# Performance Analysis of Interferometer Algorithm under Phase Measurement Error
## 4 Performance Analysis of Phase Comparison Interferometer  
In this paper, the analytic performance analysis of an interferometer algorithm for the UCA is considered. The UCA geometry is illustrated in Figure 1. The number of sensors in the UCA is equal to \(M\). The theoretical phase value of incident signal on the \(m\)-th sensor is expressed as  
\[\phi_{m}=\frac{2\pi r}{\lambda}\cos(\theta^{(0)}-\frac{2\pi m}{M})\cos(\psi^{( 0)}) \tag{1}\]  
In (1), \(\theta^{(0)}\) and \(\psi^{(0)}\) are true azimuth and true elevation, respectively. \(r\) is radius of circular array, \(\lambda\) is wavelength, and \(M\) is the number of sensors. For the UCA with \(r=\frac{\lambda}{2}\), (1) reduces to  
\[\phi_{m}=\pi\cos(\theta^{(0)}-\frac{2\pi m}{M})\cos(\psi^{(0)}) \tag{2}\]  
Since the product of two cosine functions varies from \(-1\) to \(1\), the value in (2) varies from \(-\pi\) to \(\pi\), implying that there is no phase ambiguity for \(r=\frac{\lambda}{2}\). If \(r\) is greater than \(\frac{\lambda}{2}\), the value in (2) can be greater than \(\pi\) or smaller than \(-\pi\), implying that there is phase ambiguity for \(r>\frac{\lambda}{2}\). Phase measurement error is assumed to be a Gaussian random variables or uniform random variables with zero mean, and cosine cost function is used to derive correlation value between theoretical phase values and the estimated phase of the incident signal. Cosine cost function of interferometer algorithm is expressed as  
\[J(\theta,\psi)=\sum_{m=0}^{M-1}\cos\left[\frac{2\pi r}{\lambda}\cos(\theta- \frac{2\pi m}{M})\cos(\psi)-\hat{\phi}_{m}\right] \tag{3}\]  
where \(\hat{\phi}_{m}\) is the measured phase of \(m\)-th sensor. In (3), \(\theta\) and \(\psi\) can be expressed as \(\theta^{(0)}+\Delta\theta\) and \(\psi^{(0)}+\Delta\psi\), respectively. Partial differentiations and linear approximation are used for optimization with respect to \(\theta\) and \(\psi\):  
\[\frac{\partial J(\theta,\psi)}{\partial\theta}\left| \begin{array}{ll}\theta= \hat{\theta}=\theta^{(0)}+\Delta\theta&=\sum_{m=0}^{M-1}\sin \left[\frac{2\pi r}{\lambda}\cos(\theta-\frac{2\pi m}{M})\cos(\psi)-\hat{\phi} _{m}\right]\sin(\theta-\frac{2\pi m}{M})\cos(\psi)=0.\end{array} \right. \tag{4}\]  
\[\frac{\partial J(\theta,\psi)}{\partial\psi}\left| \begin{array}{ll}\theta= \hat{\theta}=\theta^{(0)}+\Delta\theta&=\sum_{m=0}^{M-1}\sin \left[\frac{2\pi r}{\lambda}\cos(\theta-\frac{2\pi m}{M})\cos(\psi)-\hat{\phi} _{m}\right]\cos(\theta-\frac{2\pi m}{M})\sin(\psi)=0.\end{array} \right. \tag{5}\]  
\(\psi=\psi=\psi^{(0)}+\Delta\psi\)  
Analytic MSE values of azimuth and elevation are derived by solving the simultaneous equations with respect to \(\theta\) and \(\psi\). Due to \(\sin\theta\approx\theta\) for small \(\theta\), (4) can be rewritten as  
\[ \begin{array}{ll}\frac{\partial J(\theta,\psi)}{\partial\theta}\left| \begin{array}{ll}\theta=\hat{\theta}=\theta^{(0)}+\Delta\theta&=\sum_{m=0} ^{M-1}\left[\frac{2\pi r}{\lambda}\cos(\theta^{(0)}+\Delta\theta-\frac{2\pi m }{M})\cos(\psi^{(0)}+\Delta\psi)-\hat{\phi}_{m}\right]\sin(\theta^{(0)}+ \Delta\theta-\frac{2\pi m}{M})\cos(\psi^{(0)}+\Delta\psi)=0\\ \psi=\hat{\psi}=\psi^{(0)}+\Delta\psi&\end{array}\right.\end{array}  \tag{6}\]  
In (6), \(\theta^{(0)}-\frac{2\pi m}{M}\) is replaced with \(a_{m}\) and trigonometric identitiy is used:  
Figure 1: This figure shows Uniform Circular Array (UCA). The two-dimensional array can simultaneously estimate azimuth and elevation of signal source. \[\left. \begin{array}{ll}\frac{\partial J(\theta,\psi)}{\partial\theta} \bigg{|}&\theta=\hat{\theta}=\theta^{(0)}+\Delta\theta&=\sum\limits_{m=0}^{M-1} \bigg{[}\frac{2\pi r}{\lambda}\cos(a_{m}+\Delta\theta)\cos(\psi^{(0)}+\Delta \psi)-\hat{\phi}_{m}\bigg{]}\sin(a_{m}+\Delta\theta)\cos(\psi^{(0)}+\Delta \psi)=0.\\ &\psi=\hat{\psi}=\psi^{(0)}+\Delta\psi\end{array} \right. \tag{7}\]  
By distributive law, (7) can be rewritten as  
\[\left.