Recovering signals from the Short-Time Fourier Transform magnitude

The problem of recovering signals from the Short-Time Fourier Transform (STFT) magnitude is of paramount importance in many areas of engineering and physics. This problem has received a lot of attention over the last few decades, but not much is known about conditions under which the STFT magnitude is a unique signal representation. Also, the recovery techniques proposed by researchers are mostly heuristic in nature. In this work, we first show that almost all signals can be uniquely identified by their STFT magnitude under mild conditions. Then, we consider a semidefinite relaxation-based algorithm and provide the first theoretical guarantees for the same. Numerical simulations complement our theoretical analysis and provide many directions for future work.