Near-optimal bounds for signal recovery from blind phaseless periodic short-time Fourier transform

TL;DR

This paper establishes near-optimal bounds for recovering signals from phaseless short-time Fourier transform measurements, both when the window is known and unknown, advancing the theoretical understanding of phase retrieval in signal processing.

Contribution

It provides the first near-optimal theoretical bounds for signal recovery from phaseless STFT measurements, including the blind case where the window is unknown.

Findings

Known window case allows recovery from less than 4N measurements.

Blind case enables simultaneous recovery of signal and window from less than 4N+2W measurements.

Bounds are proven to be optimal up to a small constant.

Abstract

We study the problem of recovering a signal $x\in\mathbb{C}^N$ from samples of its phaseless periodic short-time Fourier transform (STFT): the magnitude of the Fourier transform of the signal multiplied by a sliding window $w\in \mathbb{C}^W$. We show that if the window $w$ is known, then a generic signal can be recovered, up to a global phase, from less than 4N phaseless STFT measurements. In the blind case, when the window is unknown, we show that the signal and the window can be determined simultaneously, up to a group of unavoidable ambiguities, from less than 4N+2W measurements. In both cases, our bounds are optimal, up to a constant smaller than two.