Matched Filtering from Limited Frequency Samples

TL;DR

This paper analyzes a compressive matched filter approach that estimates signal delay and amplitude from limited noisy frequency samples, providing probabilistic bounds and showing it requires few measurements relative to noise and bandwidth.

Contribution

It introduces a theoretical analysis of a correlation-based method for delay estimation from limited frequency samples, with bounds on measurement requirements.

Findings

The expected maximum deviation decreases sharply with more measurements.

A probabilistic tail bound on the maximum deviation is derived.

The method successfully estimates delay with high probability using few measurements.

Abstract

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.