Geometry of random Toeplitz-block sensing matrices: Bounds and implications for sparse signal processing

A rich body of literature has emerged during the last decade that seeks to exploit the sparsity of a signal for a reduction in the number of measurements required for various inference tasks. Much of the initial work in this direction has been for the case when the measurements correspond to a projection of the signal of interest onto the column space of (sub)Gaussian and subsampled Fourier matrices. The physics in a number of applications, however, dictates the use of "structured" matrices for measurement purposes. This has led to a recent push in the direction of structured measurement (or sensing) matrices for inference of sparse signals. This paper complements some of the recent work in this direction by studying the geometry of Toeplitz-block sensing matrices. Such matrices are bound to arise in any system that can be modeled as a linear, time-invariant (LTI) system with multiple inputs and single output. The reported results therefore should be of particular benefit to researchers interested in exploiting sparsity in LTI systems with multiple inputs.