Complex Valued Nonlinear Adaptive Filters

This book was written in response to the growing demand for a text that provides a unified treatment of linear and nonlinear complex valued adaptive filters, and methods for the processing of general complex signals (circular and noncircular). It brings together adaptive filtering algorithms for feedforward (transversal) and feedback architectures and the recent developments in the statistics of complex variable, under the powerful frameworks of CR (Wirtinger) calculus and augmented complex statistics. This offers a number of theoretical performance gains, which is illustrated on both stochastic gradient algorithms, such as the augmented complex least mean square (ACLMS), and those based on Kalman filters. This work is supported by a number of simulations using synthetic and real world data, including the noncircular and intermittent radar and wind signals.

Autorentext

Danilo Mandic, Department of Electrical and Electronic Engineering, Imperial College London, London
Dr Mandic is currently a Reader in Signal Processing at Imperial College, London. He is an experienced author, having written the book Recurrent Neural Networks for Prediction: Learning Algorithms, Architectures and Stability (Wiley, 2001), and more than 150 published journal and conference papers on signal and image processing. His research interests include nonlinear adaptive signal processing, multimodal signal processing and nonlinear dynamics, and he is an Associate Editor for the journals IEEE Transactions on Circuits and Systems and the International Journal of Mathematical Modelling and Algorithms. Dr Mandic is also on the IEEE Technical Committee on Machine Learning for Signal Processing, and he has produced award winning papers and products resulting from his collaboration with industry. Su-Lee Goh, Royal Dutch Shell plc, Holland
Dr Goh is currently working as a Reservoir Imaging Geophysicist at Shell in Holland. Her research interests include nonlinear signal processing, adaptive filters, complex-valued analysis, and imaging and forecasting. She received her PhD in nonlinear adaptive signal processing from Imperial College, London and is a member of the IEEE and the Society of Exploration Geophysicists.

Klappentext
The filtering of real world signals requires an adaptive mode of operation to deal with the statistically nonstationary nature of the data. Feedback and nonlinearity within filtering architectures are needed to cater for long time dependencies and possibly nonlinear signal generating mechanisms. Using the authors' original research and current established methods, this book covers the foundations of standard complex adaptive filtering and offers next generation solutions for the generality of complex valued signals. It provides a rigorous treatment of complex noncircularity and nonlinearity, thus avoiding the deficiencies inherent in several mathematical shortcuts typically used in the treatment of complex random signals. Simulations for both circular and noncircular data sources are includedfrom benchmark models to real world directional processes such as wind and radar signals. Key features:

• Provides theoretical and practical justification for converting many apparently real valued signal processing problems into the complex domain;
• Offers a unified approach to the design of complex valued adaptive filters and temporal neural networks, based on augmented complex statistics and the duality between the bivariate and complex calculus (CR calculus);
• Introduces augmented filtering algorithms based on widely linear models, making them suitable for processing both second order circular (proper) and noncircular (improper) complex signals;
• Covers adaptive stepsizes, dynamical range reduction, validity of complex representations, and data driven timefrequency decompositions;
• Includes extensive background material in appendices ranging from the theory of complex variables through to fixed point theory.
  Complex valued signals play a central role in the fields of communications, radar, sonar, array, biomedical and environmental signal processing amongst others. This book will have wide appeal to researchers and practising engineers in these and related disciplines, and can also be used as lecture material for a course on adaptive filters.

Inhalt

Preface xiii

Acknowledgements xvii

1 The Magic of Complex Numbers 1

1.1 History of Complex Numbers 2

1.2 History of Mathematical Notation 8

1.3 Development of Complex Valued Adaptive Signal Processing 9

2 Why Signal Processing in the Complex Domain? 13

2.1 Some Examples of Complex Valued Signal Processing 13

2.2 Modelling in C is Not Only Convenient But Also Natural 19

2.3 Why Complex Modelling of Real Valued Processes? 20

2.4 Exploiting the Phase Information 23

2.5 Other Applications of Complex Domain Processing of Real Valued Signals 26

2.6 Additional Benefits of Complex Domain Processing 29

3 Adaptive Filtering Architectures 33

3.1 Linear and Nonlinear Stochastic Models 34

3.2 Linear and Nonlinear Adaptive Filtering Architectures 35

3.3 State Space Representation and Canonical Forms 39

4 Complex Nonlinear Activation Functions 43

4.1 Properties of Complex Functions 43

4.2 Universal Function Approximation 46

4.3 Nonlinear Activation Functions for Complex Neural Networks 48

4.4 Generalised Splitting Activation Functions (GSAF) 53

4.5 Summary: Choice of the Complex Activation Function 54

5 Elements of CR Calculus 55

5.1 Continuous Complex Functions 56

5.2 The CauchyRiemann Equations 56

5.3 Generalised Derivatives of Functions of Complex Variable 57

5.4 CR-derivatives of Cost Functions 62

6 Complex Valued Adaptive Filters 69

6.1 Adaptive Filtering Configurations 70

6.2 The Complex Least Mean Square Algorithm 73

6.3 Nonlinear Feedforward Complex Adaptive Filters 80

6.4 Normalisation of Learning Algorithms 85

6.5 Performance of Feedforward Nonlinear Adaptive Filters 87

6.6 Summary: Choice of a Nonlinear Adaptive Filter 89

7 Adaptive Filters with Feedback 91

7.1 Training of IIR Adaptive Filters 92

7.2 Nonlinear Adaptive IIR Filters: Recurrent Perceptron 97

7.3 Training of Recurrent Neural Networks 99

7.4 Simulation Examples 102

8 Filters with an Adaptive Stepsize 107

8.1 Benveniste Type Variable Stepsize Algorithms 108

8.2 Complex Valued GNGD Algorithms 110

8.3 Simulation Examples 113

9 Filters with an Adaptive Amplitude of Nonlinearity 119

9.1 Dynamical Range Reduction 119

9.2 FIR Adaptive Filters with an Adaptive Nonlinearity 121

9.3 Recurrent Neural Networks with Trainable Amplitude of Activation Functions 122

9.4 Simulation Results 124

10 Data-reusing Algorithms for Complex Valued Adaptive Filters 129

10.1 The Data-reusing Complex Valued Least Mean Square (DRCLMS) Algorithm 129

10.2 Data-reusing Complex Nonlinear Adaptive Filters 131

10.3 Data-reusing Algorithms for Complex RNNs 134

11 Complex Mappings and M¨obius Transformations 137

11.1 Matrix Representation of a Complex Number 137

11.2 The M¨obius Transformation 140

11.3 Activation Functions and M¨obius Transformations 142

11.4 All-pass Systems as M¨obius Transformations 146

11.5 Fractional Delay Filters 147

12 Augmented Complex Statistics 151

12.1 Complex Random Variables (CRV) 152

12.2 Complex Circular Random Variables 158

12.3 Complex Signals 159

12.4 Second-order Characterisation of Complex Signals 161

13 Widely Linear Estimation and Augmented CLMS (ACLMS) 169

13.1 Minimum Mean Square Error (MMSE) Estimation in C 169

13.2 Complex White Noise 172

13.3 Autoregressive Modelling in C 173

13.4 The Augmented Complex LMS (ACLMS) Algorithm 175

13.5 Adaptive Prediction Based on ACLMS 178

**14 Duality Between Complex Valued and Real Valued F…