# Performance Analysis of Amplitude Comparison Monopulse Direction-of-Arrival Estimation
## 3 Angular Tracking of Amplitude Comparison Monopulse by Approximation  
The sum pattern (\(\Sigma\)) and the difference pattern (\(\Delta\)) of the antenna beam are represented. The noise of the incoming signal is assumed to be Gaussian-distributed. The noise in the antenna beam A that is denoted by \(n_{A}\). \(n_{B}\), \(n_{C}\) and \(n_{D}\) are similarly defined. \[\epsilon^{\prime}_{\rm azi} =A\left[ \begin{array}{c}\left\{G\left(\theta_{A}\right)+n_{A}+G \left(\theta_{D}\right)+n_{D}\right\}\\ -\left\{G\left(\theta_{B}\right)+n_{B}+G\left(\theta_{C}\right)+n_{C}\right\} \end{array} \right] \tag{11}\] \[=4 AG_{0}k_{m}\frac{\phi^{\prime}}{\theta_{\rm 3 dB}}\]  
\[\epsilon^{\prime}_{\rm ele} =A\left[ \begin{array}{c}\left\{G\left(\theta_{A}\right)+n_{A}+G \left(\theta_{D}\right)+n_{D}\right\}\\ -\left\{G\left(\theta_{B}\right)+n_{B}+G\left(\theta_{C}\right)+n_{C}\right\} \end{array} \right] \tag{12}\] \[=4 AG_{0}k_{m}\frac{\phi^{\prime}}{\theta_{\rm 3 dB}}\]  
\[\sum^{\prime} =A\left[ \begin{array}{c}G\left(\theta_{A}\right)+n_{A}+G\left( \theta_{B}\right)+n_{B}\\ +G\left(\theta_{C}\right)+n_{C}+G\left(\theta_{D}\right)+n_{D}\end{array} \right] \tag{13}\] \[=4 AG_{0}+\left(n_{A}+n_{B}+n_{C}+n_{D}\right)\]  
The azimuth estimates (\(\hat{\theta}^{\prime}\)) and the elevation estimates (\(\hat{\phi}^{\prime}\)) are given by  
\[\frac{\theta_{\rm 3 dB}\epsilon^{\prime}_{\rm azi}}{k_{m}\sum^{\prime}}=\hat{ \theta}^{\prime},\frac{\theta_{\rm 3 dB}\epsilon^{\prime}_{\rm ele}}{k_{m}\sum^{ \prime}}=\hat{\phi}^{\prime}. \tag{14}\]  
Also, the sum pattern (\(\Sigma^{o}\)) and the difference pattern (\(\Delta^{o}\)) of the antenna beam, which is defined as \(G(\theta)\) obtained from (3), can be represented that the azimuth estimates (\(\hat{\theta}^{\prime o}\)) and the elevation estimates (\(\hat{\phi}^{\prime o}\)) in (15)  
\[\frac{\theta_{\rm 3 dB}(\epsilon^{\prime})^{o}_{\rm azi}}{k_{m}(\sum^{\prime})^{o} }=\hat{\theta}^{\prime o},\frac{\theta_{\rm 3 dB}(\epsilon^{\prime})^{o}_{\rm ele}}{k_{ m}(\sum^{\prime})^{o}}=\hat{\phi}^{\prime o}. \tag{15}\]  
where \((\epsilon^{\prime})^{o}_{\rm azi}\) is different from \(\epsilon^{\prime}_{\rm azi}\) in that \(G(\theta)\) in (3) and \(G(\theta)\) in (6) are associated with used \((\epsilon^{\prime})^{o}_{\rm azi}\) and \(\epsilon^{\prime}_{\rm azi}\), respectively. Note that \(G(\theta)\) in (7) is used for defining \(\epsilon^{\prime}_{\rm azi}\) in (11). \((\epsilon^{\prime})^{o}_{\rm azi}\) is similarly defined. The azimuth angle estimate, \(\hat{\theta}^{\prime}\), and the elevation angle estimate, \(\hat{\phi}^{\prime}\), can be obtained by substituting (11)-(13):  
\[\hat{\theta}^{\prime} =\frac{\theta_{\rm 3 dB}\epsilon^{\prime}_{\rm azi}}{k_{m}\sum^{ \prime}}=\left(\frac{\theta_{\rm 3 dB}}{k_{m}}\right)\left(\frac{4 AG_{0}\frac{k_{m} \theta}{\theta_{\rm 3 dB}}+n_{A}+n_{D}-\left(n_{B}+n_{C}\right)}{4 AG_{0}+n_{A}+n_{B}+n_{C}+n_{D}}\right) \tag{16}\] \[\hat{\phi}^{\prime} =\frac{\theta_{\rm 3 dB}\epsilon^{\prime}_{\rm ele}}{k_{m}\sum^{ \prime}}=\left(\frac{\theta_{\rm 3 dB}}{k_{m}}\right)\left(\frac{4 AG_{0}\frac{k_{m} \theta}{\theta_{\rm 3 dB}}+n_{A}+n_{B}-\left(n_{C}+n_{D}\right)}{4 AG_{0}+n_{A}+n_{B}+n_{C}+n_{D}}\right) \tag{17}\]The approximate estimates of azimuth and elevation based on the first order Taylor series expansion are given by (18). \[\begin{array}{l}\hat{\theta}^{\prime\left(\text{approxi}=1\right)}\\ =\theta+\left(