用Prony方法实现动态采样中的相位恢复和系统辨识
Phase Retrieval and System Identification in Dynamical Sampling via Prony's Method
摘要：
动态采样中的相位恢复是一个新的研究方向，它要求从相对于动态坐标系的无相位测量中恢复未知信号，即由算子重复作用构造的采样向量序列。这里相位的损失将适定的动态采样变成严重的不适定逆问题。在现有文献中，所涉及的算子通常是完全已知的。本文将动态采样中的相位恢复与系统辨识相结合。例如，如果动态帧基于重复卷积，那么我们希望提前恢复未知的卷积核。利用Prony方法，我们建立了信号和系统的几个恢复保证，其证明是构造性的，并给出了解析的恢复方法。几乎所有的信号、算子和采样向量都满足所需的假设。此外，这些保证不仅适用于有限维的设置，但也结转到无限维空间。研究了解析恢复过程的灵敏度，我们还建立了复指数和近似Prony方法的误差界。
英文摘要：
Phase retrieval in dynamical sampling is a novel research direction, where an unknown signal has to be recovered from the phaseless measurements with respect to a dynamical frame, i.e. a sequence of sampling vectors constructed by the repeated action of an operator. The loss of the phase here turns the well-posed dynamical sampling into a severe ill-posed inverse problem. In the existing literature, the involved operator is usually completely known. In this paper, we combine phase retrieval in dynamical sampling with the identification of the system. For instance, if the dynamical frame is based on a repeated convolution, then we want to recover the unknown convolution kernel in advance. Using Prony's method, we establish several recovery guarantees for signal and system, whose proofs are constructive and yield analytic recovery methods. The required assumptions are satisfied by almost all signals, operators, and sampling vectors. Moreover, these guarantees not only hold for the finite-dimensional setting but also carry over to infinite-dimensional spaces. Studying the sensitivity of the analytic recovery procedures, we also establish error bounds for the applied approximate Prony method with respect to complex exponential sums.
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