Banraw: Band-Limited Radar Waveform Design Via Phase Retrieval

This paper presents a uniqueness result which states that a band-limited signal can be recovered from at least 3B measurements where B is the bandwidth from the radar ambiguity function (AF). This function is a two-dimensional mapping of the propagation delay and Doppler frequency. This formal model represents the distortion of a returned pulse due to the receiver matched filter. To estimate a time/band-limited signal from its radar AF, a trust region algorithm that minimizes a smoothed non-convex least-squares objective function is proposed. The method consists of two steps. First, we approximate the signal by an iterative spectral algorithm. Then, the attained initialization is refined based upon a sequence of gradient iterations. To the best of our knowledge this work is seminal in the sense of solving the radar phase retrieval problem for band-limited signals. Simulations results suggest that the proposed algorithm is able to estimate band-limited signals from its radar AF for both complete and incomplete radar cases. The AF is incomplete when only few shifts are considered. Numerical results show that the proposed algorithm estimates the signal with mean-squared-error of 5\times 10^{-2} for both complete and incomplete noisy cases.