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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.