Complex Valued Nonlinear Adaptive Filters
by
Danilo Mandic - Vanessa Goh
Features linear and nonlinear complex valued adaptive filters, and methods for the processing of general complex signals (circular and noncircular). This book presents adaptive filtering algorithms for feedforward (transversal) and feedback architectures and the developments in the statistics of complex variable.
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Complete description
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.
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General info
Publisher & Imprint:
Wiley-Blackwell (an imprint of John Wiley & Sons Ltd)
City:
Chicester
Pages:
344
More info:
height 250 mm
width 150 mm
weight 666 gr
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Age recommended:
Professional and scholarly
Summary
Complex Valued Nonlinear Adaptive Filters
Contents Series Editor's Foreword About the Authors 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 Mobius Transformations 11.1 Matrix Representation of a Complex Number 11.2 The Mobius Transformation 11.3 Activation Functions and Mobius Transformations 11.4 All-pass Systems as Mobius 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: Proof of 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 J.1.1 Linear Subsection Within the PRNN Appendix K: GASS Algorithms in R K.1 Gradient Adaptive Stepsize Algorithms Based on E/ mu K.2 Variable Stepsize Algorithms Based on E/ epsilon Appendix L: Derivation of Partial Derivatives from Chapter 8 L.1 Derivation of e(k)/ wn(k) L.2 Derivation of e(k)/ epsilon(k 1) L.3 Derivation of w(k)/ epsilon(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 O.2.1 Convergence in the Norm or State Space? 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
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