Polynomial Eigenvalue Decomposition for Multichannel Broadband Signal Processing: A mathematical technique offering new insights and solutions

Abstract

This article is devoted to the polynomial eigenvalue decomposition (PEVD) and its applications in broadband multichannel signal processing, motivated by the optimum solutions provided by the EVD for the narrowband case <xref ref-type="bibr" rid="ref1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[1]</xref> , <xref ref-type="bibr" rid="ref2" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[2]</xref> . In general, we would like to extend the utility of the EVD to also address broadband problems. Multichannel broadband signals arise at the core of many essential commercial applications, such as telecommunications, speech processing, health-care monitoring, astronomy and seismic surveillance, and military technologies, including radar, sonar, and communications <xref ref-type="bibr" rid="ref3" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[3]</xref> . The success of these applications often depends on the performance of signal processing tasks, including data compression <xref ref-type="bibr" rid="ref4" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[4]</xref> , source localization <xref ref-type="bibr" rid="ref5" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[5]</xref> , channel coding <xref ref-type="bibr" rid="ref6" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[6]</xref> , signal enhancement <xref ref-type="bibr" rid="ref7" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[7]</xref> , beamforming <xref ref-type="bibr" rid="ref8" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[8]</xref> , and source separation <xref ref-type="bibr" rid="ref9" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[9]</xref> . In most cases and for narrowband signals, performing an EVD is the key to the signal processing algorithm. Therefore, this article aims to introduce the PEVD as a novel mathematical technique suitable for many broadband signal processing applications.