Blind Identification and Separation of Complex-valued Signals E-Book

Eric Moreau

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 105

Veröffentlichungsjahr: 2013

Beschreibung

Blind identification consists of estimating a multi-dimensional system only through the use of its output, and source separation, the blind estimation of the inverse of the system. Estimation is generally carried out using different statistics of the output.
The authors of this book consider the blind identification and source separation problem in the complex-domain, where the available statistical properties are richer and include non-circularity of the sources – underlying components. They define identifiability conditions and present state-of-the-art algorithms that are based on algebraic methods as well as iterative algorithms based on maximum likelihood theory.

Contents

1. Mathematical Preliminaries.
2. Estimation by Joint Diagonalization.
3. Maximum Likelihood ICA.

About the Authors

Eric Moreau is Professor of Electrical Engineering at the University of Toulon, France. His research interests concern statistical signal processing, high order statistics and matrix/tensor decompositions with applications to data analysis, telecommunications and radar.
Tülay Adali is Professor of Electrical Engineering and Director of the Machine Learning for Signal Processing Laboratory at the University of Maryland, Baltimore County, USA. Her research interests concern statistical and adaptive signal processing, with an emphasis on nonlinear and complex-valued signal processing, and applications in biomedical data analysis and communications.

Blind identification consists of estimating a multidimensional system through the use of only its output. Source separation is concerned with the blind estimation of the inverse of the system. The estimation is generally performed by using different statistics of the outputs.
The authors consider the blind estimation of a multiple input/multiple output (MIMO) system that mixes a number of underlying signals of interest called sources. They also consider the case of direct estimation of the inverse system for the purpose of source separation. They then describe the estimation theory associated with the identifiability conditions and dedicated algebraic algorithms. The algorithms depend critically on (statistical and/or time frequency) properties of complex sources that will be precisely described.

Leseprobe

Contents

Preface

Acknowledgments

Chapter 1: Mathematical Preliminaries

1.1. Introduction

1.2. Linear mixing model

1.3. Problem definition

1.4. Statistics

1.5. Optimization: Wirtinger calculus

Chapter 2: Estimation By Joint Diagonalization

2.1. Introduction

2.2. Normalization, dimension reduction

2.3. Exact joint diagonalization of two matrices

2.4. Unitary approximate joint diagonalization

2.5. General approximate joint diagonalization

2.6. Summary

Chapter 3: Maximum Likelihood ICA

3.1. Introduction

3.2. Cost function choice

3.3. Algorithms

3.4. Summary

Bibliography

Index

First published 2013 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

ISTE Ltd 27-37 St George’s Road London SW19 4EU UKwww.iste.co.uk

John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USAwww.wiley.com

© ISTE Ltd 2013

The rights of Eric Moreau and Tülay Adali to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2013945047

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISSN: 2051-2481 (Print)

ISSN: 2051-249X (Online)

ISBN: 978-1-84821-459-0

Preface

Data-driven signal processing methods are based on simple latent variable models and hence can minimize the assumptions on the nature of the data. They have been promising alternatives to the traditional model-based approaches in many signal processing applications in cases where the underlying dynamics are difficult to characterize. We consider two related problems: blind identification and blind source separation. Both of these closely-related problems aim at estimating a system, with minimal information about the underlying components/variables. As such, an underlying mixing system model is the starting point. The most common latent variable model used in data-driven decompositions is a linear mixing model, where the mixture (the given set of observations) is written as a linear combination of source components, i.e., it is decomposed into a mixing matrix and a source matrix. The mixing matrix is often constrained so as to be of full rank and the interpretation of the sources depends on the application in hand. The estimated sources might correspond to actual physical quantities such as speech/audio sources in the classical cocktail party problem or they might be a set of features explaining the data when the approach is exploratory, as in most data mining applications. Independent component analysis (ICA) is the most commonly used blind source separation approach and is based on the assumption of statistical independence. Because independence is a relatively strong assumption, it can identify the underlying sources up to scaling and ordering ambiguities, as independence is invariant to these. In addition, independence is a plausible assumption in many applications. Hence, ICA has found wide application in a range of problems in areas as diverse as biomedicine, communications, finance and remote sensing [COM 10, COM 94b, HYV 01, ADA 10].

As the underlying latent variables, typically called sources within the ICA framework, are not directly observable, to achieve ICA, we have to make use of statistical properties of the signals directly linked to independence in order to make use of the diversity in the data. Most of the approaches to achieve ICA classically make use of one of the two types of diversity in the data sets: non-Gaussianity – or, alternatively, higher order statistics – and sample dependence of the underlying components. However, in the complex case, there is one more type of diversity that is due to the non-circularity – or more specifically, the second-order non-circularity – of the sources. This additional type of diversity allows relaxation of the identification conditions for the complex case and also makes the problem richer as we hope to underline in this book.

Eric MOREAUTülay ADALIAugust 2013

Acknowledgments

Tülay Adali would like to acknowledge the support from the National Science Foundation, grants NSF-IIS 1017718 and NSF-CCF 1117056. Special thanks go to Geng-Shen Fu for helping with the figures for the role of diversity in ICA.

1

Mathematical Preliminaries

1.1. Introduction

In this chapter, we first introduce the basic tools that are used in the remainder of the book: the statistical characterization and the optimization in the complex domain. For the statistical characterization, we emphasize the importance of taking full statistical information including potential non-circularity of the signals into account, and for the optimization, we review Wirtinger calculus that enables us to perform optimization in the complex domain in a manner very similar to the real-valued case, hence significantly simplifying algorithm derivation and analysis.