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 $90^\circ$ phase chirp has been made orthogonal to the $0^\circ$ phase chirp, with respect to the inner product defined by $2/S_h$. The normalizations of the chirps are also returned.

The arguments are:

ch0tilde: Input. The FFT of the zero-phase chirp $T_0$.

ch90tilde: Input/Output. The FFT of the $90^\circ$-phase chirp $T_{90}$.

twice_inv_noise: Input. Array containing $2/S_h$. 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 $90^\circ$-phase chirp.

Using the notation of () one may define an inner product of the chirps. The normalizations are defined as follows:

$${1 \over n_0^2} \equiv {1 \over 2} (Q_0,Q_0),$$ (6.18.98)

where $Q_0$ is the optimal filter defined for the zero-phase chirp $T_0$. The chirps are orthogalized internally using the Gram-Schmidt procedure. We first calculate $(Q_0,Q_0)$ and $(Q_{90},Q_{0})$ then define $\epsilon = (Q_{90},Q_{0})/(Q_0,Q_0)$. We then modify the $90^\circ$-phase chirp setting $\tilde T_{90} \rightarrow T_{90} - \epsilon T_0$. This ensures that the inner product $(Q_{90},Q_{0})$ vanishes. The normalization for this newly-defined chirp is then defined by

$${1 \over n_1^2} \equiv {1 \over 2} (Q_{90},Q_{90}).$$ (6.18.99)

Author: Bruce Allen, ballen@dirac.phys.uwm.edu

Comments: Notice that the filters $Q_0$ and $Q_{90}$ are not in general orthogonal except in the adiabatic limit as $S_h(f)$ varies very slowly with changing $f$. 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.