Acknowledgements xvii

1 The Magic of Complex Numbers 1

1.1 History of Complex Numbers 2

1.1.1 Hypercomplex Numbers 7

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

2.1.1 Duality Between Signal Representations in R and c 18

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

2.3 Why Complex Modelling of Real Valued Processes? 20

2.3.1 Phase Information in Imaging 20

2.3.2 Modelling of Directional Processes 22

2.4 Exploiting the Phase Information 23

2.4.1 Synchronisation of Real Valued Processes 24

2.4.2 Adaptive Filtering by Incorporating Phase Information 25

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

3.2.1 Feedforward Neural Networks 36

3.2.2 Recurrent Neural Networks 37

3.2.3 Neural Networks and Polynomial Filters 38

3.3 State Space Representation and Canonical Forms 39

4 Complex Nonlinear Activation Functions 43

4.1 Properties of Complex Functions 43

4.1.1 Singularities of Complex Functions 45

4.2 Universal Function Approximation 46

4.2.1 Universal Approximation in R 47

4.3 Nonlinear Activation Functions for Complex Neural Networks 48

4.3.1 Split-complex Approach 49

4.3.2 Fully Complex Nonlinear Activation Functions 51

4.4 Generalised Splitting Activation Functions (GSAF) 53

4.4.1 The Clifford Neuron 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 Cauchy–Riemann Equations 56

5.3 Generalised Derivatives of Functions of Complex Variable 57

5.3.1 CR Calculus 59

5.3.2 Link between R- and C-derivatives 60

5.4 CR-derivatives of Cost Functions 62

5.4.1 The Complex Gradient 62

5.4.2 The Complex Hessian 64

5.4.3 The Complex Jacobian and Complex Differential 64

5.4.4 Gradient of a Cost Function 65

6 Complex Valued Adaptive Filters 69

6.1 Adaptive Filtering Configurations 70

6.2 The Complex Least Mean Square Algorithm 73

6.2.1 Convergence of the CLMS Algorithm 75

6.3 Nonlinear Feedforward Complex Adaptive Filters 80

6.3.1 Fully Complex Nonlinear Adaptive Filters 80

6.3.2 Derivation of CNGD using CR calculus 82

6.3.3 Split-complex Approach 83

6.3.4 Dual Univariate Adaptive Filtering Approach (DUAF) 84

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

7.1.1 Coefficient Update for Linear Adaptive IIR Filters 93

7.1.2 Training of IIR filters with Reduced Computational Complexity 96

7.2 Nonlinear Adaptive IIR Filters: Recurrent Perceptron 97

7.3 Training of Recurrent Neural Networks 99

7.3.1 Other Learning Algorithms and Computational Complexity 102

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

8.2.1 Complex GNGD for Nonlinear Filters (CFANNGD) 112

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

10.2.1 Convergence Analysis 132

10.3 Data-reusing Algorithms for Complex RNNs 134

11 Complex Mappings and Möbius Transformations 137

11.1 Matrix Representation of a Complex Number 137

11.2 The Möbius Transformation 140

11.3 Activation Functions and Möbius Transformations 142

11.4 All-pass Systems as Möbius Transformations 146

11.5 Fractional Delay Filters 147

12 Augmented Complex Statistics 151

12.1 Complex Random Variables (CRV) 152

12.1.1 Complex Circularity 153

12.1.2 The Multivariate Complex Normal Distribution 154

12.1.3 Moments of Complex Random Variables (CRV) 157

12.2 Complex Circular Random Variables 158

12.3 Complex Signals 159

12.3.1 Wide Sense Stationarity, Multicorrelations, and Multispectra 160

12.3.2 Strict Circularity and Higher-order Statistics 161

12.4 Second-order Characterisation of Complex Signals 161

12.4.1 Augmented Statistics of Complex Signals 161

12.4.2 Second-order Complex Circularity 164

13 Widely Linear Estimation and Augmented CLMS (ACLMS) 169

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

13.1.1 Widely Linear Modelling in c 171

13.2 Complex White Noise 172

13.3 Autoregressive Modelling in c 173

13.3.1 Widely Linear Autoregressive Modelling in c 174

13.3.2 Quantifying Benefits of Widely Linear Estimation 174

13.4 The Augmented Complex LMS (ACLMS) Algorithm 175

13.5 Adaptive Prediction Based on ACLMS 178

13.5.1 Wind Forecasting Using Augmented Statistics 180

14 Duality Between Complex Valued and Real Valued Filters 183

14.1 A Dual Channel Real Valued Adaptive Filter 184

14.2 Duality Between Real and Complex Valued Filters 186

14.2.1 Operation of Standard Complex Adaptive Filters 186

14.2.2 Operation of Widely Linear Complex Filters 187

