Phase retrieval for radar waveform design.

This article focuses on the inverse problem of radar waveform design by reconstructing transmit signals from a specified ambiguity function (AF) magnitude, which characterizes radar resolution in range and Doppler velocity. It establishes uniqueness results showing that time- or band-limited complex-valued signals can be recovered from their phaseless AF measurements—up to trivial ambiguities—using at least \(3S-1\) or \(3B-1\) measurements, where \(S\) and \(B\) denote the signal's time duration and bandwidth, respectively. The authors propose a two-step algorithm combining a novel spectral initialization with a trust-region gradient descent method that minimizes a smoothed non-convex least-squares objective, providing convergence guarantees even under incomplete and noisy AF data. Extensive numerical experiments demonstrate the method's effectiveness in recovering various radar signals, including linear and nonlinear frequency modulated pulses, from both complete and undersampled AF measurements. This work advances phase retrieval theory applied to radar waveform synthesis by addressing the unique challenges posed by the AF's conjugate structure and associated ambiguities.