Phase retrieval of analytic signals from short-time Fourier transform measurements

Analytic signals belong to a widely applied class of signals especially for extracting instantaneous frequency (IF) or phase derivative which is very helpful in the characterization of ultrashort laser pulse. In this paper we investigatethe phase retrieval (PR) problem for analytic signals in $\mathbb{C}^{N}$ by their short-time Fourier transform (STFT) measurements. When the window is bandlimited, it was found that the corresponding STFT of an analytic signal enjoys nice structures. Exploiting such structures, our main results state that the STFT-PR of generic analytic signals can be achieved by their $(3\lfloor\frac{N}{2}\rfloor+1)$ measurements. Note that the generic analytic signals are $(\lfloor \frac{N}{2}\rfloor+1)$-sparse in 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 signals in $\mathbb{C}^{N}$ and of $k$-sparse (in Fourier domain) signals, respectively. Furthermore, if the length $N$ is even and the windows are also analytic then the number of measurements can be reduced to $(\frac{3 N}{2}-1)$. PR approaches are established for different cases of window bandlimit. As an application, the IF of a generic analytic signal can be exactly recovered from the PR results. Several examples are provided to demonstrate the main results and their applications.