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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 and
[see Eqs. () and ()] as
parameter space coordinates, then one's templates can be
described as a set of points
(with
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 SNR in a search. This means that if some arbitrary
signal is dropped onto the template grid, there must exist
a template, , such that

(9.7.202) 
(``'' indicates the integral on the left hand side
is to be maximized over all possible values of .) The integral
on the left is the SNR obtained when the signal is measured
using the Wiener optimal filter corresponding to the template .
The first integral on the right is the SNR obtained when is
measured with the Wiener optimal filter corresponding to a template
; the second when the signal and template are both .
(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:

(9.7.203) 
[Note: this inner product is not to be confused with the inner
product defined in Eq. ().] We will use
the convention that not including the subscript on the angle
bracket is equivalent to . Eq. () can now
be rewritten

(9.7.204) 
This motivates the definition of the match between
and :

(9.7.205) 
The match can be thought of as a distance measure between
and (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 of possible SNR be lost due to
template discreteness, then one must require adjacent templates to have
a match .
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: Function: compute_match()
Up: GRASP Routines: Template Bank
Previous: Example: area program
Contents
Bruce Allen
20001119