Formulating and solving broadband multichannel problems using matrices of functions

The analysis and design of broadband multichannel systems typically involves convolutive mixing, characterised by matrices of transfer functions. Further, many broadband multichannel problems can be formulated using space-time covariance matrices that include an explicit lag variable and thus cross-correlation sequences as entries. This is in contrast to narrowband challenges, where the problem formulation relies on standard (i.e. constant) matrices; a rich set of solutions that are optimal in various senses can be reached from these formulations by matrix factorisations such as the eigenvalue or singular value decompositions. In order to extend the utility of such linear algebraic techniques to the broadband case, the diagonalisation or factorisation of matrices of functions is key.

In this webinar, I will show that such matrices are quite ubiquitous in multichannel signal processing, review some of the theory for their factorisations, and show how with such broadband formulations and solutions directly generalise from their narrowband counterparts. I will sketch out a number of algorithms and illustrate their use in a few example applications such as beamforming, angle or arrival estimation, and signal compaction.