CmplxAntennaPatternMatrix 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
 $A_d \equiv \mathrm{Re} \sum_{X,\alpha} \widehat{a}^X_\alpha{}^* \widehat{a}^X_\alpha$
REAL8 Bd
 $B_d \equiv \mathrm{Re} \sum_{X,\alpha} \widehat{b}^X_\alpha{}^* \widehat{b}^X_\alpha$
REAL8 Cd
 $C_d \equiv \mathrm{Re} \sum_{X,\alpha} \widehat{a}^X_\alpha{}^* \widehat{b}^X_\alpha$
REAL8 Ed
 $E_d \equiv \mathrm{Im} \sum_{X,\alpha} \widehat{a}^X_\alpha{}^* \widehat{b}^X_\alpha$
REAL8 Dd
 determinant $D_d \equiv A_d B_d - C_d^2 -E_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, in the case of complex AM co\"{e}fficients, as

\begin{equation} \mathcal{M}_{\mu\nu} = \frac{1}{2}\,\mathcal{S}^{-1}\,T_\mathrm{SFT}\,\left( \begin{array}{c c c c} A_d & C_d & 0 & -E_d \\ C_d & B_d & E_d & 0 \\ 0 & E_d & A_d & C_d \\ -E_d & 0 & 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} \mathrm{Re} \widehat{a}^X_\alpha{}^* \widehat{a}^X_\alpha\,,\quad B_d \equiv \sum_{X,\alpha} \mathrm{Re} \widehat{b}^X_\alpha{}^* \widehat{b}^X_\alpha \,,\quad C_d \equiv \sum_{X,\alpha} \mathrm{Re} \widehat{a}^X_\alpha{}^* \widehat{b}^X_\alpha \,, E_d \equiv \sum_{X,\alpha} \mathrm{Im} \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, Ed 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 143 of file ComputeFstat.h.

Field Documentation

REAL8 CmplxAntennaPatternMatrix::Ad

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

Definition at line 144 of file ComputeFstat.h.

REAL8 CmplxAntennaPatternMatrix::Bd

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

Definition at line 145 of file ComputeFstat.h.

REAL8 CmplxAntennaPatternMatrix::Cd

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

Definition at line 146 of file ComputeFstat.h.

REAL8 CmplxAntennaPatternMatrix::Ed

$E_d \equiv \mathrm{Im} \sum_{X,\alpha} \widehat{a}^X_\alpha{}^* \widehat{b}^X_\alpha$

Definition at line 147 of file ComputeFstat.h.

REAL8 CmplxAntennaPatternMatrix::Dd

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

Definition at line 148 of file ComputeFstat.h.

REAL8 CmplxAntennaPatternMatrix::Sinv_Tsft

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

Definition at line 149 of file ComputeFstat.h.

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