Everything you always wanted to know about matched filters (but were afraid to ask)

TL;DR

This paper reviews the matched filter technique for detecting weak signals in noisy data, focusing on practical issues like discretization and non-Gaussian noise, with simplified explanations for better understanding.

Contribution

It provides a practical perspective on matched filters, addressing issues often overlooked in textbooks, and simplifies the mathematics to enhance comprehension of real-world applications.

Findings

Analyzes effects of signal discretization on filter performance

Examines impact of non-Gaussian noise on detection accuracy

Provides practical guidance for applying matched filters in astronomy

Abstract

In this paper we review the application of the matched filter (MF) technique and its application to detect weak, deterministic, smooth signals in a stationary, random, Gaussian noise. This is particular suitable in astronomy to detect emission lines in spectra and point-sources in two-dimensional maps. A detailed theoretical development is already available in many books (e.g. Kay 1998; Poor 1994; McNicol 2005; Hippenstiel 2002; Macmillan & Creelma 2005; Wickens 2002; Barkat 2005; Tuzlukov 2001; Levy 2008). Our aim is to examine some practical issues that are typically ignored in textbooks or even in specialized literature as, for example, the effects of the discretization of the signals and the non-Gaussian nature of the noise. To this goal we present each item in the form of answers to specific questions. The relative mathematics and its demonstration are kept to a bare simplest minimum, in the hope of a better understanding of the real performances of the MF in practical applications. For the ease of formalism, arguments will be developed for one-dimensional signals. The extension to the two-dimensional signals is trivial and will be highlighted in dedicated sections.