AntennaPatternMatrix Struct Reference

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.

\[ \newcommand{\Ad}{\widehat{A}} \newcommand{\Bd}{\widehat{B}} \newcommand{\Cd}{\widehat{C}} \newcommand{\Ed}{\widehat{E}} \newcommand{\Dd}{\widehat{D}} \newcommand{\ah}{\hat{a}} \newcommand{\bh}{\hat{b}} \newcommand{\M}{\mathcal{M}} \newcommand{\S}{\mathcal{S}} \newcommand{\Tsft}{T_{\mathrm{sft}}} \newcommand{\Nsft}{N_{\mathrm{sft}}} \]

This matrix can be shown to be generally expressible as

\begin{equation} \M_{\mu\nu} = \S^{-1}\,\Tsft\,\begin{pmatrix} \Ad & \Cd & 0 & -\Ed \\ \Cd & \Bd & \Ed & 0 \\ 0 & \Ed & \Ad & \Cd \\ -\Ed & 0 & \Cd & \Bd \\ \end{pmatrix} \end{equation}

where \( \S^{-1} \equiv \frac{1}{\Nsft}\sum_{X\alpha} S^{-1}_{X\alpha} \) characterizes the overall multi-detector noise-floor. The sum is over all detectors \( X \) and all SFTs \( \alpha \) from each detector. The nonzero matrix coefficients are expressible as

\begin{align} \Ad &\equiv \sum_{X\alpha} \left|\ah_{X\alpha}\right|^2 \,,\\ \Bd &\equiv \sum_{X\alpha} \left|\bh_{X\alpha}\right|^2 \,,\\ \Cd &\equiv \mathrm{Re} \sum_{X\alpha} \ah_{X\alpha}^{\,*} \,\bh_{X\alpha} \,,\\ \Ed &\equiv \mathrm{Im} \sum_{X\alpha} \ah_{X\alpha}^{\,*} \,\bh_{X\alpha} \,,\\ \end{align}

in terms of the noise-weighted atenna-functions are \( \ah_{X\alpha} \equiv \sqrt{w_{X\alpha}}\,a_{X\alpha} \) , and \( \bh_{X\alpha} = \sqrt{w_{X\alpha}}\,b_{X\alpha} \) , with per-SFT noise-weights \( w_{X\alpha} \equiv \frac{S^{-1}_{X\alpha}}{\S^{-1}} \) .

Note

One reason for storing the un-normalized \( \{\Ad,\,\Bd,\,\Cd,\,\Ed\} \) and the normalization-factor \( \S^{-1}\,\Tsft \) separately is that the former are of order one, while the latter is generally very large, and so it has numerical advantages for parameter-estimation to use that fact.

Definition at line 127 of file LALComputeAM.h.

Data Fields
REAL4	Ad
	\( \Ad \) More...
REAL4	Bd
	\( \Bd \) More...
REAL4	Cd
	\( \Cd \) More...
REAL4	Ed
	\( \Ed \) More...
REAL4	Dd
	determinant factor \( \Dd \equiv \Ad \Bd - \Cd^2 - \Ed^2 \) , such that \( \det\M = \Dd^2 \) More...
REAL8	Sinv_Tsft
	normalization-factor \( \S^{-1}\,\Tsft \) (using single-sided PSD!) More...

Field Documentation

Ad

REAL4 AntennaPatternMatrix::Ad

\( \Ad \)

Definition at line 128 of file LALComputeAM.h.

Bd

REAL4 AntennaPatternMatrix::Bd

\( \Bd \)

Definition at line 129 of file LALComputeAM.h.

Cd

REAL4 AntennaPatternMatrix::Cd

\( \Cd \)

Definition at line 130 of file LALComputeAM.h.

Ed

REAL4 AntennaPatternMatrix::Ed

\( \Ed \)

Definition at line 131 of file LALComputeAM.h.

Dd

REAL4 AntennaPatternMatrix::Dd

determinant factor \( \Dd \equiv \Ad \Bd - \Cd^2 - \Ed^2 \) , such that \( \det\M = \Dd^2 \)

Definition at line 132 of file LALComputeAM.h.

Sinv_Tsft

REAL8 AntennaPatternMatrix::Sinv_Tsft

normalization-factor \( \S^{-1}\,\Tsft \) (using single-sided PSD!)

Definition at line 133 of file LALComputeAM.h.