14.3 Simulations 188

15 Widely Linear Filters with Feedback 191

15.1 The Widely Linear ARMA (WL-ARMA) Model 192

15.2 Widely Linear Adaptive Filters with Feedback 192

15.2.1 Widely Linear Adaptive IIR Filters 195

15.2.2 Augmented Recurrent Perceptron Learning Rule 196

15.3 The Augmented Complex Valued RTRL (ACRTRL) Algorithm 197

15.4 The Augmented Kalman Filter Algorithm for RNNs 198

15.4.1 EKF Based Training of Complex RNNs 200

15.5 Augmented Complex Unscented Kalman Filter (ACUKF) 200

15.5.1 State Space Equations for the Complex Unscented Kalman Filter 201

15.5.2 ACUKF Based Training of Complex RNNs 202

15.6 Simulation Examples 203

16 Collaborative Adaptive Filtering 207

16.1 Parametric Signal Modality Characterisation 207

16.2 Standard Hybrid Filtering in R 209

16.3 Tracking the Linear/Nonlinear Nature of Complex Valued Signals 210

16.3.1 Signal Modality characterisation in c 211

16.4 Split vs Fully Complex Signal Natures 214

16.5 Online Assessment of the Nature of Wind Signal 216

16.5.1 Effects of Averaging on Signal Nonlinearity 216

16.6 Collaborative Filters for General Complex Signals 217

16.6.1 Hybrid Filters for Noncircular Signals 218

16.6.2 Online Test for Complex Circularity 220

17 Adaptive Filtering Based on EMD 221

17.1 The Empirical Mode Decomposition Algorithm 222

17.1.1 Empirical Mode Decomposition as a Fixed Point Iteration 223

17.1.2 Applications of Real Valued EMD 224

17.1.3 Uniqueness of the Decomposition 225

17.2 Complex Extensions of Empirical Mode Decomposition 226

17.2.1 Complex Empirical Mode Decomposition 227

17.2.2 Rotation Invariant Empirical Mode Decomposition (RIEMD) 228

17.2.3 Bivariate Empirical Mode Decomposition (BEMD) 228

17.3 Addressing the Problem of Uniqueness 230

17.4 Applications of Complex Extensions of EMD 230

18 Validation of Complex Representations – Is This Worthwhile? 233

18.1 Signal Modality Characterisation in R 234

18.1.1 Surrogate Data Methods 235

18.1.2 Test Statistics: The DVV Method 237

18.2 Testing for the Validity of Complex Representation 239

18.2.1 Complex Delay Vector Variance Method (CDVV) 240

18.3 Quantifying Benefits of Complex Valued Representation 243

18.3.1 Pros and Cons of the Complex DVV Method 244

Appendix A: Some Distinctive Properties of Calculus in C 245

Appendix B: Liouville’s Theorem 251

Appendix C: Hypercomplex and Clifford Algebras 253

C. 1 Definitions of Algebraic Notions of Group, Ring and Field 253

C. 2 Definition of a Vector Space 254

C. 3 Higher Dimension Algebras 254

C. 4 The Algebra of Quaternions 255

C. 5 Clifford Algebras 256

Appendix D: Real Valued Activation Functions 257

D. 1 Logistic Sigmoid Activation Function 257

D. 2 Hyperbolic Tangent Activation Function 258

Appendix E: Elementary Transcendental Functions (ETF) 259

Appendix F: The O Notation and Standard Vector and Matrix Differentiation 263

F. 1 The O Notation 263

F. 2 Standard Vector and Matrix Differentiation 263

Appendix G: Notions From Learning Theory 265

G. 1 Types of Learning 266

G. 2 The Bias–Variance Dilemma 266

G. 3 Recursive and Iterative Gradient Estimation Techniques 267

G. 4 Transformation of Input Data 267

Appendix H: Notions from Approximation Theory 269

Appendix I: Terminology Used in the Field of Neural Networks 273

Appendix J: Complex Valued Pipelined Recurrent Neural Network (CPRNN) 275

J.1 The Complex RTRL Algorithm (CRTRL) for CPRNN 275

J.1.1 Linear Subsection Within the PRNN 277

Appendix K: Gradient Adaptive Step Size (GASS) Algorithms in R 279

K. 1 Gradient Adaptive Stepsize Algorithms Based on ∂J/∂μ 280

K. 2 Variable Stepsize Algorithms Based on ∂J/∂ε 281

Appendix L: Derivation of Partial Derivatives from Chapter 8 283

L. 1 Derivation of ∂e(k)/∂w n (k) 283

L. 2 Derivation of ∂e ∗ (k)/∂ε(k − 1) 284

L. 3 Derivation of ∂w(k)/∂ε(k − 1) 286

Appendix M: A Posteriori Learning 287

M.1 A Posteriori Strategies in Adaptive Learning 288

Appendix N: Notions from Stability Theory 291

Appendix O: Linear Relaxation 293

O. 1 Vector and Matrix Norms 293

O. 2 Relaxation in Linear Systems 294

O.2.1 Convergence in the Norm or State Space? 297

Appendix P: Contraction Mappings, Fixed Point Iteration and Fractals 299

P. 1 Historical Perspective 303

P. 2 More on Convergence: Modified Contraction Mapping 305

P. 3 Fractals and Mandelbrot Set 308

References 309

Index 321