## 6 MSE of Direct Position Determination Method in Presence of Array Model Errors
### The Second Success Probability of Direct Position Determination  
**Definition 2**. : _If condition \"\(\sqrt{\frac{1}{L}\sum\limits_{l=1}^{L}<\widetilde{\boldsymbol{p}}>_{l}^{2}}\leq \Delta\)\" is satisfied, then the localization is successful._  
It is readily seen from Definition 2 that the second SP of positioning is equal to \(\Pr\left\{||\widetilde{\boldsymbol{p}}||_{2}^{2}\leq L\Delta^{2}\right\}\). To proceed, let us express \(\widetilde{\boldsymbol{p}}\) as \(\widetilde{\boldsymbol{p}}\overset{\mathrm{d}}{=}\boldsymbol{p}^{1/2} \widetilde{\boldsymbol{p}}_{0}\), where \(\widetilde{\boldsymbol{p}}_{0}\) is a zero-mean Gaussian random vector with covariance matrix \(\boldsymbol{I}_{L}\), and \(\overset{\mathrm{d}}{=}\) indicates that both sides have the same probability distribution. Consequently, \(||\widetilde{\boldsymbol{p}}||_{2}^{2}\) can be formulated as the quadratic form of \(\widetilde{\boldsymbol{p}}_{0}\):  
\[||\widetilde{\boldsymbol{p}}||_{2}^{2}\overset{\mathrm{d}}{=}\widetilde{ \boldsymbol{p}}_{0}^{\mathrm{T}}\boldsymbol{P}\widetilde{\boldsymbol{p}}_{0}\,. \tag{65}\]  
In light of the relationship between the cumulative distribution function and characteristic function [64], we have  
\[\Pr\{||\widetilde{\boldsymbol{p}}||_{2}^{2}\leq L\Delta^{2}\}=\frac{1}{2}- \frac{1}{\pi}\cdot\int_{0}^{+\infty}\frac{1}{t}\cdot\mathrm{Im}\{\exp\{-\mathrm{ j}L\Delta^{2}t\}\cdot\phi_{||\widetilde{\boldsymbol{p}}||_{2}^{2}}(t)\}\, \mathrm{d}t\,, \tag{66}\]  
where \(\phi_{||\widetilde{\boldsymbol{p}}||_{2}^{2}}(t)\) denotes the characteristic function of \(||\widetilde{\boldsymbol{p}}||_{2}^{2}\). Suppose that matrix \(\boldsymbol{P}\) has eigenvalues \(\gamma_{1}\), \(\gamma_{2}\), \(\cdots\), \(\gamma_{L}\). Applying the property of the characteristic function, it can be proved that  
\[\phi_{||\widetilde{\boldsymbol{p}}||_{2}^{2}}(t)=\prod\limits_{l=1}^{L}\left( 1+4\gamma_{l}^{2}\,t^{2}\right)^{-1/4}\cdot\exp\{\mathrm{j}(\arctan(2\gamma_ {l}\,t)/2)\}\,. \tag{67}\]  
The substitution of (67) into (66) produces  
\[\Pr\{||\widetilde{\boldsymbol{p}}||_{2}^{2}\leq L\Delta^{2}\}=\frac{1}{2}- \frac{1}{\pi}\cdot\int_{0}^{+\infty}\frac{1}{t}\cdot\frac{\sin(\delta_{1}(t)) }{\delta_{2}(t)}\mathrm{d}t\,, \tag{68}\]  
where  
\[\left\{ \begin{array}{l}\delta_{1}(t)=\sum\limits_{l=1}^{L}\arctan(2\gamma_{ l}t)/2-L\Delta^{2}t\,,\\ \delta_{2}(t)=\prod\limits_{l=1}^{L}\left(1+4\gamma_{l}^{2}t^{2}\right)^{1/4} \,.\end{array} \right. \tag{69}\]  
**Remark 8**. : _It is clear from (68) that a one-dimension numerical integration over \([0\,,\,+\infty)\) is required to evaluate the second SP. To this end, the values of the integrand shall be analyzed as \(\,t\to 0\) and \(\,t\to+\infty\,.\)_  
**Remark 9**. : Applying [PERSON] rule leads to  
\[\lim\limits_{t\to 0}\frac{1}{t}\cdot\frac{\sin\{\delta_{1}(t)\}}{\delta_{2}(t) }=\lim\limits_{t\to 0}\frac{\cos\{\delta_{1}(t)\}\cdot\dot{\delta}_{1}(t)}{ \delta_{2}(t)+t\delta_{2}(t)}=\dot{\delta}_{1}(0)=\sum\limits_{l=1}^{L}\gamma _{l}-L\Delta^{2}\,. \tag{70}\]  
**Remark 10**. : _The numerator of the integrand is bounded and the denominator tends to infinity when \(t\to+\infty\) and, therefore, the integrand will be arbitrarily close to zero when \(t\to+\infty\). The integral upper limit in (68) can then be replaced by a sufficiently large positive number for the sake of simplicity._  
**Remark 11**. : It can be rigorously proved that the first SP is always smaller than the second SP, provided that \(\Delta_{1}=\cdots=\Delta_{L}=\Delta\). The reason is that the first probability is computed by the numerical integral over a cube, while the second probability is equal to the integral over a circumscribed sphere of the cube. As a byproduct of (68), we can present a new method of determining the radius of circular error probable (CEP), which is first defined in [65].