# Performance Analysis of Amplitude Comparison Monopulse Direction-of-Arrival Estimation
## 3 Angular Tracking of Amplitude Comparison Monopulse by Approximation
\begin{array}{c}n_{A}\left(\frac{\left(\theta_{\text{AB}}/k_{m} \right)-\theta}{4 AC_{0}}\right)+n_{B}\left(\frac{-\left(\left(\theta_{\text{AB} }/k_{m}\right)+\theta\right)}{4 AC_{0}}\right)+n_{C}\left(\frac{-\left(\left( \theta_{\text{AB}}/k_{m}\right)+\theta\right)}{4 AC_{0}}\right)+n_{D}\left( \frac{\left(\theta_{\text{AB}}/k_{m}\right)-\theta}{4 AC_{0}}\right)\\ \hat{\phi}^{\prime\left(\text{approxi}=1\right)}\\ =\phi+\left(\begin{array}{c}n_{A}\left(\frac{\left(\theta_{\text{AB}}/k_{m} \right)-\phi}{4 AC_{0}}\right)+n_{D}\left(\frac{-\left(\left(\theta_{\text{AB} }/k_{m}\right)+\phi\right)}{4 AC_{0}}\right)+n_{C}\left(\frac{-\left(\left( \theta_{\text{AB}}/k_{m}\right)+\phi\right)}{4 AC_{0}}\right)+n_{B}\left(\frac{ \left(\theta_{\text{AB}}/k_{m}\right)-\phi}{4 AC_{0}}\right)\\ \end{array}\right)\end{array}  \tag{18}\]  
Also, the approximate estimates of azimuth and elevation based on the second order Taylor series expansion are given by  
\[\begin{array}{l}\hat{\theta}^{\prime\left(\text{approxi}=2\right)}\\ =\theta+\left( \begin{array}{c}n_{A}\left(\frac{\left(\theta_{\text{AB}}/k_{m} \right)-\theta}{4 AC_{0}}\right)+n_{B}\left(\frac{-\left(\theta_{\text{AB}}/k_{ m}\right)+\theta}{4 AC_{0}}\right)+n_{C}\left(\frac{-\left(\theta_{\text{AB}}/k_{m} \right)+\theta}{4 AC_{0}}\right)+n_{D}\left(\frac{\left(\theta_{\text{AB}}/k_{ m}\right)-\theta}{4 AC_{0}}\right)\\ +\left(\begin{array}{c}+\frac{n_{A}\theta^{2}}{2}\left(\frac{-2\left(\theta_ {\text{AB}}/k_{m}\right)-\theta}{4 AC_{0}}\right)+\frac{n_{B}\theta^{2}}{2} \left(\frac{2\left(\theta_{\text{AB}}/k_{m}\right)+\theta}{4 AC_{0}}\right)+ \frac{n_{C}^{2}}{2}\left(\frac{2\left(\theta_{\text{AB}}/k_{m}\right)+\theta}{ 4 AC_{0}}\right)+\frac{n_{D}^{2}}{2}\left(\frac{-2\left(\theta_{\text{AB}}/k_{ m}\right)-\theta}{4 AC_{0}}\right)\\ +n_{A}n_{B}\left(\frac{2\theta}{\left(4 AC_{0}\right)^{2}}\right)+n_{A}n_{C}\left( \frac{2\theta}{\left(4 AC_{0}\right)^{2}}\right)+n_{A}n_{D}\left(\frac{-2\left( \theta_{\text{AB}}/k_{m}\right)-\theta}{4 AC_{0}}\right)\\ +n_{B}n_{C}\left(\frac{2\left(\theta_{\text{AB}}/k_{m}\right)+\theta}{\left( 4 AC_{0}\right)^{2}}\right)+n_{B}n_{D}\left(\frac{2\theta}{\left(4 AC_{0} \right)^{2}}\right)+n_{C}n_{D}\left(\frac{2\theta}{\left(4 AC_{0}\right)^{2}} \right)\end{array}\right)\end{array}  \tag{19}\]  
\[\begin{array}{l}\hat{\phi}^{\prime\left(\text{approxi}=2\right)}\\ =\phi+\left( \begin{array}{c}n_{A}\left(\frac{\left(\theta_{\text{AB}}/k_{m} \right)-\phi}{4 AC_{0}}\right)+n_{D}\left(\frac{-\left(\theta_{\text{AB}}/k_{m} \right)+\phi}{4 AC_{0}}\right)+n_{C}\left(\frac{-\left(\theta_{\text{AB}}/k_{m} \right)+\phi}{4 AC_{0}}\right)+n_{B}\left(\frac{\left(\theta_{\text{AB}}/k_{m} \right)-\phi}{4 AC_{0}}\right)\\ +\left(\begin{array}{c}+\frac{n_{A}^{2}}{2}\left(\frac{-2\left(\theta_{\text{ AB}}/k_{m}\right)-\phi}{\left(4 AC_{0}\right)^{2}}\right)+\frac{n_{D} ^{2}}{2}\left(\frac{2\left(\theta_{\text{AB}}/k_{m}\right)+\phi}{4 AC_{0}} \right)+\frac{n_{C}^{2}}{2}\left(\frac{2\left(\theta_{\text{AB}}/k_{m}\right) +\phi}{4 AC_{0}}\right)+\frac{n_{B}^{2}}{2}\left(\frac{-2\left(\theta_{\text{AB} }/k_{m}\right)-\phi}{\left(4 AC_{0}\right)^{2}}\right)\\ +n_{A}n_{D}\left(\frac{2\phi}{\left(4 AC_{0}\right)^{2}}\right)+n_{A}n_{C}\left( \frac{2\phi}{\left(4 AC_{0}\right)^{2}}\right)+n_{A}n_{B}\left(\frac{-2\left( \theta_{\text{AB}}/k_{m}\right)-\phi}{\left(4 AC_{0}\right)^{2}}\right)\\ +n_{D}n_{C}\left(\frac{2\left(\theta_{\text{AB}}/k_{m}\right)+\phi}{\left(4 AC_{0} \right)^{2}}\right)+n_{B}n_{D}\left(\frac{2\phi}{\left(4 AC_{0}\right)^{2}} \right)+n_{C}n_{B}\left(\frac{2\phi}{\left(4 AC_{0}\right)^{2}}\right)\end{array} \right).\end{array}  \tag{20}\]