The abstruse meets the applicable : Some aspects of time - frequency analysis

The area of Fourier analysis connected to signal processing theory has undergone a rapid development in the last two decades. The aspect of this development that has received the most publicity is the theory of wavelets and their relatives, which involves expansions in terms of sets of functions generated from a single function by translations and dilations. However, there has also been much progress in the related area known as time-frequency analysis or Gabor analysis, which involves expansions in terms of sets of functions generated from a single function by translations and modulations. In this area there are some questions of a concrete and practical nature whose study reveals connections with aspects of harmonic and functional analysis that were previously considered quite pure and perhaps rather exotic. In this expository paper, I give a survey of some of these interactions between the abstruse and the applicable. It is based on the thematic lectures which I gave at the Ninth Discussion Meeting on Harmonic Analysis at the Harish-Chandra Research Institute in Allahabad in October 2005. 1. Background and statements of theorems In this section we develop some basic ideas of time-frequency analysis and lead up to the main results to be discussed in this paper. For a more comprehensive account of this subject we recommend the book of Gröchenig [14]; see also Daubechies [5]. For an L 2 function f on R d , the Fourier transform and its inversion formula ˆ f (ω) = f (t)e −2πiω·t dt, f (t) = ˆ f (ω)e 2πiω·t dω provide the expansion of f in terms of the pure 'sine waves' e 2πiω·t. (When d = 1 it is often appropriate to refer to t as 'time' and ω as 'frequency.' We shall sometimes use these terms even when dealing with general d, for which the basic mathematical structure is exactly the same.) The Fourier transform is a marvelous tool, but since it involves the whole function f at once, it is not an efficient way to analyze the ways in which different frequencies enter into f at different times (as is of paramount importance, for example, in music). One way to produce a 'local' Fourier analysis of a function f is to use the windowed Fourier transform, which is also known as the short-time Fourier transform and is closely related to the cross-ambiguity function of radar theory and the Fourier–Wigner transform in …