On convexity and identifiability in 1-D Fourier phase retrieval

This paper considers phase retrieval from the magnitude of 1-D oversampled Fourier measurements. We first revisit the well-known lack of identifiability in this case, and point out that there always exists a solution that is minimum phase, even though the desired signal is not. Next, we explain how...

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Published in2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) pp. 3941 - 3945
Main Authors Kejun Huang, Eldar, Yonina C., Sidiropoulos, Nicholas D.
Format Conference Proceeding Journal Article
LanguageEnglish
Published IEEE 01.03.2016
Subjects
Acoustics
auto-correlation estimation
Discrete Fourier transforms
Electronic mail
Electronics
Fourier analysis
Fourier transforms
Impulses
Manganese
minimum phase
Noise measurement
Optimization
over-sampled Fourier measurements
Phase measurement
Phase retrieval
semi-definite programming
Spectra
Speech
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Summary:This paper considers phase retrieval from the magnitude of 1-D oversampled Fourier measurements. We first revisit the well-known lack of identifiability in this case, and point out that there always exists a solution that is minimum phase, even though the desired signal is not. Next, we explain how the least-squares formulation of this problem can be optimally solved via PhaseLift followed by spectral factorization, and this solution is always minimum phase. A simple approach is then proposed to circumvent non-identifiability: adding an impulse to an arbitrary complex signal (offset to the Fourier transform) before taking the quadratic measurements, so that a minimum phase signal is constructed and thus can be uniquely estimated. Simulations with synthetic data show the effectiveness of the proposed method.
Bibliography:ObjectType-Article-2
SourceType-Scholarly Journals-1
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SourceType-Conference Papers & Proceedings-2
ISSN:2379-190X
DOI:10.1109/ICASSP.2016.7472416