scale. It allows to display current TS of the controlled object compared to the reference ellipsoid. Results of control of air humidity on RES controllable element are displayed in Fig. 11.3. Given the availability of any emergency option in the knowledge base based on the results of such a comparison, the current TS of RES will be identified. In general, multidimensional representation of CO failure with use of the offered approach shows a considerable prize in reliability of MI and eliminate of its redundancy. Thus, the results of the conducted researches show that complex idea of MI of TS-RES is increased by reliability and informational content of the obtained resultant information. Further development of researches on complex estimation of TS of autonomous RES is in numeric experiments on the basis of methods for simulation modeling, for example, Any Logic programming environment. Use of this software will allow to consider any aspect of the modelled system with any level of specification. The

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Fig. 11.3 Informational content ellipsoid: a TS of the measured sample, b “reference ellipsoid”

graphic AnyLogic interface, tools, and libraries will allow to create models for a wide range of tasks before technical implementation of TS-RES monitoring system.

11.5 Conclusions The results of a research show that for implementation of RES parameters monitoring, especially, uninhabited autonomous objects (for example, artificial Earth satellites) various instruments of control can be used. They provide the higher identification reliability and sensitivity for detection of emergency situations. The approach of complex control of TS-RES presented in the chapter on the basis of integration of indications of several types of sensors can be used for creation of the universal automated system for control of uninhabited autonomous objects of technological systems. Such control system includes the systems of computer vision and allows to estimate an operability in the wide nomenclature of REE with high reliability. Complex representation of MI taking into account its transfer at integration of RSS and TMS promotes decrease in redundancy of MI in the monitoring system and increase in reliability and accuracy of TS-RES estimation. In general, the offered approach will allow to support a decision-making in the on-line control systems of TS-RES on elimination of critical conditions. Acknowledgements The research is executed with financial support of the Russian Foundation for Basic Research within the scientific project No. 16-29-04326 ofi_m.

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