Reconstruction of Signals From Their Autocorrelation and Cross-Correlation Vectors, With Applications to Phase Retrieval and Blind Channel Estimation

Abstract

We consider the problem of reconstructing two signals from the autocorrelation and cross-correlation measurements. This inverse problem is a fundamental one in signal processing, and arises in many applications, including phase retrieval and blind channel estimation. In a typical phase retrieval setup, only the autocorrelation measurements are obtainable. We show that, when the measurements are obtained using three simple "masks", phase retrieval reduces to the aforementioned reconstruction problem.The classic solution to this problem is based on finding common factors between the z-transforms of the autocorrelation and cross-correlation vectors. This solution has enjoyed limited practical success, mainly due to the fact that it is not sufficiently stable in the noisy setting. In this work, inspired by the success of convex programming in provably and stably solving various quadratic constrained problems, we develop a semidefinite programming-based algorithm and provide theoretical guarantees. In particular, we show that almost all signals can be uniquely recovered by this algorithm (up to a global phase). Comparative numerical studies demonstrate that the proposed method significantly outperforms the classic method in the noisy setting.

Structured Signal Recovery From Quadratic Measurements: Breaking Sample Complexity Barriers via Nonconvex Optimization

“…Finally, we would like to mention that there has also been some recent publications aimed at developing theoretical guarantees for more practical models. For instance, the papers [15,32,67,42,41] develop theoretical guarantees for convex relaxation techniques for more realistic Fourier based models such as coded diffraction patterns and Ptychography. More recently, the papers [39,38,44,26,8] also develop some theoretical guarantees for faster but sometimes design-specific algorithms.…”

Fourier Phase Retrieval: Uniqueness and Algorithms

“…For some masks d 1 and d 2 , one can overcome the 'almost all' in Theorem 3.9 and obtain uniqueness of the corresponding phase retrieval problem. A different approach to exploit deterministic masks in order to overcome the ambiguity in phase retrieval is discussed in [72] and can be proven by using the characterization in Theorem 3.1. More explicitly, here the two masks…”

“…IfÑ is replaced by 2N − 1, every signal x ∈ C N is uniquely recovered up to rotation from its Fourier magnitudes y[m, k] in (2.2) with masks d 0 [n] = 1 and d i [n] = 1 + e jα i e 2π jsn/N , i = 1, 2, where α i ∈ [−π, π), and where s can be nearly every real number [14]. Several further examples of deterministic masks which allow a unique recovery are detailed in [14,30,69,72] and references therein. In Section 4.2, we consider semidefinite relaxation algorithms which stably recover the unknown signal from its masked Fourier magnitudes (2.2) under noise.…”

Short-message communication and FIR system identification using Huffman sequences

“…The blind reconstruction can hereby be re-casted as a phase retrieval problem with additional knowledge of the auto-correlations of the data and the channel at the receiver. The uniqueness of the phase retrieval problem can then be shown by constructing an explicit dual certificate in the noise free case, as was shown in [2] for almost all signals and channels. In [3] and more detailed in [4] we have shown, that the uniqueness derived in [2], holds indeed deterministically, given a particular co-prime condition is fulfilled.…”

“…In this work we will use a convex program for the channel and data reconstruction first introduced in [2] for the noise free case and show its numerical stability. The blind reconstruction can hereby be re-casted as a phase retrieval problem with additional knowledge of the auto-correlations of the data and the channel at the receiver.…”