On phase retrieval of finite-length sequences using the initial time sample

The problem of recovering a finite-length sequence from the Fourier transform magnitude and the exact value of the first time sample, which has been recently shown to be almost always possible, is shown to be NP-complete. It is shown that recovery to within a unit magnitude complex constant is possible if the magnitude of the end point is within an uncertainty interval. Larger uncertainty leads to the recovery of a set of feasible solutions that can be used to recover the sequence if additional information is available. A branch-and-bound algorithm to find all solutions which offers significant improvements over exhaustive search is presented. This algorithm can be modified to incorporate the knowledge of additional time samples or knowledge of the fact that the sequence is real to help make a unique recovery.<>