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 shift-invariant 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. The proposed algorithm is robust against bounded sampling noise and it could be implemented in a distributed manner.

References in corpus (5)

• Reconstruction of Bandlimited Functions from Unsigned Samples
• PRIME: Phase Retrieval via Majorization-Minimization
• Phase Retrieval via Structured Modulations in Paley-Wiener Spaces
• Spatially Distributed Sampling and Reconstruction
• Phase Retrieval using Lipschitz Continuous Maps

Cited by in corpus (6)

• Stable Phase Retrieval in Infinite Dimensions
• Local and Global Phaseless Sampling in Real Spline Spaces
• Phaseless Sampling and Linear Reconstruction of Functions in Spline Spaces
• Phaseless Reconstruction from Space-Time Samples
• Conjugate Phase Retrieval on ${\mathbb{C}}^M$ by real vectors
• Undersampled windowed exponentials and their applications