Blind Phaseless Short-Time Fourier Transform Recovery

TL;DR

This paper establishes that the number of measurements needed for blind phaseless short-time Fourier transform recovery is linearly proportional to the signal parameters, significantly improving previous quadratic bounds, and also explores related phase retrieval problems.

Contribution

It proves a linear measurement bound for unique recovery of generic signals in blind phaseless STFT, improving upon prior quadratic bounds, and analyzes phase retrieval with partial signal knowledge.

Findings

Linear measurement bound for blind phaseless STFT recovery

Exact measurement count for blind STFT with known phases

Bound on measurements needed when some signal entries are known

Abstract

The problem of recovering a pair of signals from their blind phaseless short-time Fourier transform measurements arises in several important phase retrieval applications, including ptychography and ultra-short pulse characterization. In this paper, we prove that in order to determine a pair of generic signals uniquely, up to trivial ambiguities, the number of phaseless measurements one needs to collect is, at most, five times the number of parameters required to describe the signals. This result improves significantly upon previous papers, which required the number of measurements to be quadratic in the number of parameters rather than linear. In addition, we consider the simpler problem of recovering a pair of generic signals from their blind short-time Fourier transform, when the phases are known. In this setting, which can be understood as a special case of the blind deconvolution problem, we show that the number of measurements required to determine the two signals, up to trivial ambiguities, equals exactly the number of parameters to be recovered. As a side result, we study the classical phase retrieval problem---that is, recovering a signal from its Fourier magnitudes---when some entries of the signal are known apriori. We derive a bound on the number of required measurements as a function of the size of the set of known entries. Specifically, we show that if most of the signal's entries are known, then only a few Fourier magnitudes are necessary to determine a signal uniquely.