Extending Narrowband Descriptions and Optimal Solutions to Broadband Sensor Arrays

Abstract

Abstract:This overview paper motivates the description of broadband sensor array problems by polynomial matrices, directly extending notation that is familiar from the characterisation of narrowband problems. To admit optimal solutions, the approach relies on extending the utility of the eigen-and singular value decompositions, by finding decompositions of such polynomial matrices. Particularly the factorisation of parahermitian polynomial matrices -including space-time covariance matrices that model the second order statistics of broadband sensor array data -is important. The paper summarises recent findings on the existence and uniqueness of the eigenvalue decomposition of such parahermitian polynomial matrices, demonstrates some algorithms that implement such factorisations, and highlights key applications where such techniques can provide advantages over state-of-the-art solutions.

Mathematical Tools for Processing Broadband Multi-Sensor Signals

“…Enforcing spectral majorisation in the case of an algebraic multiplicity greater than one as shown in Fig. 2 leads to eigenvalues that are not infinitely differentiable and to eigenvectors with discontinuities [22,51]. Current PEVD approaches based on the SBR2 and SMD algorithm families either empirically encourage or, in case of SBR2, can be shown to yield spectrally majorised eigenvalues [54].…”

“…The purpose of this paper is to provide an update on developments in the area of polynomial matrix decompositions beyond [51] and provide an example application where a broadband solution by a simple extension of narrowband results. Thus, Sec.…”