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#### Template banks based on Zn and An∗ lattices

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2102.11631.pdf

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PhysRevD.104.122007.pdf

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##### Citation

Allen, B., & Shoom, A. A. (2021). Template banks based on Zn and An∗ lattices.* Physical Review D,* *104 *(2): 122007. doi:10.1103/PhysRevD.104.122007.

Cite as: https://hdl.handle.net/21.11116/0000-0009-CB63-3

##### 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.

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.