In this paper, we investigate extreme singular values of the analysis matrix of a Gabor frame @math with a random window @math . Columns of such matrices are time and frequency shifts of @math , and @math is the set of time-frequency shift indices. Our aim is to obtain bounds on the singular values of such random time-frequency structured matrices for various choices of the frame set @math , and to investigate their dependence on the structure of @math , as well as on its cardinality. We also compare the results obtained for Gabor frame analysis matrices with the respective results for matrices with independent identically distributed entries.
To the best of our knowledge, singular values of the analysis matrix @math of a Gabor frame with random window and general @math with @math were not studied before. At the same time, the following bounds on the singular values of the synthesis matrix @math of a Gabor system @math with a Steinhaus window @math and @math have been established in @cite_6 .
@cite_6
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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.