  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.
