AntennaPatternMatrix Struct Reference

Struct holding the "antenna-pattern" matrix $\mathcal{M}_{\mu\nu} \equiv \left( \mathbf{h}_\mu|\mathbf{h}_\nu\right)$, in terms of the multi-detector scalar product. More...

#include <ComputeFstat.h>


Data Fields

REAL8 Ad
 $A_d \equiv \sum_{X,\alpha} \widehat{a}^X_\alpha \widehat{a}^X_\alpha$
REAL8 Bd
 $B_d \equiv \sum_{X,\alpha} \widehat{b}^X_\alpha \widehat{b}^X_\alpha$
REAL8 Cd
 $C_d \equiv \sum_{X,\alpha} \widehat{a}^X_\alpha \widehat{b}^X_\alpha$
REAL8 Dd
 determinant $D_d \equiv A_d B_d - C_d^2 $
REAL8 Sinv_Tsft
 normalization-factor $\mathcal{S}^{-1}\,T_\mathrm{SFT}$

Detailed Description

Struct holding the "antenna-pattern" matrix $\mathcal{M}_{\mu\nu} \equiv \left( \mathbf{h}_\mu|\mathbf{h}_\nu\right)$, in terms of the multi-detector scalar product.

This matrix can be shown to be expressible as

\begin{equation} \mathcal{M}_{\mu\nu} = \frac{1}{2}\,\mathcal{S}^{-1}\,T_\mathrm{SFT}\,\left( \begin{array}{c c} A_d & C_d \\ C_d & B_d \\\end{array}\right)\,, \end{equation}

where (here) $\mathcal{S} \equiv \frac{1}{N_\mathrm{SFT}}\sum_{X,\alpha} S_{X\alpha}$ characterizes the multi-detector noise-floor, and

\begin{equation} A_d \equiv \sum_{X,\alpha} \widehat{a}^X_\alpha \widehat{a}^X_\alpha\,,\quad B_d \equiv \sum_{X,\alpha} \widehat{b}^X_\alpha \widehat{b}^X_\alpha \,,\quad C_d \equiv \sum_{X,\alpha} \widehat{a}^X_\alpha \widehat{b}^X_\alpha \,, \end{equation}

and the noise-weighted atenna-functions $\widehat{a}^X_\alpha = \sqrt{w^X_\alpha}\,a^X_\alpha$, $\widehat{b}^X_\alpha = \sqrt{w^X_\alpha}\,b^X_\alpha$, and noise-weights $w^X_\alpha \equiv {S^{-1}_{X\alpha}/{\mathcal{S}^{-1}}$.

Note:
One reason for storing the un-normalized Ad, Bd, Cd and the normalization-factor Sinv_Tsft separately is that the former are of order unity, while Sinv_Tsft is very large, and it has numerical advantages for parameter-estimation to use that fact.

Definition at line 107 of file ComputeFstat.h.

Field Documentation

REAL8 AntennaPatternMatrix::Ad

$A_d \equiv \sum_{X,\alpha} \widehat{a}^X_\alpha \widehat{a}^X_\alpha$

Definition at line 108 of file ComputeFstat.h.

REAL8 AntennaPatternMatrix::Bd

$B_d \equiv \sum_{X,\alpha} \widehat{b}^X_\alpha \widehat{b}^X_\alpha$

Definition at line 109 of file ComputeFstat.h.

REAL8 AntennaPatternMatrix::Cd

$C_d \equiv \sum_{X,\alpha} \widehat{a}^X_\alpha \widehat{b}^X_\alpha$

Definition at line 110 of file ComputeFstat.h.

REAL8 AntennaPatternMatrix::Dd

determinant $D_d \equiv A_d B_d - C_d^2 $

Definition at line 111 of file ComputeFstat.h.

REAL8 AntennaPatternMatrix::Sinv_Tsft

normalization-factor $\mathcal{S}^{-1}\,T_\mathrm{SFT}$

Definition at line 112 of file ComputeFstat.h.

The documentation for this struct was generated from the following file:
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