Complex Valued Nonlinear Adaptive Filters

Complex Valued Nonlinear Adaptive Filters: Noncircularity, Widely Linear and Neural Models (Adaptive and Learning Systems for Signal Processing, Communications and Control Series)
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.

Table of Contents

Chapter 1 The Magic of Complex Numbers.
Chapter 2 Why Signal Processing in the Complex Domain?
Chapter 3 Adaptive Filtering Architectures.
Chapter 4 Complex Nonlinear Activation Functions.
Chapter 5 Elements of CR Calculus.
Chapter 6 Complex Valued Adaptive Filters.
Chapter 7 Adaptive Filters with Feedback.
Chapter 8 Filters with an Adaptive Stepsize.
Chapter 9 Filters with an Adaptive Amplitude of Nonlinearity.
Chapter 10 Data-reusing Algorithms for Complex Valued Adaptive Filters.
Chapter 11 Complex Mappings and M¨obius Transformations.
Chapter 12 Augmented Complex Statistics.
Chapter 13 Widely Linear Estimation and Augmented CLMS (ACLMS).
Chapter 14 Duality Between Complex Valued and Real Valued Filters.
Chapter 15 Widely Linear Filters with Feedback.
Chapter 16 Collaborative Adaptive Filtering.
Chapter 17 Adaptive Filtering Based on EMD.
Chapter 18 Validation of Complex Representations – Is This Worthwhile?

Appendix A: Some Distinctive Properties of Calculus in C.
Appendix B: Liouville’s Theorem.
Appendix C: Hypercomplex and Clifford Algebras.
Appendix D: Real Valued Activation Functions.
Appendix E: Elementary Transcendental Functions (ETF).
Appendix F: The O Notation and Standard Vector and Matrix Differentiation.
Appendix G: Notions From Learning Theory.
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).
Appendix K: Gradient Adaptive Step Size (GASS) Algorithms in R.
Appendix L: Derivation of Partial Derivatives from Chapter 8.
Appendix M: A Posteriori Learning.
Appendix N: Notions from Stability Theory.
Appendix O: Linear Relaxation.
Appendix P: Contraction Mappings, Fixed Point Iteration and Fractals.