论文标题
来自短时傅立叶变换测量的分析信号的相位检索
Phase retrieval of analytic signals from short-time Fourier transform measurements
论文作者
论文摘要
分析信号构成了一类信号，这些信号广泛应用于时频分析中，例如提取瞬时频率（如果）或相位导数在超短激光脉冲的表征中。本文的目的是通过短期傅立叶变换（STFT）测量结果调查$ \ mathbb {C}^{n} $中分析信号的相位检索问题（PR）问题，因为它们享受了一些非常好的结构。由于通用分析信号通常在时间域中并不稀疏，因此稀疏（时域）信号的现有PR结果不适用于分析信号。我们将使用通常具有完整支撑长度$ n $的带限制窗口，这使我们能够在低频上获得更好的分辨率。更准确地说，通过利用STFT的结构进行分析信号，我们证明，基于STFT的相位检索（STFT-PR短）可以通过其$（3 \ lfloor \ frac {n} {n} {2} {2} {2} \ rfloor+1）$测量来实现。由于通用的分析信号为$（\ lfloor \ frac {n} {2} \ rfloor+1）$ - 在傅立叶域中稀疏，因此此类测量值低于$ 4N+\ hbox {o}（O}（O}（O}（1）$和$ \ hbox {o}（o} o}（k^{3} $），该文献的文献和n. $ k^{2} $ - 分别在$ \ mathbb {c}^{n^{2}} $中的缩写（在傅立叶域中）信号。此外，我们还证明，如果长度$ n $均匀并且窗口也是分析性的，则可以将测量数减少到$（\ frac {3 n} {2} -1）$。作为此应用，我们可以从STFT测量中准确恢复通用分析信号的瞬时频率（IF）。
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.
