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.

Preface.Acknowledgements.1 The Magic of Complex Numbers.1.1 History of Complex Numbers.1.2 History of Mathematical Notation.1.3 Development of Complex Valued Adaptive Signal Processing.2 Why Signal Processing in the Complex Domain?2.1 Some Examples of Complex Valued Signal Processing.2.2 Modelling in C is Not Only Convenient But Also Natural.2.3 Why Complex Modelling of Real Valued Processes?2.4 Exploiting the Phase Information.2.5 Other Applications of Complex Domain Processing of Real Valued Signals.2.6 Additional Benefits of Complex Domain Processing.3 Adaptive Filtering Architectures.3.1 Linear and Nonlinear Stochastic Models.3.2 Linear and Nonlinear Adaptive Filtering Architectures.3.3 State Space Representation and Canonical Forms.4 Complex Nonlinear Activation Functions.4.1 Properties of Complex Functions.4.2 Universal Function Approximation.4.3 Nonlinear Activation Functions for Complex Neural Networks.4.4 Generalised Splitting Activation Functions (GSAF).4.5 Summary: Choice of the Complex Activation Function.5 Elements of CR Calculus.5.1 Continuous Complex Functions.5.2 The Cauchy-Riemann Equations.5.3 Generalised Derivatives of Functions of Complex Variable.5.4 CR-derivatives of Cost Functions.6 Complex Valued Adaptive Filters.6.1 Adaptive Filtering Configurations.6.2 The Complex Least Mean Square Algorithm.6.3 Nonlinear Feedforward Complex Adaptive Filters.6.4 Normalisation of Learning Algorithms.6.5 Performance of Feedforward Nonlinear Adaptive Filters.6.6 Summary: Choice of a Nonlinear Adaptive Filter.7 Adaptive Filters with Feedback.7.1 Training of IIR Adaptive Filters.7.2 Nonlinear Adaptive IIR Filters: Recurrent Perceptron.7.3 Training of Recurrent Neural Networks.7.4 Simulation Examples.8 Filters with an Adaptive Stepsize.8.1 Benveniste Type Variable Stepsize Algorithms.8.2 Complex Valued GNGD Algorithms.8.3 Simulation Examples.9 Filters with an Adaptive Amplitude of Nonlinearity.9.1 Dynamical Range Reduction.9.2 FIR Adaptive Filters with an Adaptive Nonlinearity.9.3 Recurrent Neural Networks with Trainable Amplitude of Activation Functions.9.4 Simulation Results.10 Data-reusing Algorithms for Complex Valued Adaptive Filters.10.1 The Data-reusing Complex Valued Least Mean Square (DRCLMS) Algorithm.10.2 Data-reusing Complex Nonlinear Adaptive Filters.10.3 Data-reusing Algorithms for Complex RNNs.11 Complex Mappings and M¿obius Transformations.11.1 Matrix Representation of a Complex Number.11.2 The M¿obius Transformation.11.3 Activation Functions and M¿obius Transformations.11.4 All-pass Systems as M¿obius Transformations.11.5 Fractional Delay Filters.12 Augmented Complex Statistics.12.1 Complex Random Variables (CRV).12.2 Complex Circular Random Variables.12.3 Complex Signals.12.4 Second-order Characterisation of Complex Signals.13 Widely Linear Estimation and Augmented CLMS (ACLMS).13.1 Minimum Mean Square Error (MMSE) Estimation in C.13.2 Complex White Noise.13.3 Autoregressive Modelling in C.13.4 The Augmented Complex LMS (ACLMS) Algorithm.13.5 Adaptive Prediction Based on ACLMS.14 Duality Between Complex Valued and Real Valued Filters.14.1 A Dual Channel Real Valued Adaptive Filter.14.2 Duality Between Real and Complex Valued Filters.14.3 Simulations.15 Widely Linear Filters with Feedback.15.1 The Widely Linear ARMA (WL-ARMA) Model.15.2 Widely Linear Adaptive Filters with Feedback.15.3 The Augmented Complex Valued RTRL (ACRTRL) Algorithm.15.4 The Augmented Kalman Filter Algorithm for RNNs.15.5 Augmented Complex Unscented Kalman Filter (ACUKF).15.6 Simulation Examples.16 Collaborative Adaptive Filtering.16.1 Parametric Signal Modality Characterisation.16.2 Standard Hybrid Filtering in R.16.3 Tracking the Linear/Nonlinear Nature of Complex Valued Signals.16.4 Split vs Fully Complex Signal Natures.16.5 Online Assessment of the Nature of Wind Signal.16.6 Collaborative Filters for General Complex Signals.17 Adaptive Filtering Based on EMD.17.1 The Empirical Mode Decomposition Algorithm.17.2 Complex Extensions of Empirical Mode Decomposition.17.3 Addressing the Problem of Uniqueness.17.4 Applications of Complex Extensions of EMD.18 Validation of Complex Representations - Is This Worthwhile?18.1 Signal Modality Characterisation in R.18.2 Testing for the Validity of Complex Representation.18.3 Quantifying Benefits of Complex Valued Representation.Appendix A: Some Distinctive Properties of Calculus in C.Appendix B: Liouville's Theorem.Appendix C: Hypercomplex and Clifford Algebras.C.1 Definitions of Algebraic Notions of Group, Ring and Field.C.2 Definition of a Vector Space.C.3 Higher Dimension Algebras.C.4 The Algebra of Quaternions.C.5 Clifford Algebras.Appendix D: Real Valued Activation Functions.D.1 Logistic Sigmoid Activation Function.D.2 Hyperbolic Tangent Activation Function.Appendix E: Elementary Transcendental Functions (ETF).Appendix F: The O Notation and Standard Vector and Matrix Differentiation.F.1 The O Notation.F.2 Standard Vector and Matrix Differentiation.Appendix G: Notions From Learning Theory.G.1 Types of Learning.G.2 The Bias-Variance Dilemma.G.3 Recursive and Iterative Gradient Estimation Techniques.G.4 Transformation of Input Data.Appendix H: Notions from Approximation Theory.Appendix I: Terminology Used in the Field of Neural Networks.Appendix J: Complex Valued Pipelined Recurrent Neural Network (CPRNN).J.1 The Complex RTRL Algorithm (CRTRL) for CPRNN.Appendix K: Gradient Adaptive Step Size (GASS) Algorithms in R.K.1 Gradient Adaptive Stepsize Algorithms Based on dJ/d1/4.K.2 Variable Stepsize Algorithms Based on dJ/dµ.Appendix L: Derivation of Partial Derivatives from Chapter 8.L.1 Derivation of de(k)/dwn(k).L.2 Derivation of de*(k)/dµ(k - 1).L.3 Derivation of dw(k)/dµ(k - 1).Appendix M: A Posteriori Learning.M.1 A Posteriori Strategies in Adaptive Learning.Appendix N: Notions from Stability Theory.Appendix O: Linear Relaxation.O.1 Vector and Matrix Norms.O.2 Relaxation in Linear Systems.Appendix P: Contraction Mappings, Fixed Point Iteration and Fractals.P.1 Historical Perspective.P.2 More on Convergence: Modified Contraction Mapping.P.3 Fractals and Mandelbrot Set.References.Index.