Multi-Window Approaches for Direct and Stable STFT Phase Retrieval

TL;DR

This paper introduces multi-window methods using fractional Fourier and Hermite functions to achieve stable phase retrieval from STFT measurements, overcoming instability issues of single-window approaches.

Contribution

It proposes novel multi-window techniques that ensure broad coverage of the time-frequency plane, enabling stable STFT phase retrieval with explicit inversion formulas.

Findings

Achieves stable phase retrieval using multi-window approaches.

Broadens time-frequency coverage with fractional Fourier and Hermite functions.

Demonstrates practical stability in phase retrieval process.

Abstract

Phase retrieval from phaseless short-time Fourier transform (STFT) measurements is known to be inherently unstable when measurements are taken with respect to a single window. While an explicit inversion formula exists, it is useless in practice due to its instability. In this paper, we overcome this lack of stability by presenting two multi-window approaches that rely on a "good coverage" of the time-frequency plane by the ambiguity functions of the windows. The first is to use the fractional Fourier transform of a dilated Gauss function with various angles as window functions. The essential support of a superposition of the ambiguity function from such window functions is of a "daffodil shape", which converges to a large disc as more angles are used, yielding a much broader coverage in the time-frequency domain. The second approach uses Hermite functions of various degrees as the window functions. The larger the degree, the wider the ambiguity function but with zeros on circles in the time-frequency domain. Combining Hermite functions of different degrees, we can achieve a wide coverage with zeros compensated by the essential support of the ambiguity function from other Hermite windows. Taking advantage of these multi-window procedures, we can stably perform STFT phase retrieval using the direct inversion formula.