Next: Dirty details of optimal
Up: GRASP Routines: Gravitational Radiation
Previous: Function: avg_inv_spec()
Contents
Function: orthonormalize()
0
void orthonormalize(float* ch0tilde, float* ch90tilde, float* twice_inv_noise, int n, float* n0, float* n90)
This function takes as input the (positive frequency parts of the) FFT
of a pair of chirp signals. Upon return, the phase chirp
has been made orthogonal to the phase chirp, with respect to
the inner product defined by . The normalizations of the chirps
are also returned.
The arguments are:
 ch0tilde: Input. The FFT of the zerophase chirp .
 ch90tilde: Input/Output. The FFT of the phase chirp .
 twice_inv_noise: Input. Array containing .
The array element twice_inv_noise[0] contains
the DC value, and the array element twice_inv_noise[n/2]
contains the value at the Nyquist frequency.
 n: Input. Defines the length of the arrays: ch0tilde[0..n1], ch90tilde[0..n1],
and twice_inv_noise[0..n/2].
 n0: Output. The normalization of the 0phase chirp.
 n90: Output. The normalization of the phase chirp.
Using the notation of () one may define an inner product of the
chirps. The normalizations are defined as follows:

(6.18.98) 
where is the optimal filter defined for the zerophase chirp .
The chirps are orthogalized internally using the GramSchmidt procedure.
We first calculate and
then define
. We then modify the phase chirp
setting
. This ensures that the
inner product
vanishes. The normalization for this newlydefined
chirp is then defined by

(6.18.99) 
 Author:
Bruce Allen, ballen@dirac.phys.uwm.edu
 Comments:
Notice that the filters and are not in general
orthogonal except in the adiabatic limit as varies very slowly
with changing . Our approach to this is to construct a
slightlymodified ninetydegree phase signal. Note however that this
may introduce small errors in the determination of the orbital phase.
This should be quantified.
Next: Dirty details of optimal
Up: GRASP Routines: Gravitational Radiation
Previous: Function: avg_inv_spec()
Contents
Bruce Allen
20001119