Tutorials

Polynomial Matrix Decompositions With Applications

Dr. Stephan Weiss, University of Strathclyde, Scotland, UK and

Prof. John G. McWhirter, University of Cardiff, Wales, UK

Abstract:

Many broadband multichannel problems can be elegantly formulated using polynomial matrices, representing a straightforward generalisation of well-known narrowband approaches. In a broadband formulation, e.g. a space-time covariance matrix will no longer just contain complex multipliers representing gain and phase shifts, but will comprise of general auto- and cross-correlation sequences with explicit lag elements. To solve such broadband problems requires to extend the utility of optimal narrowband techniques, often relying on eigen-(EVD) and singular value decompositions (SVD), to the domain of polynomial matrices. This generalisation centrally leads to a polynomial matrix EVD (PEVD), which is capable of diagonalising a space-time covariance matrix at all lags by means of lossless polynomial matrix factors. The purpose of this tutorial is to introduce participants to polynomial matrices, including parahermitian and paraunitary matrix properties, and the characteristics of a PEVD. We will review some iterative algorithms for its approximation, such as McWhirter's original second order sequential best rotation (SBR2) algorithm and recent developments. Applications ranging from optimal subband coding, where the PEVD for the first time permitted the design of filter banks with more than two channels, to broadband source separation, angle of arrival estimation and broadband beamforming will be demonstrated to the participants. The algorithms and many examples are contained in a free Matlab toolbox available at pevd-toolbox.eee.strath.ac.uk.

Biographies:

Stephan Weiss received a Dipl.-Ing.~degree from the University of Erlangen-Nurnberg, Erlangen, Germany, in 1995, and a Ph.D.~degree from the University of Strathclyde, Glasgow, Scotland, in 1998, both in electronic and electrical engineering. He is a Reader in the Department of EEE at Strathclyde. From 1999 until 2006, he held lecturer and senior lecturer positions within the School of Electronics and Computer Science at the University of Southampton. Prior to this, he was a Visiting Lecturer at the University of Strathclyde in 1998/99 and a Visiting Scholar at the University of Southern California in 1996/97. His research interests lie in adaptive, multirate, and array signal processing with applications in communications, audio, and biomedical signal processing, where he has published more than 250 technical papers. For his work in biomedical signal processing, he was a co-recipient of the 2001 research award of the German society on hearing aids. In 2011 he was a co-recipient of the VTC-Spring best paper award in the MIMO systems track. Dr. Weiss is a member of EURASIP and a senior member of the IEEE. He was the technical co-chair for EUSIPCO 2009 , and general chair for ISPLC 2014 in Glasgow.

John G.~McWhirter received the first class honors degree in mathematics and the Ph.D.~degree in theoretical physics from the Queen's University of Belfast, Belfast, Ireland, in 1970 and 1973, respectively. He joined the Royal Radar Establishment in Malvern (later to become the Royal Signals and Radar Establishment, and now part of QinetiQ Ltd.) in 1973, where he became a Senior Fellow in the Centre for Signal and Information Processing Group. In 2007, he left QinetiQ to take up his current post as Distinguished Research Professor in the Engineering Department, Cardiff University, Cardiff, U.K. He is also a Visiting Professor in Electrical Engineering at the Queen's University of Belfast. He has been carrying out research on adaptive signal processing since 1980 and was awarded the J.J. Thomson Medal by the Institution of Electrical Engineers in 1994 for his research on systolic arrays. He has published more than 180 research papers and holds numerous patents. His current research is devoted to broadband sensor arrays, convolutive blind signal separation, and polynomial matrix techniques. The signal processing group which he built up in Malvern over many years, received the EURASIP Group Technical Achievement Award for 2003. Dr. McWhirter was elected as a Fellow of the Royal Academy of Engineering in 1996 and as a Fellow of the Royal Society in 1999. He is a Fellow of the Institute of Mathematics and its Applications (IMA) and served as President of the IMA in 2002 and 2003. He is also a Fellow of the Institute of Electrical Engineers, a Fellow of the Institute of Physics, and a member of the London Mathematical Society.