## 6 The Analysis of Direction Finding Error  
In this section, the direction finding error is derived in order to clarify the factors that affect the performance of direction finding. For simplicity, three elements are arranged on the surface of a cylinder as shown in Figure 9.  
The cylindrical coordinate is adopted here. The transform from Cartesian coordinate system to cylindrical coordinates system can be expressed as  
\[x_{i}=r\cdot\cos\phi_{i},y_{i}=r\cdot\sin\phi_{i},z_{i}=z_{i}(i=\rm{A},\rm{B}, \rm{C}) \tag{22}\]  
where \(r\) stands for the radius of the antenna plate; \(\phi_{i}\) is angle between each element's projection on the XOY plane and X-axis. Differentiating the two sides of Equations (4) and (5), they can be expressed as:  
\[ \begin{array}{l}\Delta\phi_{AB}=\frac{2\pi}{\lambda}\left[-(x_{B}-x_{A})\sin \alpha+(y_{B}-y_{A})\cos\alpha\right]\cos\beta\Delta\alpha\\ \quad\quad-\frac{2\pi}{\lambda}\left[-(x_{B}-x_{A})\sin\beta\cos\alpha+(y_{B} -y_{A})\sin\beta\sin\alpha-(z_{B}-z_{A})\cos\beta\right]\Delta\beta\end{array}  \tag{23}\]  
\[ \begin{array}{l}\Delta\phi_{AC}=\frac{2\pi}{\lambda}\left[-(x_{C}-x_{A})\sin \alpha+(y_{C}-y_{A})\cos\alpha\right]\cos\beta\Delta\alpha\\ \quad\quad-\frac{2\pi}{\lambda}\left[-(x_{C}-x_{A})\sin\beta\cos\alpha+(y_{C} -y_{A})\sin\beta\sin\alpha-(z_{C}-z_{A})\cos\beta\right]\Delta\beta\end{array}  \tag{24}\]  
Figure 9: The model of antenna placement. Based on Equation (22), \(\Delta\alpha\) and \(\Delta\beta\) can be represented as:  
\[\Delta\alpha=\frac{m}{n} \tag{25}\]  
where  
\[ \begin{array}{c}m=\left[(x_{C}-x_{A})\cos\alpha+(y_{C}-y_{A})\sin\alpha\right] \Delta\phi_{AB}-\\ \left[(x_{B}-x_{A})\cos\alpha+(y_{B}-y_{A})\sin\alpha\right]\Delta\phi_{AC} \end{array}  \tag{26}\]  
\[ \begin{array}{c}n=\frac{2\pi}{\lambda}\left\{z_{\rm A}\cot\beta\left[\sin( \psi_{\rm B}-\alpha)-\sin(\psi_{\rm C}-\alpha)\right]+\right.\\ \left.(x_{C}-x_{A})(y_{B}-y_{A})-(x_{B}-x_{A})(y_{C}-y_{A})\right\}\cos\beta \end{array}  \tag{27}\]  
and  
\[\Delta\beta=\frac{o}{p} \tag{28}\]  
\[ \begin{array}{c}o=\left[(y_{B}-y_{A})\cos\alpha-(x_{B}-x_{A})\sin\alpha \right]\Delta\phi_{AC}-\\ \left[(y_{C}-y_{A})\cos\alpha-(x_{C}-x_{A})\sin\alpha\right]\Delta\phi_{AB} \end{array}  \tag{29}\]  
\[ \begin{array}{c}p=\frac{2\pi}{\lambda}\left\{z_{\rm A}\cot\beta\left[\sin( \psi_{\rm B}-\alpha)-\sin(\psi_{\rm C}-\alpha)\right]+\right.\\ \left.(x_{C}-x_{A})(y_{B}-y_{A})-(x_{B}-x_{A})(y_{C}-y_{A})\right\}\sin\beta \end{array}  \tag{30}\]  
where \(\Delta\alpha\) and \(\Delta\beta\) are the errors of azimuth and elevation respectively; \(\Delta\phi_{AC}\) and \(\Delta\phi_{AB}\) represent the measure errors of \(\phi_{AC}\) and \(\phi_{AB}\) respectively. The average number of phase difference is \(N\). Usually, it is reasonable to assume that \(\Delta\phi_{AC}=\Delta\phi_{AB}\approx 1/\sqrt{N\times SNR}\) (here SNR stands for signal to noise ratio). Using cylindrical coordinates, Equations (25) and (28) can be represented as  
\[\Delta\alpha{=}\frac{1}{\frac{2\pi}{\lambda}\cos\beta\sqrt{N\times SNR}}\times \frac{e}{f} \tag{31}\]  
\[e=\cos(\phi_{\rm C}-\alpha)-\cos(\phi_{\rm B}-\alpha) \tag{32}\]  
\[f=\left\{z_{\rm A}\cot\beta\left[\sin(\phi_{\rm B}-\alpha)-\sin(\phi_{\rm C} -\alpha)\right]+\right. \tag{33}\]  
\[\left.r\left[\sin(\phi_{\rm B}-\phi_{\rm C})+\sin(\phi_{\rm C}-\phi_{\rm A})+ \sin(\phi_{\rm A}-\phi_{\rm B})\right]\right\}\]  
\[\Delta\beta{=}\frac{1}{\frac{2\pi}{\lambda}\sin\beta\sqrt{N\times SNR}}\times \frac{g}{h} \tag{34}\]  
\[g=\sin(\phi_{\rm C}-\alpha)-\sin(\phi_{\rm B}-\alpha) \tag{35}\]  
\[h=\left\{z_{\rm A}\cot\beta\left[\sin(\phi_{\rm B}-\alpha)-\sin(\phi_{\rm C} -\alpha)\right]+\right. \tag{36}\]  
When \(z_{A}\) is zero, the 3D model deteriorates into 2D model.