On statistical restricted isometry property of a new class of deterministic partial Fourier compressed sensing matrices

Compressed sensing is a novel technique where one can recover sparse signals from the undersampled measurements. In this paper, a new class of partial Fourier matrices is studied for deterministic compressed sensing. A basic partial Fourier matrix is constructed by choosing the rows deterministically from the inverse discrete Fourier transform (DFT) matrix. By a column rearrangement, the matrix is represented as a concatenation of DFT-based submatrices. Then, a full or a part of columns of the concatenated matrix is used to form a K × N sensing matrix for deterministic compressed sensing. It is shown that the sensing matrix forms a tight frame with nearly optimal coherence. Theoretically, the sensing matrix turns out to have the statistical restricted isometry property (StRIP) for unique sparse recovery guarantee.