# Distributed Information Compression for Target Tracking in Cluster-Based Wireless Sensor Networks
## 4 Performance Analysis
### Analysis of Positioning Accuracy  
As the Cramer-Rao lower bound (CRLB) tests the performance of the proposed measurement mechanism, this subsection describes the achievable tracking accuracy of joint TOA/AOA positioning. The measurements at the reference sensor can be modeled as:  
\[\hat{\tau} = \tau+\delta_{\tau} \tag{3}\] \[\hat{\phi} = \phi+\delta_{\phi} \tag{4}\]  
where \(\tau\) is the true propagation time and \(\phi\) is the true angle information. Note that \(\delta_{\tau}\) and \(\delta_{\phi}\) are uncorrelated Gaussian noises with the distributions \(\delta_{\tau}\sim\mathcal{N}(0,\sigma_{\tau}^{2})\) and \(\delta_{\phi}\sim\mathcal{N}(0,\sigma_{\phi}^{2})\). Assuming that the direct path exists between the sensor and the target, the estimated target position is given by:  
\[\hat{x} = x_{s}+v\hat{\tau}\cos(\hat{\phi})=x_{s}+\hat{r}\cos(\hat{\phi}) \tag{5}\] \[\hat{y} = y_{s}+v\hat{\tau}\sin(\hat{\phi})=y_{s}+\hat{r}\sin(\hat{\phi}) \tag{6}\]  
where \(\hat{r}\) is the distance measurement (_i.e._, \(\hat{r}=v\hat{\tau}=r+v\delta_{\tau}\)), \((x_{s},y_{s})\) is the true position of the sensor and \(v\) is the speed of the signal. Assuming \(\delta_{\tau}\) and \(\delta_{\phi}\) are sufficiently small, the variance of the position estimation \(\hat{p}\) is approximated by:  
\[\sigma_{p}^{2}\approx v^{2}\sigma_{\tau}^{2}+d^{2}\sigma_{\phi}^{2}=\sigma_{r} ^{2}+d^{2}\sigma_{\phi}^{2} \tag{7}\]  
Given the above assumptions [19], the CRLB with a single sensor is derived as follows.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c} \hline \hline
**Value** & & & & & & & & & \(-\)**5.678** & & & & & & & & & \\ \hline Code & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 1 \\ Meaning & \(\pm\) & \(2^{2}\) & \(2^{1}\) & \(2^{0}\) & \(2^{2}\) & \(2^{1}\) & \(2^{0}\) &. & 2\({}^{-1}\) & \(2^{-2}\) & \(2^{-3}\) & \(2^{-4}\) & \(2^{-5}\)Assuming that a line-of-sight path exists between a sensor and the target, the joint density function of the distance measurement \(\hat{r}\) and the angle measurement \(\hat{\phi}\) yields:  
\[\begin{array}{l}f(\mathbf{g};x,y)=\frac{1}{\sqrt{2\pi\sigma_{\theta}^{2}}} \cdot\exp\left[-\frac{1}{2\sigma_{r}^{2}}\left(\hat{r}-d\right)^{2}\right]\\ \cdot\frac{1}{\sqrt{2\pi\sigma_{\phi}^{2}}}\cdot\exp\left[-\frac{1}{2\sigma_{ \phi}^{2}}\left(\hat{\phi}-\arctan\left(\frac{y-y_{s}}{x-x_{s}}\right)\right)^{ 2}\right]\end{array} \tag{8}\]  
where \(\mathbf{g}=[\hat{r},\hat{\phi}]\), \((x_{s},y_{s})\) is the true sensor position and \(d\) is the true distance between pairwise sensors. Thus, the Fisher information matrix is given by:  
\[I(\mathbf{x}^{(t)})=\left[\begin{array}{cc}\frac{\cos^{2}(\phi)}{\sigma_{r} ^{2}}+\frac{\sin^{2}(\phi)}{d^{2}\sigma_{\phi}^{2}}&\frac{\sin(2\phi)}{2}\left[ \frac{1}{\sigma_{r}^{2}}-\frac{1}{d\sigma_{\phi}^{2}}\right]\\ \frac{\sin(2\phi)}{2}\left[\frac{1}{\sigma_{r}^{2}}-\frac{1}{d^{2}\sigma_{ \phi}^{2}}\right]&\frac{\sin^{2}(\phi)}{\sigma_{r}^{2}}+\frac{\cos^{2}(\phi)} {d^{2}\sigma_{\phi}^{2}}\end{array}\right] \tag{9}\]  
which leads to the CRLB for the best possible estimation for target tracking. Figure 12 shows a typical run of the conceptual-stochastic distribution of the position estimation. The readers may refer to [19] for details.