# Performance Analysis of Amplitude Comparison Monopulse Direction-of-Arrival Estimation
## 2 Angular Tracking of Amplitude Comparison Monopulse  
The amplitude comparison monopulse radar overlays the two antenna beams to detect the angular error and estimate the angle of the target. Four antenna beams are required to detect each angular error in the azimuth and the elevation. The beam pattern of the basic monopulse radar consists of two patterns, which include one azimuth pattern and one elevation pattern. The radar transmitter emits a signal in a total pattern, and the receiver receives the signal in a sum pattern (\(\Sigma\)) and a difference pattern (\(\Delta\)). The error signal can be expressed as  
\[e=\frac{\Delta}{\sum}\cos\theta \tag{1}\]  
where \(\Delta\), \(\sum\) and \(\theta\) denote the magnitude of the signal in the sum pattern, the magnitude of the signal in the difference pattern and the phase difference between the sum pattern and the difference pattern, respectively. Figure 1a illustrates the case where the target is located on the track centre axis, and the magnitude of the pulses received from the four beams are the same. Figure 1b-d shows the cases where the target is not on the track centre axis, which results in different received signals [21]. The beam pattern of the antenna is assumed to be a Gaussian pattern [21] :  
\[G\left(\theta\right)=G_{0}\exp\left[-2\ln 2(\theta/\theta_{\rm 3 dB})^{2}\right], \tag{2}\]  
where \(\theta_{\rm 3 dB}\) denotes a 3-dB beam width. The four beams of the antenna are placed equally on each quadrant, and the target positions at the antenna beam A are shown in (3). The beam pattern can be expressed as  
\[ \begin{array}{l}G\left(\theta\right)=G_{0}\exp\left[-2\ln 2(\theta/\theta_{ \rm 3 dB})^{2}\right]\\ =G_{0}\exp\left[-2\ln 2\left\{\left(\frac{\left(\theta_{S}/\sqrt{2}\right)- \theta_{\rm axi}}{\theta_{\rm 3 dB}}\right)^{2}+\left(\frac{\left(\theta_{S}/ \sqrt{2}\right)-\theta_{\rm ele}}{\theta_{\rm 3 dB}}\right)^{2}\right\}\right]\end{array}  \tag{3}\]  
where \(\theta_{S}\) is the squint angle of the antenna, \(G_{0}\) is the voltage gain of the antenna beam at the track axis and \(\left(\theta_{\rm azi},\theta_{\rm ele}\right)\) is the value of the angle component of the target position relative to the track centre axis. \[ \begin{array}{l}\theta_{A}{}^{2}=\left(\frac{\theta_{S}}{\sqrt{2}}-\theta_ {\rm azi}\right)^{2}+\left(\frac{\theta_{S}}{\sqrt{2}}-\theta_{\rm ele}\right) ^{2}\\ =\theta_{S}{}^{2}-\sqrt{2}\theta_{S}\left(\theta_{\rm ele}+\theta_{\rm azi} \right)+\theta_{\rm azi}{}^{2}+\theta_{\rm ele}{}^{2}\end{array}  \tag{4}\]  
For small \(\theta_{\rm azi}\) and \(\theta_{\rm ele}\), \(\theta_{A}\) can be approximated as  
\[\theta_{A}{}^{2}\approx\theta_{S}{}^{2}-\sqrt{2}\theta_{S}\left(\theta_{\rm azi }+\theta_{\rm ele}\right) \tag{5}\]  
Figure 1: Illustration of monopulse concept. The voltage obtained from the target in the antenna beam A defined in (5) can be written as (6). \[ \begin{array}{l}G\left(\theta_{A}\right)\approx G_{0}\exp\left[-2\ln 2\left( \frac{\theta_{S}}{\theta_{\text{3 dB}}}\right)^{2}\left[\theta_{S}{}^{2}-\sqrt{2} \theta_{S}\left(\theta_{\text{azi}}+\theta_{\text{ele}}\right)\right]\right]\\ =G_{0}\exp\left[-2\ln 2\left(\frac{\theta_{S}}{\theta_{\text{3 dB}}}\right)^{2} \right]\exp\left[+2\sqrt{2}\ln 2\left(\frac{1}{\theta_{\text{3 dB}}}\right)^{2} \theta_{S}\left(\theta_{\text{azi}}+\theta_{\text{ele}}\right)\right]\\ =G\left(\theta_{S}\right)\exp\left[+2\sqrt{2}\ln 2\left(\frac{\theta_{S}}{\theta_{ \text{3 dB}}}\right)\left(\frac{\theta_{\text{azi}}+\theta_{\text{ele}}}{ \theta_{\text{3 dB}}}\right)\right]\end{array}  \tag{6}\]  
As the coefficient of monopulse error is \(k_{m}=2\sqrt{2}\ln 2\theta_{S}/\theta_{\text{3 dB}}\) for \(k_{m}\left(\theta_{\text{azi}}-\theta_{\text{ele}}\right)/\theta_{\text{3 dB}}\ll 1\), \(G\left(\theta_{S}\right)\) is approximately equal to \(G_{0}\), which results in  
\[