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This program reads the 40meter data stream, and then filters it though
a chirp template corresponding to a pair of inspiraling
neutron stars.
The correspondence between different arrays in this program, and the
quantities discussed previously in this section, is given below. In these
equations,
is the sample time in seconds, and
is the size of a
frequency bin, in Hz. Here
is the number of points in
the data stream which are being optimally filtered in one pass.
Chirp templates (in frequency space) for the two polarizations are
related to the arrays chirp0[ ] and chirp1[ ] by
where the elements chirp0[2j] and chirp0[2j+1] are the real
and imaginary parts at frequency (with the exception of
the Nyquist frequency, stored in chirp0[1]). Note that to ensure
that quantities within the code remain within the dynamic range of
floating point numbers, we have scaled up the template strain by a
constant factor HSCALE; we also scale up the interferometer
output by the same factor, so that all program output (such as
signaltonoise ratios) is independent of the value of HSCALE.
If you're not comfortable with this, go ahead and change HSCALE to 1.
It won't change anything, provided that you don't overflow the dynamic
range of the floating point variables!
The scaled interferometer response function is

(6.30.147) 
where the function is defined by equation ().
The Fourier transform of the dimensionless strain is
obtained by multiplying and the FFT of channel.0 by
response[ ], yielding

(6.30.148) 
The onesided noise power spectrum is the average of

(6.30.149) 
The power spectrum is averaged using the same exponential
averaging technique described for the routine avg_spec(). This
average is stored as

(6.30.150) 
Twice the inverse of this average is stored in the array twice_inv_noise[ ], so that

(6.30.151) 
The expected noisesquared for the plus polarization is given by
equation ():
where the subscript on the inverse FFT means ``at zero lag", and
``" means ``returned by the call to the function f". We
have chosen a distance for the system producing the ``chirp" so that the expected value of
.
In similar fashion, the signal at lag is given
by
where now the subscript on the FFT means ``at lag
".
You might wonder why we have been so careful  after all, both the
signal and the noise, as we've defined them, are dimensionless, so it's
not surprising that all of the factors of drop out of the
final formulae for the signal and the expected noisesquared. The main
reason we've been so long winded is to show exactly how the units cancel out,
and to demonstrate that there aren't any missing dimensionless constants,
like npoint, left out of the program. Some sample output from
this program is shown in the next section.
Includes/optimal.tex
Next: Some output from the
Up: GRASP Routines: Gravitational Radiation
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Bruce Allen
20001119