On discrete short-time Fourier analysis

Weyl–Heisenberg frames are a principal tool of short-time Fourier analysis. In this paper, we present a comprehensive study of Weyl–Heisenberg frames in 2( ), with a focus on frames that are tight. A number of properties of these frames are derived. A complete parameterization of finite-length windows for tightWeyl–Heisenberg frames in 2( ) is described. Design of windows for tight Weyl–Heisenberg frames requires optimization of their frequency characteristics under nonlinear constraints. We propose an efficient design method based on expansions with respect to prolate spheroidal sequences. The advantages of the proposed method over standard optimization procedures include a reduction in computational complexity and the ability to provide long windows that can be specified concisely using only a few parameters; these advantages become increasingly pronounced as the frame redundancy increases. The resilience of overcomplete Weyl–Heisenberg expansions to additive noise and quantization is also studied. We show that manifestations of degradation due to uncorrelated zero-mean additive noise are inversely proportional to the expansion redundancy, whereas the quantization error is for a given quantization step inversely proportional to the square of the expansion redundancy.