Template banks based on $\mathbb{Z}^n$ and $A_n^*$ lattices

TL;DR

This paper analyzes the properties of template banks based on $ ext{Z}^n$ and $A_n^*$ lattices for signal detection, showing that high mismatch thresholds can still yield effective searches and that $A_n^*$ offers limited advantages.

Contribution

It provides a detailed comparison of $ ext{Z}^n$ and $A_n^*$ lattice-based template banks, including mismatch distribution analysis and large-$n$ limits, with practical implications for search efficiency.

Findings

Effective searches possible at high mismatch values.

Minimal advantage of $A_n^*$ over $ ext{Z}^n$ at large mismatches.

Large-$n$ limit estimates are accurate even for small $n$.

Abstract

Matched filtering is a traditional method used to search a data stream for signals. If the source (and hence its $n$ parameters) are unknown, many filters must be employed. These form a grid in the $n$-dimensional parameter space, known as a template bank. It is often convenient to construct these grids as a lattice. Here, we examine some of the properties of these template banks for $\mathbb{Z}^n$ and $A_n^*$ lattices. In particular, we focus on the distribution of the mismatch function, both in the traditional quadratic approximation and in the recently-proposed spherical approximation. The fraction of signals which are lost is determined by the even moments of this distribution, which we calculate. Many of these quantities we examine have a simple and well-defined $n\to\infty$ limit, which often gives an accurate estimate even for small $n$. Our main conclusions are the following: (i) a fairly effective template-based search can be constructed at mismatch values that are shockingly high in the quadratic approximation; (ii) the minor advantage offered by an $A_n^*$ template bank (compared to $\mathbb{Z}^n$) at small template separation becomes even less significant at large mismatch. So there is little motivation for using template banks based on the $A_n^*$ lattice.