CR Bound analysis

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

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

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

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

Let the FIM be given by

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

A standard CR bound analysis [64] produces

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

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

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

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