WaveMax: FrFT-Based Convex Phase Retrieval for Radar Waveform Design

Abstract

We consider the recovery of a complex band-limited radar waveform from the magnitude of the fractional Fourier transform (FrFT) formulation of its ambiguity function (AF). This is essentially a phase retrieval (PR) problem applied to radar waveform design. The FrFT-based AF is mathematically obtained by correlating the signal with its frequency-rotated, Doppler-shifted, and delayed replicas. It completely characterizes the radar's capability to discriminate closely-spaced targets in the delay-Doppler plane. Unlike prior works which largely involved analytical approaches, our method WaveMax formulates the recovery of the waveform via the FrFT-based AF PR as a convex optimization problem. Specifically, we retrieve the signal by solving a basis pursuit that requires a designed approximation of the radar signal obtained by extracting the leading eigenvector of a matrix depending on the AF. Our theoretical analysis shows that unique waveform reconstruction is possible using signal samples no more than thrice the number of signal frequencies or time samples. Numerical experiments demonstrate that our method recovers band-limited signals from both even-sparse and random samples of the AFs with a mean squared error of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$1\times 10^{-6}$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$5\times 10^{-2}$</tex> for full noiseless samples and sparse noisy samples, respectively.