Phase retrieval of analytic signals from short-time Fourier transform 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 $\mathbb{C}^{N}$ by short-time Fourier transform (STFT) measurements since they enjoy some very nice structures. Since generic analytic signals are generally not sparse in the time domain, the existing PR results for sparse (in time domain) signals do not apply to analytic signals. We will use bandlimited windows that usually have the full support length $N$ which allows us to get much better resolutions on low frequencies. More precisely, by exploiting the structure of the STFT for analytic signals, we prove that the STFT based phase retrieval (STFT-PR for short) of generic analytic signals can be achieved by their $(3\lfloor\frac{N}{2}\rfloor+1)$ measurements. Since the generic analytic signals are $(\lfloor \frac{N}{2}\rfloor+1)$-sparse in the Fourier domain, such a number of measurements is lower than $4N+\hbox{O}(1)$ and $\hbox{O}(k^{3})$ which are required in the literature for STFT-PR of all signals and of $k^{2}$-sparse (in the Fourier domain) signals in $\mathbb{C}^{N^{2}}$, 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 $(\frac{3 N}{2}-1)$. 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.