A PROBABILISTIC ANALYSIS OF THE COMPRESSIVE MATCHED FILTER

In this paper we study the “compressive matched filter,” a correlation-based technique for estimating the unknown delay and amplitude of a signal using only a small number of randomly chosen (and possibly noisy) frequency-domain samples of that signal. To study the performance of this estimator, we model its output as a random process and—borrowing from analytical techniques that have been used to derive state-of-the-art signal recovery bounds in the field of compressive sensing— we derive a lower bound on the number of samples needed to guarantee successful operation of the compressive matched filter. Our analysis allows the roles of time and frequency to be exchanged, and we study the particular problem of estimating the frequency of a pure sinusoidal tone from a small number of random samples in the time domain. Thus, for signals parameterized by an unknown translation in either the time or frequency domain, our theoretical bounds and experimental results confirm that random measurements provide an economical means for capturing and recovering such information. Keywords— Compressive sensing, Matched filtering, Tone estimation, Random processes