Finding Subsampling Index Sets for Kronecker Product of Unitary Matrices for Incoherent Tight Frames

Abstract

Frames are recognized for their importance in many fields of communications, signal processing, quantum physics, and so on. In this paper, we design an incoherent tight frame by selecting some rows of a matrix that is the Kronecker product of Fourier and unitary matrices. The Kronecker-product-based frame allows its elements to have a small number of phases, regardless of the frame length, which is suitable for low-cost implementation. To obtain the Kronecker-product-based frame with low mutual coherence, we first derive an objective function by transforming the Gram matrix expression to compute the coherence. If the Hadamard matrix is employed as a unitary matrix, the objective function can be computed efficiently with low complexity. Then, we find a subsampling index set for the Kronecker-product-based frame by minimizing the objective function. In simulations, we show that the Kronecker-product-based frames can achieve similar mutual coherence to optimized harmonic frames of a large number of phases. We apply the frames to compressed sensing (CS) as the measurement matrices, where the Kronecker-product-based frames demonstrate reliable performance of sparse signal recovery.