A General Approach for Construction of Deterministic Compressive Sensing Matrices

Abstract

In this paper, deterministic construction of measurement matrices in Compressive Sensing (CS) is considered. First, by employing the column replacement concept, a theorem for construction of large minimum distance linear codes containing all-one codewords is proposed. Then, by applying an existing theorem over these linear codes, deterministic sensing matrices are constructed. To evaluate this procedure, two examples of constructed sensing matrices are presented. The first example contains a matrix of size ${{p}^{2}}\times {{p}^{3}}$ and coherence ${1}/{p}\;$, and the second one comprises a matrix with the size $p\left( p-1 \right)\times {{p}^{3}}$ and coherence ${1}/{\left( p-1 \right)}\;$, where $p$ is a prime integer. Based on the Welch bound, both examples asymptotically achieve optimal results. Moreover, by presenting a new theorem, the column replacement is used for resizing any sensing matrix to a greater-size sensing matrix whose coherence is calculated. Then, using an example, the outperformance of the proposed method is compared to a well-known method. Simulation results show the satisfying performance of the column replacement method either in created or resized sensing matrices.