Toeplitz Block Matrices in Compressed Sensing

Recent work in compressed sensing theory shows that $n\times N$ independent and identically distributed (IID) sensing matrices whose entries are drawn independently from certain probability distributions guarantee exact recovery of a sparse signal with high probability even if $n\ll N$. Motivated by signal processing applications, random filtering with Toeplitz sensing matrices whose elements are drawn from the same distributions were considered and shown to also be sufficient to recover a sparse signal from reduced samples exactly with high probability. This paper considers Toeplitz block matrices as sensing matrices. They naturally arise in multichannel and multidimensional filtering applications and include Toeplitz matrices as special cases. It is shown that the probability of exact reconstruction is also high. Their performance is validated using simulations.