One- and two-dimensional minimum and nonminimum phase retrieval by solvinglinear systems of equations

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.