Conventions and terminology
\[\def\ld{£}
\def\rn{\mathbb{R}^n}
\def\half{\frac{1}{2}}
\def\dd{\ \!\mathrm{d}}
\def\dvvp#1#2{\frac{\partial #1}{\partial #2}}
\def\dvvpn#1#2#3{\frac{\partial^#3 #1}{\partial #2^#3}}
\def\of#1{\tilde{#1}}
\def\ten#1{\mathsf{#1}}
\def\diag#1{\mathrm{diag}{#1}}
\def\abs#1{\left| #1 \right|}
\def\pdv#1{\frac{\partial}{\partial #1}}
\def\dv#1{\frac{\dd}{\dd #1}}
\def\ddv#1#2{\frac{\dd #1}{\dd #2}}
\def\vdot{\mathbf{\cdot}}\]
Strain
- characteristic squared amplitude
The characteristic squared amplitude, \(|| h^{2} ||\), is defined as
\[|| h^{2} || = \int_{\infty}^{\infty} | h(t)|^{2} \dd{t} = \int_{\infty}^{\infty} | \tilde{h}(f) |^{2} \dd{f},\]
for \(h(t)\) and \(\tilde{h}(f)\) respectively the strain in the time, \(t\), and frequency \(f\) representations :cite:`2008CQGra..25d5002B`.
Noise