Complex valued nonlinear adaptive filters : noncircularity, widely linear, and neural models / Danilo P. Mandic, Vanessa Su Lee Goh.

Format:

Book

Author/Creator:

Mandic, Danilo P.

Contributor:

Goh, Vanessa Su Lee.

Series:

Adaptive and Learning Systems for Signal Processing, Communications and Control Series

Adaptive and Learning Systems for Signal Processing, Communications and Control Series ; v.59

Language:

English

Subjects (All):

Functions of complex variables.

Adaptive filters--Mathematical models.

Adaptive filters.

Filters (Mathematics).

Nonlinear theories.

Neural networks (Computer science).

Physical Description:

1 online resource (345 p.)

Edition:

1st ed.

Place of Publication:

Hoboken, N.J. : Wiley, c2009.

Language Note:

English

Summary:

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 stochast

Contents:

Complex Valued Nonlinear Adaptive Filters: Noncircularity, Widely Linear and Neural Models; Series Page; Contents; Preface; Acknowledgements; 1 The Magic of Complex Numbers; 1.1 History of Complex Numbers; 1.1.1 Hypercomplex 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; 2.1.1 Duality Between Signal Representations in R and C; 2.2 Modelling in C is Not Only Convenient But Also Natural

2.3 Why Complex Modelling of Real Valued Processes?2.3.1 Phase Information in Imaging; 2.3.2 Modelling of Directional Processes; 2.4 Exploiting the Phase Information; 2.4.1 Synchronisation of Real Valued Processes; 2.4.2 Adaptive Filtering by Incorporating 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; 3.2.1 Feedforward Neural Networks

3.2.2 Recurrent Neural Networks3.2.3 Neural Networks and Polynomial Filters; 3.3 State Space Representation and Canonical Forms; 4 Complex Nonlinear Activation Functions; 4.1 Properties of Complex Functions; 4.1.1 Singularities of Complex Functions; 4.2 Universal Function Approximation; 4.2.1 Universal Approximation in R; 4.3 Nonlinear Activation Functions for Complex Neural Networks; 4.3.1 Split-complex Approach; 4.3.2 Fully Complex Nonlinear Activation Functions; 4.4 Generalised Splitting Activation Functions (GSAF); 4.4.1 The Clifford Neuron

4.5 Summary: Choice of the Complex Activation Function5 Elements of CR Calculus; 5.1 Continuous Complex Functions; 5.2 The Cauchy-Riemann Equations; 5.3 Generalised Derivatives of Functions of Complex Variable; 5.3.1 CR Calculus; 5.3.2 Link between R- and C-derivatives; 5.4 CR-derivatives of Cost Functions; 5.4.1 The Complex Gradient; 5.4.2 The Complex Hessian; 5.4.3 The Complex Jacobian and Complex Differential; 5.4.4 Gradient of a Cost Function; 6 Complex Valued Adaptive Filters; 6.1 Adaptive Filtering Configurations; 6.2 The Complex Least Mean Square Algorithm

6.2.1 Convergence of the CLMS Algorithm6.3 Nonlinear Feedforward Complex Adaptive Filters; 6.3.1 Fully Complex Nonlinear Adaptive Filters; 6.3.2 Derivation of CNGD using CR calculus; 6.3.3 Split-complex Approach; 6.3.4 Dual Univariate Adaptive Filtering Approach (DUAF); 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; 7.1.1 Coefficient Update for Linear Adaptive IIR Filters

7.1.2 Training of IIR filters with Reduced Computational Complexity

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

ISBN:

9786612123375

9781282123373

1282123378

9780470742624

0470742623

9780470742631

0470742631

OCLC:

352829734