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Next: Function: compute_match() Up: GRASP Routines: Template Bank Previous: Example: area program   Contents

The match between two templates

When one performs a search for a gravitational wave signal in noisy instrumental data, one lays a grid of templates out in parameter space. For instance, if one uses $\tau_0$ and $\tau_1$ [see Eqs. ([*]) and ([*])] as parameter space coordinates, then one's templates can be described as a set of points $(\tau_0^i,\tau_1^i)$ (with $i$ ranging from 1 to the total number of templates). One requires these points to be spaced such that no more than some a priori fraction of SNR is lost due to the discreteness of the template family.

Suppose one has decided that a set templates can lose no more than $3\%$ SNR in a search. This means that if some arbitrary signal $b(t)$ is dropped onto the template grid, there must exist a template, $a(t)$, such that

\begin{displaymath}
\max_{t_0}\int_{-\infty}^{\infty} df {{\tilde b}(f){\tilde a...
...infty} df
{\vert{\tilde a}(f)\vert^2\over S_h(f)}\right]^{1/2}
\end{displaymath} (9.7.202)

(``$\max t_0$'' indicates the integral on the left hand side is to be maximized over all possible values of $t_0$.) The integral on the left is the SNR obtained when the signal $b(t)$ is measured using the Wiener optimal filter corresponding to the template $a(t)$. The first integral on the right is the SNR obtained when $b(t)$ is measured with the Wiener optimal filter corresponding to a template $b(t)$; the second when the signal and template are both $a(t)$. (The integrals on the right hand side, in other words, describe the situation in which the template exactly matches the signal). For a detailed discussion of Wiener filtering, see Section [*].

To simplify this discussion, let us introduce the following inner product:

\begin{displaymath}
\langle a,b\rangle_{t_0} \equiv
\int_{-\infty}^{\infty} df {{\tilde a}^*(f){\tilde b}(f)\over S_h(f)}
e^{-2\pi ift_0}.
\end{displaymath} (9.7.203)

[Note: this inner product is not to be confused with the inner product $(a,b)$ defined in Eq. ([*]).] We will use the convention that not including the $t_0$ subscript on the angle bracket is equivalent to $t_0=0$. Eq. ([*]) can now be rewritten
\begin{displaymath}
\max_{t_0}\quad\langle a,b\rangle_{t_0} \ge .97\sqrt{\langle a,a\rangle
\langle b,b\rangle}.
\end{displaymath} (9.7.204)

This motivates the definition of the match between $a(t)$ and $b(t)$:
\begin{displaymath}
\mu \equiv \max_{t_0}{\langle a,b \rangle_{t_0}\over\sqrt{\langle a,a\rangle
\langle b,b\rangle}}.
\end{displaymath} (9.7.205)

The match can be thought of as a distance measure between $a(t)$ and $b(t)$ (it is in fact one of the starting points for the metric that Owen defines in [5]). One uses the match function as a means of determining how one must space templates on the parameter space. If one requires that no more than $3\%$ of possible SNR be lost due to template discreteness, then one must require adjacent templates to have a match $\mu = .97$.

The next few functions described in this manual are tools that can be used for calculating the match function and understanding how it varies over one's parameter space.


next up previous contents
Next: Function: compute_match() Up: GRASP Routines: Template Bank Previous: Example: area program   Contents
Bruce Allen 2000-11-19