Generalized Haar condition-based phaseless random sampling for compactly supported functions in shift-invariant spaces

08/15/2019
by   Youfa Li, et al.
0

It is proved that the phase retrieval (PR) in the linear-phase modulated shift-invariant space (SIS) V(e^iα·φ), α≠0, is impossible even though the real-valued φ enjoys the full spark property (so does e^iα·φ). Stated another way, the PR in the complex-generated SISs is essentially different from that in the real-generated ones. Motivated by this, we first establish the condition on the complex-valued ϕ such that the PR of compactly supported and nonseparable (CSN) functions in V(ϕ) can be achieved by random phaseless sampling. The condition is established from the perspective of the Lebesgue measure of the zero set of a related function system, or more precisely from the generalized Haar condition (GHC). Based on the proposed reconstruction approach, it is proved that if the GHC holds, then the PR of CSN functions in the complex-generated SISs can be achieved with probability 1, provided that the phaseless random sampling density (SD) ≥3. For the real-generated case we also prove that, if the GHC holds then the PR of real-valued CSN functions can be achieved with the same probability if the random SD ≥2. Recall that the deterministic SD for PR depends on Haar condition(measured in terms of the cardinality of the corresponding zero set). Compared with deterministic sampling, the proposed random sampling enjoys not only the greater sampling flexibility but the lower SD. For the lower SD, the highly oscillatory signals such as chirps can be efficiently reconstructed. To verify our results, numerical simulations were conducted to reconstruct CSN functions in the chirp-modulated SISs.

READ FULL TEXT

page 1

page 2

page 3

page 4

01/08/2018

Sampling Almost Periodic and related Functions

We consider certain finite sets of circle-valued functions defined on in...
06/02/2020

A Randomized Algorithm to Reduce the Support of Discrete Measures

Given a discrete probability measure supported on N atoms and a set of n...
02/26/2021

Single-angle Radon samples based reconstruction of functions in refinable shift-invariant space

The traditional approaches to computerized tomography (CT) depend on the...
09/04/2019

Phase retrieval of complex and vector-valued functions

The phase retrieval problem in the classical setting is to reconstruct r...
09/24/2021

Linear convergence of randomized Kaczmarz method for solving complex-valued phaseless equations

A randomized Kaczmarz method was recently proposed for phase retrieval, ...
08/20/2021

Structure and Interleavings of Relative Interlevel Set Cohomology

The relative interlevel set cohomology (RISC) is an invariant of real-va...
12/06/2021

Associative Memories Using Complex-Valued Hopfield Networks Based on Spin-Torque Oscillator Arrays

Simulations of complex-valued Hopfield networks based on spin-torque osc...