Applying numerical linear algebra techniques to analyzing algorithms in signal processing

Abstract

Without theoretical evidence to support the use of an algorithm, one cannot be certain that if it solves one problem successfully it will necessarily solve all problems successfully. One can analyze an algorithm from many different settings, each having the possibility for different focal points for results and insights. In signal processing, an analysis into the stability of an algorithm can have as its focal point stability in the sense of Lyapunov or of numerical linear algebra (backward, forward), etc. Unfortunately, without a clear vocabulary to apply toward the focal point of interest, the interpretation is ambiguous. To avoid this problem, a clear distinction must be made that delineates each significant contribution to the process of solving the problem, from the definition of a problem to the derivation of the algorithm. It requires that perturbations assumed at each juncture be described in a clear language that offers no chance for confusion. We offer such a terminology and apply it to the fast transversal filter and its (existing) analyses, an algorithm that has long been known to diverge yet has nonetheless received much interest regarding the reasons behind its divergence as well as inquiries into how it might be modified to one that enjoys more dependable and robust behavior.