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E-raamat: Complex Valued Nonlinear Adaptive Filters: Noncircularity, Widely Linear and Neural Models

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(Imperial College, London), (Shell EP Europe)
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Complex Valued Nonlinear Adaptive Filters: Noncircularity, Widely Linear and Neural ModelsSuurem pilt
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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 xiii
Acknowledgements xvii
The Magic of Complex Numbers
1(12)
History of Complex Numbers
2(6)
Hypercomplex Numbers
7(1)
History of Mathematical Notation
8(1)
Development of Complex Valued Adaptive Signal Processing
9(4)
Why Signal Processing in the Complex Domain?
13(20)
Some Examples of Complex Valued Signal Processing
13(6)
Duality Between Signal Representations in R and C
18(1)
Modelling in C is Not Only Convenient But Also Natural
19(1)
Why Complex Modelling of Real Valued Processes?
20(3)
Phase Information in Imaging
20(2)
Modelling of Directional Processes
22(1)
Exploiting the Phase Information
23(3)
Synchronisation of Real Valued Processes
24(1)
Adaptive Filtering by Incorporating Phase Information
25(1)
Other Applications of Complex Domain Processing of Real Valued Signals
26(3)
Additional Benefits of Complex Domain Processing
29(4)
Adaptive Filtering Architectures
33(10)
Linear and Nonlinear Stochastic Models
34(1)
Linear and Nonlinear Adaptive Filtering Architectures
35(4)
Feedforward Neural Networks
36(1)
Recurrent Neural Networks
37(1)
Neural Networks and Polynomial Filters
38(1)
State Space Representation and Canonical Forms
39(4)
Complex Nonlinear Activation Functions
43(12)
Properties of Complex Functions
43(3)
Singularities of Complex Functions
45(1)
Universal Function Approximation
46(2)
Universal Approximation in R
47(1)
Nonlinear Activation Functions for Complex Neural Networks
48(5)
Split-complex Approach
49(2)
Fully Complex Nonlinear Activation Functions
51(2)
Generalised Splitting Activation Functions (GSAF)
53(1)
The Clifford Neuron
53(1)
Summary: Choice of the Complex Activation Function
54(1)
Elements of CR Calculus
55(14)
Continuous Complex Functions
56(1)
The Cauchy-Riemann Equations
56(1)
Generalised Derivatives of Functions of Complex Variable
57(5)
CR Calculus
59(1)
Link between R- and C-derivatives
60(2)
CR-derivatives of Cost Functions
62(7)
The Complex Gradient
62(2)
The Complex Hessian
64(1)
The Complex Jacobian and Complex Differential
64(1)
Gradient of a Cost Function
65(4)
Complex Valued Adaptive Filters
69(22)
Adaptive Filtering Configurations
70(3)
The Complex Least Mean Square Algorithm
73(7)
Convergence of the CLMS Algorithm
75(5)
Nonlinear Feedforward Complex Adaptive Filters
80(5)
Fully Complex Nonlinear Adaptive Filters
80(2)
Derivation of CNGD using CR calculus
82(1)
Split-complex Approach
83(1)
Dual Univariate Adaptive Filtering Approach (DUAF)
84(1)
Normalisation of Learning Algorithms
85(2)
Performance of Feedforward Nonlinear Adaptive Filters
87(2)
Summary: Choice of a Nonlinear Adaptive Filter
89(2)
Adaptive Filters with Feedback
91(16)
Training of IIR Adaptive Filters
92(5)
Coefficient Update for Linear Adaptive IIR Filters
93(3)
Training of IIR filters with Reduced Computational Complexity
96(1)
Nonlinear Adaptive IIR Filters: Recurrent Perceptron
97(2)
Training of Recurrent Neural Networks
99(3)
Other Learning Algorithms and Computational Complexity
102(1)
Simulation Examples
102(5)
Filters with an Adaptive Stepsize
107(12)
Benveniste Type Variable Stepsize Algorithms
108(2)
Complex Valued GNGD Algorithms
110(3)
Complex GNGD for Nonlinear Filters (CFANNGD)
112(1)
Simulation Examples
113(6)
Filters with an Adaptive Amplitude of Nonlinearity
