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## 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 zero-phase 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..n-1], ch90tilde[0..n-1], and twice_inv_noise[0..n/2].
n0: Output. The normalization of the 0-phase 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 zero-phase chirp . The chirps are orthogalized internally using the Gram-Schmidt 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 newly-defined 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 slightly-modified ninety-degree 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 2000-11-19