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

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.
The filtering of real world signals requires an adaptive mode of operation to deal with the statistically nonstationary nature of the data. Feedback and nonlinearity within filtering architectures are needed to cater for long time dependencies and possibly nonlinear signal generating mechanisms. Using the authors' original research and current established methods, this book covers the foundations of standard complex adaptive filtering and offers next generation solutions for the generality of complex valued signals. It provides a rigorous treatment of complex noncircularity and nonlinearity, thus avoiding the deficiencies inherent in several mathematical shortcuts typically used in the treatment of complex random signals. Simulations for both circular and noncircular data sources are included - from benchmark models to real world directional processes such as wind and radar signals.
Key features:

* Provides theoretical and practical justification for converting many apparently real valued signal processing problems into the complex domain;
* Offers a unified approach to the design of complex valued adaptive filters and temporal neural networks, based on augmented complex statistics and the duality between the bivariate and complex calculus (CR calculus);
* Introduces augmented filtering algorithms based on widely linear models, making them suitable for processing both second order circular (proper) and noncircular (improper) complex signals;
* Covers adaptive stepsizes, dynamical range reduction, validity of complex representations, and data driven time-frequency decompositions;
* Includes extensive background material in appendices ranging from the theory of complex variables through to fixed point theory.

Complex valued signals play a central role in the fields of communications, radar, sonar, array, biomedical and environmental signal processing amongst others. This book will have wide appeal to researchers and practising engineers in these and related disciplines, and can also be used as lecture material for a course on adaptive filters.

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