Multiwindow Approaches for Direct and Stable STFT Phase Retrieval

SIAM Journal on Applied Mathematics, vol. 85, pp. 2376–2398

Abstract

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 multiwindow approaches that rely on “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 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 is, the wider the ambiguity function is but with zeros on circles in the time-frequency domain. Combining Hermite functions of different degrees, we can achieve wide coverage with zeros compensated by the essential support of the ambiguity function from other Hermite windows. Taking advantage of these multi-multiwindow procedures, we can stably perform STFT phase retrieval using the direct inversion formula.

Authors 2

1. Rima Alaifari Aachen Department of Mathematics

   RWTH Aachen University

   Affiliation as printed

   Department of Mathematics, RWTH Aachen University, Germany

2. Yunan Yang

   Cornell University

   Affiliation as printed

   Department of Mathematics, Cornell University, Ithaca, NY 14853 USA