Phase Retrieval for $$L^2([-\pi ,\pi ])$$ via the Provably Accurate and Noise Robust Numerical Inversion of Spectrogram Measurements

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.