On the Injectivity of STFT Phase Retrieval with super-exponential decaying window function

TL;DR

This paper addresses the injectivity of STFT phase retrieval for signals in $L^2(\mathbb{R})$ when the window's Fourier transform decays super-exponentially, i.e., $|\hat{g}(\xi)| \le C e^{-a|\xi|^m}$ with $m>1$. It develops a Paley–Wiener–type framework to determine the analytic order $\rho=\frac{m}{m-1}$ and type $\tau=\frac{m-1}{m}(2\pi)^{m/(m-1)}(am)^{-1/(m-1)}$, and extends the STFT to an entire function in two complex variables to enable a discrete uniqueness analysis. The main result shows that a explicitly constructed lattice $\Lambda=\{(\pm\tau_1 n^{(m-1)/m},\pm\tau_2 n^{1/m})\}$, with bounds on $\tau_1$ and $\tau_2$, yields a uniqueness set: if $|V_g f(\lambda)|=|V_g h(\lambda)|$ for all $\lambda\in\Lambda$, then $f= e^{i\alpha}h$ for some $\alpha$. The approach generalizes prior Gaussian-window injectivity results and identifies a sharp, constructive sampling regime for STFT phase retrieval with super-exponentially decaying windows, with potential impact on practical signal reconstruction from discrete spectrogram measurements.

Abstract

We investigate the uniqueness of short-time Fourier transform phase retrieval problems in $L^2(\mathbb{R})$. In particular, for underlying window functions whose Fourier transform decay faster than any exponential function, we derive sufficient conditions on discrete sampling sets for unique phase retrieval from the spectrogram. This result generalizes previous uniqueness guarantees on sampling sets for Gaussian windows.

Figures (1)

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Theorems & Definitions (22)

• theorem 1
• Theorem A
• lemma 1
• proof
• remark 1
• corollary 1
• proof
• Theorem B
• lemma 2
• proof