4.7
Suboptimal filtering
To extract signals from the noise one very often uses filters
that are not optimal. We may have to choose an approximate,
suboptimal filter because we do not know the exact form of the
signal (this is almost always the case in practice) or in order
to reduce the computational cost and to simplify the analysis.
The most natural and simplest way to proceed is to use as our
statistic the
-statistic where the filters
are the approximate ones instead of the optimal ones matched to
the signal. In general the functions
will be different from the functions
used in optimal filtering, and also the set of parameters
will be different from the set of parameters
in optimal filters. We call this procedure the
suboptimal filtering
and we denote the suboptimal statistic by
.
We need a measure of how well a given
suboptimal filter performs. To find such a measure we calculate
the expectation value of the suboptimal statistic. We get
where
Let us rewrite the expectation value
in the following form,
where
is the optimal signal-to-noise ratio. The expectation value
reaches its maximum equal to
when the filter is perfectly matched to the signal. A natural
measure of the performance of a suboptimal filter is the quantity
FF defined by
We call the quantity FF the
generalized fitting factor
.
In the case of a signal given by
the generalized fitting factor defined above reduces to the
fitting factor introduced by Apostolatos
[7]
:
The fitting factor is the ratio of the maximal signal-to-noise
ratio that can be achieved with suboptimal filtering to the
signal-to-noise ratio obtained when we use a perfectly matched,
optimal filter. We note that for the signal given by
Equation (74), FF is independent of the value of the amplitude
. For the general signal with 4 constant amplitudes it follows
from the Rayleigh principle that the fitting factor is the
maximum of the largest eigenvalue of the matrix
over the intrinsic parameters of the signal.
For the case of a signal of the form
where
is a constant phase, the maximum over
in Equation (75) can be obtained analytically. Moreover, assuming that over the
bandwidth of the signal the spectral density of the noise is
constant and that over the observation time
oscillates rapidly, the fitting factor is approximately given by
In designing suboptimal filters one faces the
issue of how small a fitting factor one can accept. A popular
rule of thumb is accepting
. Assuming that the amplitude of the signal and consequently the
signal-to-noise ratio decreases inversely proportional to the
distance from the source this corresponds to 10% loss of the
signals that would be detected by a matched filter.
Proposals for good suboptimal (search)
templates for the case of coalescing binaries are given in
[22, 78
]
and for the case spinning neutron stars in
[40
, 13]
.