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_8: We show that for any random matrix (X ij ) with independent mean zero entries E∥(X ij )∥ ≤ C(max √ΣEX 2 ij + max √ΣEX 2 ij + 4√ΣEX 4 ij ), where C is some universal constant.
summary: Before we study the case when @math is a Gabor frame, we include here a short overwiew of the results on the singular values for random frames with independent entries, such as Gaussian matrices. The largest singular value of the analysis matrix @math of a random frame with independent entries can be estimated using Latala’s theorem @cite_8 . The following result implies that, with high probability, @math .