# Performance Analysis of Interferometer Algorithm under Phase Measurement Error
## 4 Performance Analysis of Phase Comparison Interferometer
\begin{array}{c}-\frac{2\pi r}{\lambda}\frac{M}{2} \sin(\psi^{(0)})\left\{\cos(\psi^{(0)})\right\}^{2}\frac{M-1}{2}\Delta\phi_{m} \cos(a_{m})-\left[\sin(\psi^{(0)})\cos(\psi^{(0)})\right]\\ \left[\begin{array}{c}\left\{\sum\limits_{m=0}^{M-1}\Delta\phi_{m}\cos(a_{m} )\right\}^{2}+\left\{\sum\limits_{m=0}^{M-1}\Delta\phi_{m}\sin(a_{m})\right\}^ {2}\right]\end{array}\right]}\\ \hline\left\{\begin{array}{c}\left\{\frac{2\pi r}{\lambda}\frac{M}{2}\right\}^ {2}\left\{\sin(\psi^{(0)})\right\}^{2}\left\{\cos(\psi^{(0)})\right\}^{2}+ \frac{2\pi r}{\lambda}\frac{M}{2}\left\{\cos(\psi^{(0)})\right\}^{3}\sum \limits_{m=0}^{M-1}\Delta\phi_{m}\cos(a_{m})\\ +\frac{2\pi r}{\lambda}\frac{M}{2}\left\{\cos(\psi^{(0)})\right\}^{2}\sin( \psi^{(0)})\sum\limits_{m=0}^{M-1}\Delta\phi_{m}\cos(a_{m})+\left\{\cos(\psi^ {(0)})\sum\limits_{m=0}^{M-1}\Delta\phi_{m}\cos(a_{m})\right\}^{2}\\ -\sin(\psi^{(0)})\cos(\psi^{(0)})\cdot\sum\limits_{m=0}^{M-1}\Delta\phi_{m} \sin(a_{m})\sum\limits_{m=0}^{M-1}\Delta\phi_{m}\cos(a_{m})\end{array}\right] \end{array} \right]} \tag{14}\]  
\[\text{Since }-\frac{2\pi r}{\lambda}\frac{M}{2}\left\{\sin(\psi^{(0)})\right\}^{2} \cos(\psi^{(0)})\sum\limits_{m=0}^{M-1}\Delta\phi_{m}\sin(a_{m})\text{ in the numerator of \eqref{eq:21} and }\left\{\frac{2\pi r}{\lambda}\frac{M}{2}\right\}^{2}\left\{\sin(\psi^{(0)}) \right\}^{2}\left\{\cos(\psi^{(0)})\right\}^{2}\text{ in the denominator of \eqref{eq:21} is larger than other terms, \eqref{eq:22} and \eqref{eq:223} can be approximated as  
\[\Delta\theta=\frac{-\frac{2\pi r}{\lambda}\frac{M}{2}\left\{\sin(\psi^{(0)}) \right\}^{2}\cos(\psi^{(0)})\sum\limits_{m=0}^{M-1}\Delta\phi_{m}\sin(a_{m})}{ \left\{\frac{2\pi r}{\lambda}\frac{M}{2}\right\}^{2}\left\{\sin(\psi^{(0)}) \right\}^{2}\left\{\cos(\psi^{(0)})\right\}^{2}} \tag{15}\]  
\[\Delta\psi=\frac{-\frac{2\pi r}{\lambda}\frac{M}{2}\sin(\psi^{(0)})\left\{\cos( \psi^{(0)})\right\}^{2}\sum\limits_{m=0}^{M-1}\Delta\phi_{m}\cos(a_{m})}{ \left\{\frac{2\pi r}{\lambda}\frac{M}{2}\right\}^{2}\left\{\sin(\psi^{(0)}) \right\}^{2}\left\{\cos(\psi^{(0)})\right\}^{2}}. \tag{16}\]  
Since the phase errors in the different sensors are uncorrelated, we have  
\[ \begin{split} E\Bigg{[}\left\{\sum\limits_{m=0}^{M-1}\Delta\varphi_ {m}\sin(a_{m})\right\}^{2}\Bigg{]}&=\sigma_{\phi}^{2}\frac{M}{2} \\ E\Bigg{[}\left\{\sum\limits_{m=0}^{M-1}\Delta\varphi_{m}\cos(a_{m}) \right\}^{2}\Bigg{]}&=\sigma_{\phi}^{2}\frac{M}{2}.\end{split}  \tag{17}\]  
(17) is derived in Appendix C.  
Applying the first equation of (17) to (15), the azimuth MSE can be written as \[E\Big{[}(\Delta\theta)^{2}\Big{]}=\left[ \begin{array}{c}-\frac{2\pi r}{\lambda} \frac{M}{2}\Big{\{}\sin(\psi^{(0)})\Big{\}}^{2}\cos(\psi^{(0)})\\ \Big{\{}\frac{2\pi r}{\lambda}\frac{M}{2}\Big{\}}^{2}\big{\{}\sin(\psi^{(0)}) \big{\}}^{2}\big{\{}\cos(\psi^{(0)})\big{\}}^{2}\end{array} \right]^{2}E \left[\left(\sum_{m=0}^{M-1}\Delta\phi_{m}\sin(a_{m})\right)^{2}\right]=\frac{ \sigma_{\varphi}^{2}}{\frac{M}{2}\big{\{}\frac{2\pi r}{\lambda}\big{\}}^{2} \big{\{}\cos(\psi^{(0)})\big{\}}^{2}}. \tag{18}\]  
Similarly, applying the second equation of (17) to (16), the elevation MSE can be expressed as  
\[E\Big{[}(\Delta\psi)^{2}\Big{]}=\left[ \begin{array}{c}-\frac{2\pi r}{\lambda} \frac{M}{2}\Big{\{}\sin(\psi^{(0)})\Big{\}}^{2}\cos(\psi^{(0)})\\ \Big{\{}\frac{2\pi r}{\lambda}\frac{M}{2}\Big{\}}^{2}\big{\{}\sin(\psi^{(0)}) \big{\}}^{2}\big{\{}\cos(\psi^{(0)})\big{\}}^{2}\end{array} \right]^{2}E\left[ \left(\sum_{m=0}^{M-1}\Delta\phi_{m}\cos(a_{m})\right)^{2}\right]=\frac{ \sigma_{\varphi}^{2}}{\frac{M}{2}\big{\{}\frac{2\pi r}{\lambda}\big{\}}^{2} \big{\{}\sin(\psi^{(0)})\big{\}}^{2}}. \tag{19}\]  
From (18) and (19), for small \(\psi\), \(E[(\Delta\psi)^{2}]\) is much greater than \(E[(\Delta\theta)^{2}]\). That is, for a signal with small incident elevation, elevation error is much larger than the azimuth error. This could be confirmed by simulation in [13]. We are concerned with how the azimuth MSE is dependent on the true azimuth or the true elevation.