Many audio signal processing methods are formulated in the time-frequency (T-F) domain which is obtained by the short-time Fourier transform (STFT). The properties of the STFT are fully characterized by window function, number of frequency channels, and time-shift. Thus, designing a better window is important for improving the performance of the processing especially when a less redundant T-F representation is desirable. While many window functions have been proposed in the literature, they are designed to have a good frequency response for analysis, which may not perform well in terms of signal processing. The window design must take the effect of the reconstruction (from the T-F domain into the time domain) into account for improving the performance. In this paper, an optimization-based design method of a nearly tight window is proposed to obtain a window performing well for the T-F domain signal processing.
Weyl-Heisenberg frames are a principal tool of short-time Fourier analysis. We present a comprehensive study of Weyl-Heisenberg frames in l sup 2 (Z), 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 tight Weyl-Heisenberg frames in l sup 2 (Z) 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. Redundant Gabor frames admit an infinite number of dual frames, yet only the canonical dual Gabor system, constructed from the minimal l2-norm dual window, is widely used. This window function however, might lack desirable properties, e.g. good time–frequency concentration, small support or smoothness. We employ convex optimization methods to design dual windows satisfying the Wexler–Raz equations and optimizing various constraints. Numerical experiments suggest that alternate dual windows with considerably improved features can be found.