  In this paper, we study a simple correlation-based strategy for estimating
the unknown delay and amplitude of a signal based on a small number of noisy,
randomly chosen frequency-domain samples. We model the output of this
"compressive matched filter" as a random process whose mean equals the scaled,
shifted autocorrelation function of the template signal. Using tools from the
theory of empirical processes, we prove that the expected maximum deviation of
this process from its mean decreases sharply as the number of measurements
increases, and we also derive a probabilistic tail bound on the maximum
deviation. Putting all of this together, we bound the minimum number of
measurements required to guarantee that the empirical maximum of this random
process occurs sufficiently close to the true peak of its mean function. We
conclude that for broad classes of signals, this compressive matched filter
will successfully estimate the unknown delay (with high probability, and within
a prescribed tolerance) using a number of random frequency-domain samples that
scales inversely with the signal-to-noise ratio and only logarithmically in the
in the observation bandwidth and the possible range of delays.
