Off-the-grid recovery of time and frequency shifts with multiple measurement vectors

Abstract

We address the problem of estimating time and frequency shifts of a known waveform in the presence of multiple measurement vectors (MMVs). This problem naturally arises in radar imaging and wireless communications. Specifically, a signal ensemble is observed, where each signal of the ensemble is formed by a superposition of a small number of scaled, time-delayed, and frequency shifted versions of a known waveform sharing the same continuous-valued time and frequency components. The goal is to recover the continuous-valued time-frequency pairs from a small number of observations. In this work, we propose a semidefinite programming which exactly recovers s pairs of time-frequency shifts from L regularly spaced samples per measurement vector under a minimum separation condition between the time-frequency shifts. Moreover, we prove that the number s of time-frequency shifts scales linearly with the number L of samples up to a log-factor. Extensive numerical results are also provided to validate the effectiveness of the proposed method over the single measurement vectors (SMVs) problem and MUSIC. In particular, we find that our approach leads to a relaxed minimum separation condition and reduced number of required samples.