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.

翻译：解析信号是一类广泛应用于时频分析的信号，例如在超短激光脉冲表征中提取瞬时频率（IF）或相位导数。本文旨在研究基于短时傅里叶变换（STFT）测量的解析信号在$\mathbb{C}^{N}$中的相位恢复（PR）问题，因为此类信号具有优良的结构特性。由于一般解析信号在时域中并不稀疏，现有的针对（时域）稀疏信号的相位恢复结果不适用于解析信号。我们将采用通常具有完整支撑长度$N$的带限窗函数，这使我们能够获得更好的低频分辨率。更具体地，通过利用解析信号STFT的结构特性，我们证明基于STFT的相位恢复（简称STFT-PR）可通过其$(3\lfloor\frac{N}{2}\rfloor+1)$个测量实现一般解析信号的相位重建。由于一般解析信号在傅里叶域中具有$(\lfloor \frac{N}{2}\rfloor+1)$-稀疏性，该测量数低于文献中STFT-PR对所有信号所需的$4N+\hbox{O}(1)$以及对$\mathbb{C}^{N^{2}}$中$k^{2}$-稀疏（傅里叶域）信号所需的$\hbox{O}(k^{3})$。此外，我们还证明：若长度$N$为偶数且窗函数为解析的，则测量数可减少至$(\frac{3 N}{2}-1)$。作为该结果的应用，我们得出一般解析信号的瞬时频率（IF）可通过STFT测量精确重建。