AUTHOR=Beinert Robert , Plonka Gerlind 

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

JOURNAL=Frontiers in Applied Mathematics and Statistics

VOLUME=3

YEAR=2017

URL=https://www.frontiersin.org/journals/applied-mathematics-and-statistics/articles/10.3389/fams.2017.00005

DOI=10.3389/fams.2017.00005

ISSN=2297-4687

ABSTRACT=<p>In this paper, we show that sparse signals <italic>f</italic> representable as a linear combination of a finite number <italic>N</italic> of spikes at arbitrary real locations or as a finite linear combination of B-splines of order <italic>m</italic> with arbitrary real knots can be almost surely recovered from <inline-formula><mml:math id="M1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi mathvariant="-tex-caligraphic">O</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula> intensity measurements <inline-formula><mml:math id="M2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mi mathvariant="-tex-caligraphic">F</mml:mi></mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>ω</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>|</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> 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 <italic>f</italic> are derived. Moreover, we present an algorithm to evaluate <italic>f</italic> from its Fourier intensities and illustrate it at different numerical examples.</p>