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.

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.

Contents
Series Editor's Foreword
About the Authors
Preface
Acknowledgements
The Magic of Complex Numbers
History of Complex Numbers
History of Mathematical Notation
Development of Complex Valued Adaptive Signal Processing
Why Signal Processing in the Complex Domain?
Some Examples of Complex Valued Signal Processing
Modelling in C is Not Only Convenient But Also Natural
Why Complex Modelling of Real Valued Processes?
Exploiting the Phase Information
Other Applications of Complex Domain Processing of Real Valued Signals
Additional Benefits of Complex Domain Processing
Adaptive Filtering Architectures
Linear and Nonlinear Stochastic Models
Linear and Nonlinear Adaptive Filtering Architectures
State Space Representation and Canonical Forms
Complex Nonlinear Activation Functions
Properties of Complex Functions
Universal Function Approximation
Nonlinear Activation Functions for Complex Neural Networks
Generalised Splitting Activation Functions (GSAF
Summary: Choice of the Complex Activation Function
Elements of CR Calculus
Continuous Complex Functions
The Cauchy-Riemann Equations
Generalised Derivatives of Functions of Complex Variable
CR-derivatives of Cost Functions
Complex Valued Adaptive Filters
Adaptive Filtering Configurations
The Complex Least Mean Square Algorithm
Nonlinear Feedforward Complex Adaptive Filters
Normalisation of Learning Algorithms
Performance of Feedforward Nonlinear Adaptive Filters
Summary: Choice of a Nonlinear Adaptive Filter
Adaptive Filters with Feedback
Training of IIR Adaptive Filters
Nonlinear Adaptive IIR Filters: Recurrent Perceptron
Training of Recurrent Neural Networks
Simulation Examples
Filters with an Adaptive Stepsize
Benveniste Type Variable Stepsize Algorithms
Complex Valued GNGD Algorithms
Simulation Examples
Filters with an Adaptive Amplitude of Nonlinearity
Dynamical Range Reduction
FIR Adaptive Filters with an Adaptive Nonlinearity
Recurrent Neural Networks with Trainable Amplitude of Activation Functions
Simulation Results
Data-reusing Algorithms for Complex Valued Adaptive Filters
The Data-reusing Complex Valued Least Mean Square (DRCLMS) Algorithm
Data-reusing Complex Nonlinear Adaptive Filters
Data-reusing Algorithms for Complex RNNs
Complex Mappings and Möbius Transformations
Matrix Representation of a Complex Number
The Möbius Transformation
Activation Functions and Möbius Transformations
All-pass Systems as Möbius Transformations
Fractional Delay Filters
Augmented Complex Statistics
Complex Random Variables (CRV
Complex Circular Random Variables
Complex Signals
Second-order Characterisation of Complex Signals
Widely Linear Estimation and Augmented CLMS (ACLMS
Minimum Mean Square Error (MMSE) Estimation in C
Complex White Noise
Autoregressive Modelling in C
The Augmented Complex LMS (ACLMS) Algorithm
Adaptive Prediction Based on ACLMS
Duality Between Complex Valued and Real Valued Filters
A Dual Channel Real Valued Adaptive Filter
Duality Between Real and Complex Valued Filters
Simulations
Widely Linear Filters with Feedback
The Widely Linear ARMA (WL-ARMA) Model
Widely Linear Adaptive Filters with Feedback
The Augmented Complex Valued RTRL (ACRTRL) Algorithm
The Augmented Kalman Filter Algorithm for RNNs
Augmented Complex Unscented Kalman Filter (ACUKF
Simulation Examples
Collaborative Adaptive Filtering
Parametric Signal Modality Characterisation
Standard Hybrid Filtering in R
Tracking the Linear/Nonlinear Nature of Complex Valued Signals
Split vs Fully Complex Signal Natures
Online Assessment of the Nature of Wind Signal
Collaborative Filters for General Complex Signals
Adaptive Filtering Based on EMD
The Empirical Mode Decomposition Algorithm
Complex Extensions of Empirical Mode Decomposition
Addressing the Problem of Uniqueness
Applications of Complex Extensions of EMD
Validation of Complex Representations - Is This Worthwhile?
Signal Modality Characterisation in R
Testing for the Validity of Complex Representation
Quantifying Benefits of Complex Valued Representation
Some Distinctive Properties of Calculus in C
Proof of Liouville's Theorem
Hypercomplex and Clifford Algebras
Definitions of Algebraic Notions of Group, Ring and Field
Definition of a Vector Space
Higher Dimension Algebras
The Algebra of Quaternions
Clifford Algebras
Real Valued Activation Functions
Logistic Sigmoid Activation Function
Hyperbolic Tangent Activation Function
Elementary Transcendental Functions (ETF
The O Notation and Standard Vector and Matrix Differentiation
The O Notation
Standard Vector and Matrix Differentiation
Notions From Learning Theory
Types of Learning
The Bias-Variance Dilemma
Recursive and Iterative Gradient Estimation Techniques
Transformation of Input Data
Notions from Approximation Theory
Terminology Used in the Field of Neural Networks
Complex Valued Pipelined Recurrent Neural Network (CPRNN
The Complex RTRL Algorithm (CRTRL) for CPRNN
Linear Subsection Within the PRNN
GASS Algorithms in R
Gradient Adaptive Stepsize Algorithms Based on ?E/??
Variable Stepsize Algorithms Based on ?E/??
Derivation of Partial Derivatives from Chapter 8
Derivation of ?e(k)/?wn(k
Derivation of ?e∗(k)/??(k − 1
Derivation of ?w(k)/??(k − 1
A Posteriori Learning
A Posteriori Strategies in Adaptive Learning
Notions from Stability Theory
Linear Relaxation
Vector and Matrix Norms
Relaxation in Linear Systems
Convergence in the Norm or State Space?
Contraction Mappings, Fixed Point Iteration and Fractals
Historical Perspective
More on Convergence: Modified Contraction Mapping
Fractals and Mandelbrot Set
References
Index
Table of Contents provided by Publisher. All Rights Reserved.

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Complex Valued Nonlinear Adaptive Filters Noncircularity, Widely Linear and Neural Models

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