119(10)
Dynamical Range Reduction
119(2)
FIR Adaptive Filters with an Adaptive Nonlinearity
121(1)
Recurrent Neural Networks with Trainable Amplitude of Activation Functions
122(2)
Simulation Results
124(5)
Data-reusing Algorithms for Complex Valued Adaptive Filters
129(8)
The Data-reusing Complex Valued Least Mean Square (DRCLMS) Algorithm
129(2)
Data-reusing Complex Nonlinear Adaptive Filters
131(3)
Convergence Analysis
132(2)
Data-reusing Algorithms for Complex RNNs
134(3)
Complex Mappings and Mobius Transformations
137(14)
Matrix Representation of a Complex Number
137(3)
The Mobius Transformation
140(2)
Activation Functions and Mobius Transformations
142(4)
All-pass Systems as Mobius Transformations
146(1)
Fractional Delay Filters
147(4)
Augmented Complex Statistics
151(18)
Complex Random Variables (CRV)
152(6)
Complex Circularity
153(1)
The Multivariate Complex Normal Distribution
154(3)
Moments of Complex Random Variables (CRV)
157(1)
Complex Circular Random Variables
158(1)
Complex Signals
159(2)
Wide Sense Stationarity, Multicorrelations, and Multispectra
160(1)
Strict Circularity and Higher-order Statistics
161(1)
Second-order Characterisation of Complex Signals
161(8)
Augmented Statistics of Complex Signals
161(3)
Second-order Complex Circularity
164(5)
Widely Linear Estimation and Augmented CLMS (ACLMS)
169(14)
Minimum Mean Square Error (MMSE) Estimation in C
169(3)
Widely Linear Modelling in C
171(1)
Complex White Noise
172(1)
Autoregressive Modelling in C
173(2)
Widely Linear Autoregressive Modelling in C
174(1)
Quantifying Benefits of Widely Linear Estimation
174(1)
The Augmented Complex LMS (ACLMS) Algorithm
175(3)
Adaptive Prediction Based on ACLMS
178(5)
Wind Forecasting Using Augmented Statistics
180(3)
Duality Between Complex Valued and Real Valued Filters
183(8)
A Dual Channel Real Valued Adaptive Filter
184(2)
Duality Between Real and Complex Valued Filters
186(2)
Operation of Standard Complex Adaptive Filters
186(1)
Operation of Widely Linear Complex Filters
187(1)
Simulations
188(3)
Widely Linear Filters with Feedback
191(16)
The Widely Linear ARMA (WL-ARMA) Model
192(1)
Widely Linear Adaptive Filters with Feedback
192(5)
Widely Linear Adaptive IIR Filters
195(1)
Augmented Recurrent Perceptron Learning Rule
196(1)
The Augmented Complex Valued RTRL (ACRTRL) Algorithm
197(1)
The Augmented Kalman Filter Algorithm for RNNs
198(2)
EKF Based Training of Complex RNNs
200(1)
Augmented Complex Unscented Kalman Filter (ACUKF)
200(3)
State Space Equations for the Complex Unscented Kalman Filter
201(1)
ACUKF Based Training of Complex RNNs
202(1)
Simulation Examples
203(4)
Collaborative Adaptive Filtering
207(14)
Parametric Signal Modality Characterisation
207(2)
Standard Hybrid Filtering in R
209(1)
Tracking the Linear/Nonlinear Nature of Complex Valued Signals
210(4)
Signal Modality Characterisation in C
211(3)
Split vs Fully Complex Signal Natures
214(2)
Online Assessment of the Nature of Wind Signal
216(1)
Effects of Averaging on Signal Nonlinearity
216(1)
Collaborative Filters for General Complex Signals
217(4)
Hybrid Filters for Noncircular Signals
218(2)
Online Test for Complex Circularity
220(1)
Adaptive Filtering Based on EMD
221(12)
The Empirical Mode Decomposition Algorithm
222(4)
Empirical Mode Decomposition as a Fixed Point Iteration
223(1)
Applications of Real Valued EMD
224(1)
Uniqueness of the Decomposition
225(1)
Complex Extensions of Empirical Mode Decomposition
226(4)
Complex Empirical Mode Decomposition
227(1)
Rotation Invariant Empirical Mode Decomposition (RIEMD)
228(1)
Bivariate Empirical Mode Decomposition (BEMD)
228(2)
Addressing the Problem of Uniqueness
230(1)
Applications of Complex Extensions of EMD
230(3)
Validation of Complex Representations - Is This Worthwhile?
