Super-resolution radar

Abstract

In this paper, we study the identification of a time-varying linear system from its response to a known input signal. More specifically, we consider systems whose response to the input signal is given by a weighted superposition of delayed and Doppler-shifted versions of the input. This problem arises in a multitude of applications such as wireless communications and radar imaging. Due to practical constraints, the input signal has finite bandwidth |$B$|⁠, and the received signal is observed over a finite time interval of length |$T$| only. This gives rise to a delay and Doppler resolution of |$1/B$| and |$1/T$|⁠. We show that this resolution limit can be overcome, i.e., we can exactly recover the continuous delay–Doppler pairs and the corresponding attenuation factors, by solving a convex optimization problem. This result holds provided that the distance between the delay–Doppler pairs is at least |$2.37/B$| in time or |$2.37/T$| in frequency. Furthermore, this result allows the total number of delay–Doppler pairs to be linear up to a log-factor in |$BT$|⁠, the dimensionality of the response of the system, and thereby the limit for identifiability. Stated differently, we show that we can estimate the time–frequency components of a signal that is |$S$|-sparse in the continuous dictionary of time–frequency shifts of a random window function, from a number of measurements that is linear up to a log-factor in |$S$|⁠.