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.