The Uniqueness of Phase Retrieval of Analytic Signals from Very Few STFT Measurements.

Analytic signals constitute a class of signals that are widely applied in time–frequency analysis such as extracting instantaneous frequency (IF) or phase derivative in the characterization of ultrashort laser pulse. The purpose of this paper is to investigate the phase retrieval (PR) problem for analytic signals in C N by short-time Fourier transform (STFT) measurements since such measurements enjoy some nice structures. Since generic analytic signals are not sparse in the time domain, the existing PR results for sparse (in time domain) signals do not apply to analytic signals. In this paper, the windows are required to be bandlimited such that they usually have the full support length N and allow us to get much better resolutions on low frequencies. By exploiting the structure of the STFT associated with bandlimited windows for analytic signals, we prove that the STFT-based phase retrieval (STFT-PR for short) of generic analytic signals can be achieved by their (3 ⌊ N 2 ⌋ + 1) measurements, where ⌊ x ⌋ denotes the largest integer that is not larger than x. Since the generic analytic signals are (⌊ N 2 ⌋ + 1) -sparse in the Fourier domain, such a number of measurements is lower than 4 N + O (1) and O (k 3) which are required in the literature for STFT-PR of all signals and of k 2 -sparse (in the Fourier domain) signals in C N 2 , respectively. Moreover, we also prove that if the length N is even and the windows are also analytic, then the number of measurements can be reduced to (3 N 2 - 1) . As an application of this, we get that the instantaneous frequency (IF) of a generic analytic signal can be exactly recovered from the STFT measurements. [ABSTRACT FROM AUTHOR]