Sparse Phase Retrieval of One-Dimensional Signals by Prony's Method

Abstract

In this paper, we show that sparse signals f representable as a linear combination of a finite number N of spikes at arbitrary real locations or as a finite linear combination of B-splines of order m with arbitrary real knots can be almost surely recovered from2 up to trivial ambiguities. The constructive proof consists of two steps, where in the first step Prony's method is applied to recover all parameters of the autocorrelation function and in the second step the parameters of f are derived. Moreover, we present an algorithm to evaluate f from its Fourier intensities and illustrate it at different numerical examples.

Super Resolution Phase Retrieval for Sparse Signals

“…Perhaps surprisingly, the continuous sparse phase retrieval problem has received little attention in the literature. During the completion of this manuscript, we became aware of the work of Beinert et al [20], [21]. They propose a super-resolution approach based on the finite rate of innovation (FRI) framework, which is also one of the building blocks of the proposed algorithm in this paper.…”

“…Then, we compute the probability that the support recovery algorithm picks an element from C k instead of an element from W, when searching for the solution of (13). This happens if the cost of C k is smaller than the one of W measured via (20),…”

“…A few algorithms are designed [20] or can be adapted [14] to work with continuous supports, but they fall short when the measured support D is noisy. For example, TSPR [14] relies on the triangle equality between locations to recover the support; as soon as the locations are corrupted with noise, these equalities do not hold anymore.…”

Sparse phase retrieval of structured signals by Prony's method

“…However, if the signal f is of the form (1), than the non-trivial ambiguities of the phase retrieval problem can be entirely avoided under certain assumptions on the knots T j and coefficients c j of the signal. In order to derive these assumptions and to guarantee uniqueness of the recovery of f (up to the trivial ambiguities presented above), we follow the approach in [7], which is closely related to our setting. Theorem 2.1 Let f be a signal of the form (1), whose knot differences T j − T k differ pairwise for j, k ∈ {1, .…”

“…Note that the mapping (j, k) → is unknown and also has to be recovered. In a first step, we apply an adapted version of Prony's method [7], to determine all frequency differences τ and the corresponding coefficients γ of (2) from the equispaced samples P (h ) with = 0, . .…”