On signal reconstruction from its spectrogram

This paper presents a framework for discrete-time signal reconstruction from absolute values of its short-time Fourier coefficients. Our approach has two steps. In step one we reconstruct a band-diagonal matrix associated to the rank-one operator K_x = xx*. In step two we recover the signal x by solving an optimization problem. The two steps are somewhat independent, and one purpose of this paper is to present a framework that decouples the two problems. The solution to the first step is connected to the problem of constructing frames for spaces of Hilbert-Schmidt operators. The second step is somewhat more elusive. Due to inherent redundancy in recovering x from its associated rank-one operator K_x, the reconstruction problem allows for imposing supplemental conditions. In this paper we make one such choice that yields a fast and robust reconstruction. However this choice may not necessarily be optimal in other situations. It is worth mentioning that this second step is related to the problem of finding a rank-one approximation to a matrix with missing data.