Phase retrieval of real-valued signals in a shift-invariant space

Abstract

Phase retrieval arises in various fields of science and engineering and it is well studied in a finite-dimensional setting. In this paper, we consider an infinite-dimensional phase retrieval problem to reconstruct real-valued signals living in a shiftinvariant space from its phaseless samples taken either on the whole line or on a set with finite sampling rate. We find the equivalence between nonseparability of signals in a linear space and its phase retrievability with phaseless samples taken on the whole line. For a spline signal of order N , we show that it can be well approximated, up to a sign, from its noisy phaseless samples taken on a set with sampling rate 2N −1. We propose an algorithm to reconstruct nonseparable signals in a shift-invariant space generated by a compactly supported continuous function φ. The proposed algorithm is robust against bounded sampling noise and it could be implemented in a distributed manner.

Phaseless inverse discrete Hilbert transform and determination of signals in shift‐invariant space

“…Please refer to Cheng et al, Sun et al, and Sun for the related developments of phaseless sampling in a shift‐invariant space . Recall that in previous studies,() the generator φ for V ( φ ) is required to be compactly supported. On the other hand, the Hilbert transform of a compactly supported function is commonly not compactly supported.…”

“…Consequently, if ϕ is compactly supported, then the phaseless sampling results in Cheng et al and Sun do not hold for in . Unlike the methods in previous studies,() we do not intend to directly determine the target signal in by its phaseless sampling. Instead, we first determine the Hilbert transform of the target.…”

“…It aims at reconstructing a real-valued signal f from samples of |f|. Please refer to Cheng et al, 16 Sun et al, 21 and Sun 17 for the related developments of phaseless sampling in a…”

“…Recall that in previous studies, 16,17,21 the generator for V( ) is required to be compactly supported. On the other hand, the Hilbert transform of a compactly supported function is commonly not compactly supported.…”

Phaseless Sampling and Reconstruction of Real-Valued Signals in Shift-Invariant Spaces

“…A spatial signal f on a domain D is defined by its evaluations f (x), x ∈ D. In this paper, we consider the problem whether and how a real-valued signal f can be reconstructed, up to a global sign, from magnitude information |f (x)|, x ∈ D, or from its phaseless samples |f (γ)|, γ ∈ Γ, taken on a discrete set Γ ⊂ D in a stable way. The above problem has been discussed for bandlimited signals [39] and wavelet signals residing in a principal shiftinvariant space [13,14,38]. It is a nonlinear ill-posed problem which can be solved only if we have some extra information about the signal f .…”