Solving multiple windowed STFT phase retrieval problems in phase and amplitude respectively

We study the Phase Retrieval (PR) problem under the phaseless short-time Fourier transform (STFT) measurements. This paper proposes a novel algorithm named PAR to solve the STFT PR problem in phase and amplitude respectively with a milder retrieval condition compared with the original methods. First, a symmetric undirected graph of signals is proposed for the computation of the relative phase. Then the retrieval conditions of STFT PR problem are discussed for a single window case and some weaker retrieval conditions are proposed compared with the LS method. We also discuss STFT PR problem in multiple windows and establish retrieval theorems without restrictions of sliding step-size L. We give some numerical results of the PAR algorithm.