text: 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.
  @cite_6: We consider signals and operators in finite dimension which have sparse time-frequency representations. As main result we show that an S-sparse Gabor representation in ℂ n with respect to a random unimodular window can be recovered by Basis Pursuit with high probability provided that S≤Cn log (n). Our results are applicable to the channel estimation problem in wireless communications and they establish the usefulness of a class of measurement matrices for compressive sensing.
summary: 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 .