
Software & Hardware
Software
NonConvex Phase Retrieval from STFT Measurements
T. Bendory and Y. C. Eldar
Introduction
The problem of recovering a signal from its Fourier transform magnitude, called phase retrieval, arises in many
areas in engineering and science, such as optics, Xray crystallography, speech recognition, blind channel estimation
and astronomy. Phase retrieval for onedimensional signals is an illposed problem in most cases. We consider the
closelyrelated problem of recovering a signal from its phaseless shorttime Fourier transform (STFT) measurements.
This problem arises naturally in several applications, such as ultrashort pulse measurements and ptychography.
In contrast to previous works in the field of phase retrieval, we aim at developing a phase retrieval algorithm
that reflects a practical setup, is computationally efficient and enjoys theoretical guarantees.
Main Idea
The algorithm begins by taking the onedimensional DFT of the acquired information with respect to the frequency
variable (the second variable of the STFT). This transformation reveals the underlying structure of the data and
greatly simplifies the analysis. Then, we suggest recovering the signal by minimizing a nonconvex loss function
(frequently called empirical risk or nonlinear leastsquares) using a gradient algorithm. We propose to initialize
the gradient algorithm by the principle eigenvector (with proper normalization) of an approximation matrix that
approximates the correlation matrix of the underlying signal. This approximation matrix is constructed as the
solution of a simple leastsquares problem. For a detailed description and analysis of the algorithm,
see "NonConvex Phase Retrieval from STFT Measurements"
side view
view from above
The figure presents the twodimensional plane (first two variables) of the nonconvex
loss function to be minimized of the signal x=[0.2,0.2,0,0,0].
A Representative Example
The following figures show a representative example of the algorithm's performance. The experiment was
conducted on a signal of length N = 23 with L=1, a rectangular window of length W=7 in a noisy environment
of SNR= 20 db. The upperleft and upperright figures present the initialization and recovered signal versus
the underlying signal, respectively. The figure below presents the error and objective function curves
as a function of the iterations.
References
Software Download
Installation:
1. Unzip.
2. Follow the instructions in the Readme.txt

