# Estimation of the Measurement Accuracy of Wireless Passive Resonance Sensors
## Appendix A Calculation of the Fischer Information Matrix and Its Inverse  
The lower Cramer-Rao bounds are calculated using the Fisher information matrix \(J\). In the case where we have additive white Gaussian noise, unbiased complex data and unknown real parameters in \(\chi\), the Fisher information matrix can be calculated using [50] (p. 49). \[J_{ij}(x)=\frac{2}{N_{p}}\Re\left\{\sum_{n=0}^{N-1}\left[\frac{\partial\mu_{n }{}^{*}}{\partial\chi_{i}}\frac{\partial\mu_{n}}{\partial\chi_{j}}\right]\right\} \tag{11}\]  
with  
\[\mu_{n}=\tilde{b}e^{n(-\alpha+j\tilde{\omega})T+j\tilde{\varphi}},\qquad \qquad\chi=\left( \begin{array}{c}\tilde{\omega}\\ \tilde{\varphi}\\ \alpha\\ \tilde{b}\end{array} \right), \tag{12}\]  
and \(\Re[x]\) being the real part of \(x\). The partial derivatives with respect to the variables in \(\chi\) are  
\[\frac{\partial\mu_{n}}{\partial\tilde{\omega}} = jnT\tilde{b}e^{(-\alpha+j\tilde{\omega})nT+j\tilde{\varphi}}\] \[\frac{\partial\mu_{n}}{\partial\tilde{\varphi}} = j\tilde{b}e^{(-\alpha+j\tilde{\omega})nT+j\tilde{\varphi}}\] \[\frac{\partial\mu_{n}}{\partial\alpha} = -nT\tilde{b}e^{(-\alpha+j\tilde{\omega})nT+j\tilde{\varphi}}\] \[\frac{\partial\mu_{n}}{\partial\tilde{b}} = e^{(-\alpha+j\tilde{\omega})nT+j\tilde{\varphi}} \tag{13}\]  
\[= \frac{2}{N_{p}}\mathfrak{R}\left(\sum_{n=0}^{N-1}\left[\frac{\partial \mu_{n}{}^{*}}{\partial\alpha}\frac{\partial\mu_{n}}{\partial\bar{b}}\right] \right)=\frac{2}{N_{p}}\mathfrak{R}\left(\sum_{n=0}^{N-1}\left(-nT\bar{b}e^{(- \alpha-j\hat{\omega})nT-j\hat{\varphi}}\right)\left(e^{(-\alpha+j\hat{\omega}) nT+j\hat{\varphi}}\right)\right\}\] \[= \frac{2}{N_{p}}\mathfrak{R}\left(\sum_{n=0}^{N-1}-nT\bar{b}e^{-2 \alpha nT}\right)=-\frac{2T\bar{b}}{N_{p}}\left(\sum_{n=0}^{N-1}ne^{-2\alpha nT }\right)=-\frac{2T\bar{b}}{N_{p}}Q \tag{12}\] \[J_{44} = \frac{2}{N_{p}}\mathfrak{R}\left(\sum_{n=0}^{N-1}\left[\frac{ \partial\mu_{n}{}^{*}}{\partial\bar{b}}\frac{\partial\mu_{n}}{\partial\bar{b}} \right]\right)=\frac{2}{N_{p}}\mathfrak{R}\left(\sum_{n=0}^{N-1}\left(e^{(- \alpha-j\hat{\omega})nT-j\hat{\varphi}}\right)\left(e^{(-\alpha+j\hat{\omega} )nT+j\hat{\varphi}}\right)\right)\] (13) \[= \frac{2}{N_{p}}\mathfrak{R}\left(\sum_{n=0}^{N-1}e^{-2\alpha nT }\right)=\frac{2}{N_{p}}P\] (14) \[J(\chi) = \frac{2}{N_{p}}\left[ \begin{array}{cccc}T^{2}\bar{b}^{2}R&T \bar{b}^{2}Q&0&0\\ T\bar{b}^{2}Q&\bar{b}^{2}P&0&0\\ 0&0&T^{2}\bar{b}^{2}R&-T\bar{b}Q\\ 0&0&-T\bar{b}Q&P\end{array} \right] \tag{15}\]  
The matrix \(J\) consists of two \(2\times 2\) submatrices, \(A_{1}\) and \(A_{2}\), each of which can be inverted on its own:  
\[A=\left[ \begin{array}{cc}A_{1}&0\\ 0&A_{2}\end{array} \right]\to A^{-1}=\left[ \begin{array}{cc}A_{1}^{-1}&0\\ 0&A_{2}^{-1}\end{array} \right] \tag{16}\]  
The inverse of a \(2\times 2\) matrix A is given by  
\[A=\left[ \begin{array}{cc}a_{11}&a_{12}\\ a_{21}&a_{22}\end{array} \right]\to A^{-1}=\frac{1}{a_{11}a_{22}-a_{21}a_{12}} \left[ \begin{array}{cc}a_{22}&-a_{12}\\ -a_{21}&a_{11}\end{array} \right] \tag{17}\]  
We obtain for \(A_{1}\)  
\[A_{1}{}^{-1} = \frac{1}{T^{2}\bar{b}^{2}PR-T^{2}\bar{b}^{2}Q^{2}}\left[ \begin{array} []{cc}\bar{b}^{2}P&-T\bar{b}^{2}Q\\ -T\bar{b}^{2}Q&T^{2}\bar{b}^{2}R\end{array} \right] \tag{18}\] \[= \frac{1}{T^{2}\bar{b}^{2}(PR-Q^{2})}\left[ \begin{array}{cc}P&-T \bar{b}Q\\ -T\bar{b}Q&T^{2}\bar{b}^{2}R\end{array} \right]\]  
And for \(A_{2}\)  
\[A_{2}{}^{-1} = \frac{1}{T^{2}\bar{b}^{2}PR-T^{2}\bar{b}^{2}Q^{2}}\left[ \begin{array} []{cc}P&T\bar{b}Q\\ T\bar{b}Q&T^{2}\bar{b}^{2}R\end{array} \right] \tag{19}\] \[= \frac{1}{T^{2}\bar{b}^{2}(PR-Q^{2})}\left[ \begin{array}{cc}P&T \bar{b}Q\\ T\bar{b}Q&T^{2}\bar{b}^{2}R\end{array} \right]\]  
This results for the inverse Fisher information matrix in  
\[J^{-1}(\chi)=\frac{N_{p}}{2T^{2}\bar{b}^{2}(PR-Q^{2})}\left[ \begin{array}{ cccc}\mathrm{P}&-TQ&0&0\\ -TQ&T^{2}R&0&0\\ 0&0&P&T\bar{b}Q\\ 0&0&T\bar{b}Q&T^{2}\bar{b}^{2}R\end{array}