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Function: calculate_var()

0

double calculate_var(int n, float delta_f, float omega_0, float f_low, float f_high, float t, double *gamma12, double *power1, double *power2)
This function calculates the theoretical variance $\sigma^2$ of the stochastic background cross-correlation signal $S$.

The arguments of calculate_var() are:

n: Input. The number $N$ of discrete frequency values at which the spectra are to be evaluated.
delta_f: Input. The spacing $\Delta f$ (in Hz) between two adjacent discrete frequency values: $\Delta f:=f_{i+1}-f_i$.
omega_0: Input. The constant value $\Omega_0$ (dimensionless) of the frequency spectrum $\Omega_{\rm gw}(f)$ for the stochastic background:

\begin{displaymath}
\Omega_{\rm gw}(f)=\left\{
\begin{array}{cl}
\Omega_0 & \qua...
... f_{\rm high}\0 & \quad {\rm otherwise.}
\end{array}\right.
\end{displaymath}

$\Omega_0$ should be greater than or equal to zero.
f_low: Input. The frequency $f_{\rm low}$ (in Hz) below which the spectrum $\Omega_{\rm gw}(f)$ of the stochastic background is zero. $f_{\rm low}$ should lie in the range $0\le f_{\rm low}\le f_{\rm Nyquist}$, where $f_{\rm Nyquist}$ is the Nyquist critical frequency. (The Nyquist critical frequency is defined by $f_{\rm Nyquist}:=1/(2\Delta t)$, where $\Delta t$ is the sampling period of the detector.) $f_{\rm low}$ should also be less than or equal to $f_{\rm high}$.
f_high: Input. The frequency $f_{\rm high}$ (in Hz) above which the spectrum $\Omega_{\rm gw}(f)$ of the stochastic background is zero. $f_{\rm high}$ should lie in the range $0\le f_{\rm high}\le f_{\rm Nyquist}$. It should also be greater than or equal to $f_{\rm low}$.
t: Input. The observation time $T$ (in sec) of the measurement.
gamma12: Input. gamma12[0..n-1] is an array of double precision variables containing the values of the overlap reduction function $\gamma(f)$ for the two detector sites. These variables are dimensionless. gamma12[i] contains the value of $\gamma(f)$ evaluated at the discrete frequency $f_i=i\Delta f$, where $i=0,1,\cdots,N-1$.
power1: Input. power1[0..n-1] is an array of double precision variables containing the values of the noise power spectrum $P_1(f)$ of the first detector. These variables have units of ${\rm strain}^2/{\rm Hz}$ (or seconds). power1[i] contains the value of $P_1(f)$ evaluated at the discrete frequency $f_i=i\Delta f$, where $i=0,1,\cdots,N-1$.
power2: Input. power2[0..n-1] is an array of double precision variables containing the values of the noise power spectrum $P_2(f)$ of the second detector, in exactly the same format as the previous argument.

The double precision value returned by calculate_var() is the theoretical variance $\sigma^2$ given by Eq. ([*]) of Sec. [*]. As discussed in that section, Eq. ([*]) for $\sigma^2$ makes no assumption about the relative strengths of the stochastic background and detector noise signal, but it does use Eq. ([*]) for the filter function $\tilde Q(f)$, which is optimal only for the large detector noise case. For stochastic background simulations, $\Omega_0$ is usually chosen to equal some known non-zero value. This is the value that should be passed as a parameter to calculate_var(). For stochastic background searches (where $\Omega_0$ is not known a priori) the value of of the parameter $\Omega_0$ should be set to zero. The variance for this case is given by Eq. ([*]).

Authors: Bruce Allen, ballen@dirac.phys.uwm.edu, and Joseph Romano, romano@csd.uwm.edu
Comments: None.


next up previous contents
Next: Example: snr program Up: GRASP Routines: Stochastic background Previous: Discussion: Theoretical signal-to-noise ratio   Contents
Bruce Allen 2000-11-19