Compressed Sensing Matrices From Fourier Matrices

The class of Fourier matrices is of special importance in compressed sensing (CS). This paper concerns deterministic construction of CS matrices from Fourier matrices. Using Katz' character sum estimation, we are able to design a deterministic procedure to select rows from a Fourier matrix to form a good CS matrix for sparse recovery. The sparsity bound in our construction is similar to that of binary CS matrices constructed by DeVore, which greatly improves previous results for CS matrices from Fourier matrices. Our approach also provides more flexibility in terms of the dimension of CS matrices. This paper also contains a useful improvement to Katz' character sum estimation for quadratic extensions, with an elementary and transparent proof. Based on this improvement, we construct a class of special CS matrices consisting of partial Fourier matrices whose columns are a union of orthonormal bases. As a consequence, our construction yields an approximately mutually unbiased bases from Fourier matrices which is of particular interest to quantum information theory. Some numerical examples are also included.