Numerical algorithms for discrete Gabor expansions

Abstract

We present a unifying approach to discrete Gabor analysis, based on unitary matrix factorization. The factorization point of view is notably useful for the design of efficient numerical algorithms. This presentation is the first systematic account of its kind. In particular, it is shown that different algorithms for the computation of the dual window correspond to different factorizations of the frame operator. Simple number theoretic conditions on the time-frequency lattice parameters imply additional structural properties of the frame operator, which do not appear in an infinite-dimensional setting. Further the computation of adaptive dual windows is discussed. We point out why the conjugate gradient method is particularly useful in connection with Gabor expansions and discuss approximate solutions and preconditioners.

“In every mathematical investigation, the question will arise whether we can apply our mathematical results to the real world.” V. I. Arnold, 1983