233(12)
Signal Modality Characterisation in R
234(5)
Surrogate Data Methods
235(2)
Test Statistics: The DVV Method
237(2)
Testing for the Validity of Complex Representation
239(4)
Complex Delay Vector Variance Method (CDVV)
240(3)
Quantifying Benefits of Complex Valued Representation
243(2)
Pros and Cons of the Complex DVV Method
244(1)
Appendix A: Some Distinctive Properties of Calculus in C
245(6)
Appendix B: Liouville's Theorem
251(2)
Appendix C: Hypercomplex and Clifford Algebras
253(4)
Definitions of Algebraic Notions of Group, Ring and Field
253(1)
Definition of a Vector Space
254(1)
Higher Dimension Algebras
254(1)
The Algebra of Quaternions
255(1)
Clifford Algebras
256(1)
Appendix D: Real Valued Activation Functions
257(2)
Logistic Sigmoid Activation Function
257(1)
Hyperbolic Tangent Activation Function
258(1)
Appendix E: Elementary Transcendental Functions (ETF)
259(4)
Appendix F: The Notation and Standard Vector and Matrix Differentiation
263(2)
The Notation
263(1)
Standard Vector and Matrix Differentiation
263(2)
Appendix G: Notions From Learning Theory
265(4)
Types of Learning
266(1)
The Bias-Variance Dilemma
266(1)
Recursive and Iterative Gradient Estimation Techniques
267(1)
Transformation of Input Data
267(2)
Appendix H: Notions from Approximation Theory
269(4)
Appendix I: Terminology Used in the Field of Neural Networks
273(2)
Appendix J: Complex Valued Pipelined Recurrent Neural Network (CPRNN)
275(4)
The Complex RTRL Algorithm (CRTRL) for CPRNN
275(4)
Linear Subsection Within the PRNN
277(2)
Appendix K: Gradient Adaptive Step Size (GASS) Algorithms in R
279(4)
Gradient Adaptive Stepsize Algorithms Based on ∂J/∂μ
280(1)
Variable Stepsize Algorithms Based on ∂J/∂ε
281(2)
Appendix L: Derivation of Partial Derivatives from
Chapter 8
283(4)
Derivation of ∂e(k)/∂n(k)
283(1)
Derivation of ∂e*(k)/∂ε(k -- 1)
284(2)
Derivation of ∂w(k)/∂ε(k - 1)
286(1)
Appendix M: A Posteriori Learning
287(4)
A Posteriori Strategies in Adaptive Learning
288(3)
Appendix N: Notions from Stability Theory
291(2)
Appendix O: Linear Relaxation
293(6)
Vector and Matrix Norms
293(1)
Relaxation in Linear Systems
294(5)
Convergence in the Norm or State Space?
297(2)
Appendix P: Contraction Mappings, Fixed Point Iteration and Fractals
299(10)
Historical Perspective
303(2)
More on Convergence: Modified Contraction Mapping
305(3)
Fractals and Mandelbrot Set
308(1)
References 309(12)
Index 321
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.
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