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Comparison of signal detectability for singlephase and twophase searches
The previous Section described optimal filtering searches in two cases 
looking for:
 A signal of known phase, proportional to , and
 A signal of unknown phase, which is some linear
combinations of and .
With the choice of filter normalizations made previously, the expected
signal produced by a source would be the same for both
searches, but the expected (noise) was higher in the twophase case.
One might wonder if this reduced SNR means that a twophase search reduces
ones ability to identify signals. The answer turns out to be ``not significantly".
The reason for this is that the distribution of signal values produced
by detector noise alone in the single and twophase cases are quite
different. In order to answer the question: ``what is the smallest
signal detectable" we need to fix a false alarm rate. For a given
timeduration of data, this is equivalent to fixing a false alarm
probability. Let us assume that this probability has been fixed to be
a small value , and compare the single and twophase searches.
In the singlephase case, in the absence of a source, the values of
the signal () are Gaussian random variables with a
meansquared value of 1. Hence the threshold determined by the
false alarm rate must be set so that there is probability
of falling outside the range . This means that

(6.15.88) 
The solution to this equation is the threshold as a function of the false alarm
probability:

(6.15.89) 
Thus, for example, to obtain a false alarm probability of
we need to set a threshold
. In this case, our
minimum detectable signal has amplitude
.
In the twophase case, the probability distribution of the signal in
the absence of a source is different, because in this case the signal
() is described by the probability distribution of
a random variable , where
and and are
independent random Gaussian variables with unit rms. Here, and
are the real and imaginary parts of the signal (
in the absence of a source. Their probability distribution is:
In the final line, we have integrated over the irrelevant angular variable
. So in the twophase case, as before, the threshold value of the
signal is set by requiring that the false alarm probability be :

(6.15.92) 
The solution here is that the threshold is
. For example, to obtain a false alarm probability of
we need to set a threshold
. In this case,
our minimum detectable (0phase) signal has amplitude
,
which is only slightly higher than in the singlephase case.
It is not a coincidence that for a given false alarm rate, the amplitude
of the minimum detectable signals are almost the same. Although the
expected value of the singlephase signal in the absence of a source
is smaller than the expected value of the twophase
in the absence of a source, the tails of the two probability
distributions are almost identical. For the same false alarm probability
the thresholds in the two instances are related by

(6.15.93) 
But for thresholds of reasonable size (small ) both integrals
are dominated by the region just to the right of , and in this neighborhood the
integrands differ by a small factor of approximately
.
Since varies exponentially with the threshold, there is a logarithmically
small difference between the thresholds
and
.
For a fixed false alarm probability, we can write the the twophase
threshold
as a function of the
onephase threshold
:

(6.15.94) 
The plot of this relationship in Figure shows clearly
that once the thresholds are reasonably large, they are very nearly equal.
Figure:
The threshold for a twophase search
is shown as a function of the threshold
for the singlephase search
which
gives the same false alarm rate. When the false alarm rates are small,
they are very nearly equal.

Next: Function: correlate()
Up: GRASP Routines: Gravitational Radiation
Previous: Wiener (optimal) filtering
Contents
Bruce Allen
20001119