On the Relation between Block Diagonal Matrices and Compressive Toeplitz Matrices

In a typical communications problem, Toeplitz matrices Φ arise when modeling the task of determining an unknown impulse response a from a given probe signal φ. When a is sparse, then whenever Φ formed from the probe signal φ satisfy the Restricted Isometry Property (RIP), a can be robustly recovered from its measurements via l1-minimization. In this paper, we derived the RIP for compressive Toeplitz matrices whose number of rows of the matrices J is much smaller than the number of columns N . We show that J should scale like J ∼ S log(N), where S is the sparsity of the impulse response. While this is marginally worse than the state-of-the-art scaling currently achieved in the literature, the novelty of this work comes from making the relation between the Toeplitz matrix of interest to a block diagonal matrix. The proof of the RIP then follows from using recent results on the concentration of measure inequalities of block diagonal matrices, together with a standard covering-and-counting argument.