# Performance Analysis of Interferometer Algorithm under Phase Measurement Error
## 4 Performance Analysis of Phase Comparison Interferometer
\begin{array}{ll}\frac{\partial J(\theta,\psi)}{\partial\theta}\bigg{|} &\theta=\hat{\theta}=\theta^{(0)}+\Delta\theta&=\sum\limits_{m=0}^{M-1} \bigg{[}\frac{2\pi r}{\lambda}\sin(2a_{m}+2\Delta\theta)\cos^{2}\big{(}\psi^{(0 )}+\Delta\psi\big{)}-\hat{\phi}_{m}\sin(a_{m}+\Delta\theta)\cos(\psi^{(0)}+ \Delta\psi\big{)}\bigg{]}=0.\\ &\psi=\hat{\psi}=\psi^{(0)}+\Delta\psi\end{array} \right. \tag{8}\]  
In Appendix B, it is shown why \(\sum\limits_{m=0}^{M-1}\sin(2a_{m}+2\Delta\theta)\) is zero:  
\[\left. \begin{array}{ll}\frac{\partial J(\theta,\psi)}{\partial\theta}\bigg{|} &\theta=\hat{\theta}=\theta^{(0)}+\Delta\theta&=\sum\limits_{m=0}^{M-1} \Big{[}(\phi_{m}+\Delta\phi_{m})\sin(a_{m}+\Delta\theta)\cos(\psi^{(0)}+ \Delta\psi)\Big{]}=0.\\ &\psi=\hat{\psi}=\psi^{(0)}+\Delta\psi\end{array} \right. \tag{9}\]  
Using \(\cos(\Delta\theta)\approx 1\), \(\sin(\Delta\theta)\approx\theta\), (9) can be rewritten as  
\[\left. \begin{array}{ll}\frac{\partial J(\theta,\psi)}{\partial\theta}\Big{|} &\theta=\hat{\theta}=\theta^{(0)}+\Delta\theta&=\sum\limits_{m=0}^{M-1} \Big{[}(\phi_{m}+\Delta\phi_{m})\sin(a_{m}+\Delta\theta)\cos(\psi^{(0)}+ \Delta\psi)\Big{]}\\ &\psi=\hat{\psi}=\psi^{(0)}+\Delta\psi\end{array} \right. \tag{10}\]  
By using \(\sum\limits_{m=0}^{M-1}\cos^{2}(a_{m})=\sum\limits_{m=0}^{M-1}\sin^{2}(a_{m}) =\frac{M}{2}\) which is derived in Appendix B, the right side of (10) is expressed as  
\[\Delta\theta\bigg{[}-\frac{2\pi r}{\lambda}\frac{M}{2}\Big{\{} \cos(\psi^{(0)})\Big{\}}^{2}-\cos(\psi^{(0)})\sum\limits_{m=0}^{M-1}\Delta \phi_{m}\cos(a_{m})\bigg{]}+\Delta\psi\bigg{[}\sin(\psi^{(0)})\sum\limits_{m= 0}^{M-1}\Delta\phi_{m}\sin(a_{m})\bigg{]} \tag{11}\] \[=\cos(\psi^{(0)})\sum\limits_{m=0}^{M-1}\Delta\phi_{m}\sin(a_{m})\]  
In Appendix A, the partial derivative of cost function with respect to the elevation is derived. Applying a few trigonometric identities to (A3) yields  
\[\Delta\theta\bigg{[}\sum\limits_{m=0}^{M-1}\Delta\phi_{m}\sin(a_{ m})\sin(\psi^{(0)})\bigg{]}+\Delta\psi\bigg{[}-\frac{2\pi r}{\lambda}\frac{M}{2} \Big{\{}\sin(\psi^{(0)})\Big{\}}^{2}-\cos(\psi^{(0)})\sum\limits_{m=0}^{M-1} \Delta\phi_{m}\cos(a_{m}))\bigg{]} \tag{12}\] \[=\sin(\psi^{(0)})\sum\limits_{m=0}^{M-1}\Delta\phi_{m}\cos(a_{m})\]  
From (11) and (12), \(\Delta\theta\) and \(\Delta\psi\) can be rewritten as \[\Delta\theta=\frac{\left[\begin{array}{c}-\frac{2\pi r}{\lambda}\frac{M}{2}\left\{ \sin(\psi^{(0)})\right\}^{2}\cos(\psi^{(0)})\sum\limits_{m=0}^{M-1}\Delta \phi_{m}\sin(a_{m})\\ -\left\{\cos(\psi^{(0)})\right\}^{2}+\cos(\psi^{(0)})\sin(\psi^{(0)})\right] \\ \cdot\frac{\int\limits_{m=0}^{M-1}\Delta\phi_{m}\cos(a_{m})}{\left\{\sum \limits_{m=0}^{M-1}\Delta\phi_{m}\sin(a_{m})\right\}}\\ \hline\left[ \begin{array}{c}\left\{\frac{2\pi r}{\lambda}\frac{M}{2}\right\}^ {2}\left\{\sin(\psi^{(0)})\right\}^{2}\left\{\cos(\psi^{(0)})\right\}^{2}+ \frac{2\pi r}{\lambda}\frac{M}{2}\left\{\cos(\psi^{(0)})\right\}^{3}\sum \limits_{m=0}^{M-1}\Delta\phi_{m}\cos(a_{m})\\ +\frac{2\pi r}{\lambda}\frac{M}{2}\left\{\cos(\psi^{(0)})\right\}^{2}\sin( \psi^{(0)})\sum\limits_{m=0}^{M-1}\Delta\phi_{m}\cos(a_{m})+\left\{\cos(\psi^ {(0)})\sum\limits_{m=0}^{M-1}\Delta\phi_{m}\cos(a_{m})\right\}^{2}\\ -\sin(\psi^{(0)})\cos(\psi^{(0)})+\cdot\sum\limits_{m=0}^{M-1}\Delta\phi_{m} \sin(a_{m})\sum\limits_{m=0}^{M-1}\Delta\phi_{m}\cos(a_{m})\end{array} \right] \tag{13}\]  
\[\Delta\psi=\frac{\left[