Signal Processing and the Jacobi Group

Abstract

The recent concern with advancing our understanding of both analog and digital signal processing has lead to two approaches which have been in competition for several years. One approach, the “window Fourier transform” , has the advantage of being amenable to theoretical analysis to the extent that we now understand its workings quite well. In fact we have a full description of its shortcomings too, largely evident in the ill conditioning of the inverse transform. The newer approach, “wavelets”, seems, at least experimentally, to avoid the conditioning difficulties of its predecessor. It suffers, however, from a less well developed theoretical basis, and furthermore has no easily describable inverse transform, so that one relies heavily on numerical techniques if one chooses this path. This paper introduces a new method which, combining the sampling techniques of both of its predecessors, promises someday to offer the advantages of both, as well as an added advantage of generalizability discussed in the conclusion of this paper.