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.