Ae. Yagle et Ae. Bell, One- and two-dimensional minimum and nonminimum phase retrieval by solvinglinear systems of equations, IEEE SIGNAL, 47(11), 1999, pp. 2978-2989
The discrete phase retrieval problem is to reconstruct a discrete time sign
al whose support is known and compact from the magnitude of its discrete Fo
urier transform. We formulate the problem as a linear system of equations;
our methods do not require polynomial rooting, tracking zero curves of alge
braic functions, or any sort of iteration like previous methods. Our soluti
ons obviate the stagnation problems associated with iterative algorithms, a
nd our solutions are computationally simpler and more stable than alternati
ve noniterative algorithms. Furthermore, our methods can explicitly accommo
date noisy Fourier magnitude information through the use of total least squ
ares type techniques. We assume either of the following tno types of a prio
ri knowledge of the signal: 1) a band of known values (which may be zeros)
or 2) some known values of a subminimum phase signal (whose zeros lie insid
e a disk of radius greater than unity). We illustrate our methods with nonm
inimum-phase one-dimensional (1-D) and two-dimensional (2-D) signals.