Phase Retrieval for L([-π, π]) via the Provably Accurate and Noise Robust Numerical Inversion of Spectrogram Measurements

Mark A. Iwen, Michael Perlmutter, Nada Sissouno, Aditya Viswanathan

Abstract: In this paper, we focus on the approximation of smooth functions $f: [-\pi, \pi] \rightarrow \mathbb{C}$, up to an unresolvable global phase ambiguity, from a finite set of Short Time Fourier Transform (STFT) magnitude (i.e., spectrogram) measurements. Two algorithms are developed for approximately inverting such measurements, each with theoretical error guarantees establishing their correctness. A detailed numerical study also demonstrates that both algorithms work well in practice and have good numerical convergence behavior.