Computing the Matched Filter in Linear Time

TL;DR

This paper introduces a novel group-theoretic approach to designing waveforms that enable the matched filter algorithm to solve the time-frequency shift problem in linear time, significantly improving computational efficiency.

Contribution

The paper presents new waveforms designed via group representation theory that allow the matched filter algorithm to operate in O(p log p) time, a substantial improvement over previous methods.

Findings

Achieved linearithmic complexity for the matched filter algorithm.

Developed two new fast matched filter algorithms: the flag and cross algorithms.

Applied the algorithms to mobile communication, GPS, and radar.

Abstract

A fundamental problem in wireless communication is the time-frequency shift (TFS) problem: Find the time-frequency shift of a signal in a noisy environment. The shift is the result of time asynchronization of a sender with a receiver, and of non-zero speed of a sender with respect to a receiver. A classical solution of a discrete analog of the TFS problem is called the matched filter algorithm. It uses a pseudo-random waveform S(t) of the length p, and its arithmetic complexity is O(p^{2} \cdot log (p)), using fast Fourier transform. In these notes we introduce a novel approach of designing new waveforms that allow faster matched filter algorithm. We use techniques from group representation theory to design waveforms S(t), which enable us to introduce two fast matched filter (FMF) algorithms, called the flag algorithm, and the cross algorithm. These methods solve the TFS problem in O(p\cdot log (p)) operations. We discuss applications of the algorithms to mobile communication, GPS, and radar.