Extreme Singular Values of Random Time-Frequency Structured Matrices

Abstract: In this paper, we investigate extreme singular values of the analysis matrix of a Gabor frame $(g, \Lambda)$ with a random window $g$. Columns of such matrices are time and frequency shifts of $g$, and $\Lambda\subset \mathbb{Z}_M\times\mathbb{Z}_M$ 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 $\Lambda$, and to investigate their dependence on the structure of $\Lambda$, 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.