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Details of Normalization
The Fourier transform convensions of Numerical Recipes are used here.
In particular, suppose that the time series is sampled at intervals
of
and these samples are stored in the
array array[0..n1] where n is the number of samples taken
(thus, the total observation time is
. Then, the
Fourier transform
is related to
the FFT of array by

(13.2.306) 
where atilde[0..n1] is produced from array[0..n1] by
realft(...,1). Note that the frequency where
. Define the
onesided mean power spectrum of by

(13.2.307) 
and similarly define

(13.2.308) 
Notice that has dimensions of time while power[j] is,
of course, dimensionless. These two power spectra are related by

(13.2.309) 
A well known ``feature'' of the inverse FFT produced by realft(...,1)
is that

(13.2.310) 
We can express the correlation between two time series, and
, weighted by twice the inverse power spectrum, as
[cf. equation ()]
where
and
.
Notice that all the factors of 2, n, and srate are accounted
for. However, the correlation defined in the first line is one half of the
correlation defined in equation ().
Next: Function: strain_spec()
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Bruce Allen
20001119