CR Bound analysis

The log-PDF of the data $ {\bf b}$ is

$\displaystyle \ln p({\bf b};d,w) =
-\frac{1}{2} \ln (2\pi\sigma^{2}) - \frac{1...
...i} - A \exp\left\{ -0.346 (d-d_{i})^{2}
\frac{4}{w^{2}}\right\} \right] ^{2},
$

where $ \sigma^{2}$ is the variance of the additive independent Gaussian noise. The components of the Fisher Information Matrix (FIM) for PDF parameters $ \phi_{i},\phi_{j}$ are given by

$\displaystyle F_{\phi_i,\phi_j}=-{\bf E}\left(\frac{\partial^{2}
\ln p({\bf b};\phi_{i},\phi_{j})
}{\partial\phi_{i}\partial\phi_{j}}\right)
$

Let the FIM be given by

$\displaystyle F(d,w)= \left[ \begin{array}{ll}
F_{dd} & F_{dw}\\
F_{wd} & F_{ww} \\
\end{array} \right].
$

A standard CR bound analysis [56] produces

$\displaystyle F_{dd} =
\frac{A^{2}}{\sigma^{2}}\left(0.346
\frac{8}{w^{2}}\ri...
...displaystyle \sum}_{i=1}^{M} \left( (d-d_{i})
\exp( -\omega_{i} ) \right)^{2}
$

$\displaystyle F_{ww} =
\frac{A^{2}}{\sigma^{2}}\left(0.346
\frac{8}{w^{3}}\ri...
...laystyle \sum}_{i=1}^{M} \left( (d-d_{i})^{2}
\exp( -\omega_{i} ) \right)^{2}
$

$\displaystyle F_{dw} = F_{wd} =
\frac{A^{2}}{w \sigma^{2}}\left(0.346
\frac{8...
...laystyle \sum}_{i=1}^{M} \left( (d-d_{i})^{2}
\exp( -\omega_{i} ) \right)^{2}
$

where $ \omega_{i} = 0.346 (d - d_{i})^{2}\frac{4}{w^{2}}$. The CR bound matrix is given by $ {\bf C}(d,w)={\bf F}^{-1}(d,w)$.



Baggenstoss 2017-05-19