# A Study on the Amplitude Comparison Monopulse Algorithm
## 2 Conventional Amplitude Comparison Monopulse Algorithm  
Measurement of the target direction angle or target location is required for target tracking. Amplitude comparison monopulse radar operates by comparing the amplitude of signal returns of four squinted sub-beams steered symmetrically around the expected target direction as shown in Figure 1[1]. The sum, horizontal difference, and vertical difference channels of the incoming signal are used to estimate target location, which is represented by the azimuth and elevation angle deviations in the track axis. Figure 1: Beam pattern of a monopulse radar. Each antenna pattern can be modeled near the center of the track axis as a squinted Gaussian beam. Each one-way voltage gain function is given by  
\[g(\theta)=g_{0}\exp\left(-2\left(\frac{\theta}{\theta_{3 dB}}\right)^{2}\log 2\right) \tag{1}\]  
where \(\theta\) is the total angle from the target location to the beam peak, \(\theta_{3 dB}\) is the beam width, and \(g_{0}\) is the gain at the beam peak [12]. The total angle, \(\theta\), can be computed from the target location defined in Figure 2, which presents the field of view as seen from the monopulse radar. For a beam with its maximum in quadrant 1, the target location from the beam peak can be written in terms of azimuth and elevation angle deviations (\(\theta_{az},\ \theta_{el}\)) as  
\[\theta^{2}=\left(\frac{\theta_{s}}{\sqrt{2}}-\theta_{az}\right)^{2}+\left( \frac{\theta_{s}}{\sqrt{2}}-\theta_{el}\right)^{2} \tag{2}\]  
and the quadrant 1 voltage gain is then  
\[g_{1}(\theta)=g_{1}(\theta_{az},\theta_{el})=g_{0}\exp\left[-2\left(\frac{ \left(\frac{\theta_{s}}{\sqrt{2}}-\theta_{az}\right)^{2}+\left(\frac{\theta_{ s}}{\sqrt{2}}-\theta_{el}\right)^{2}}{\theta_{3 dB}^{2}}\right)\log 2\right] \tag{3}\]  
where \(\theta_{s}\) is the squint angle of the antenna. Using the above gain functions and a received signal voltage of amplitude \(A\), the so-called pseudorange equations are given as follows:  
\[\left[ \begin{array}{c}s_{1}\\ s_{2}\\ s_{3}\\ s_{4}\end{array} \right]=A\left[ \begin{array}{c}g_{1}(\theta_{az},\theta_{el })\\ g_{2}(\theta_{az},\theta_{el})\\ g_{3}(\theta_{az},\theta_{el})\\ g_{4}(\theta_{az},\theta_{el})\end{array} \right]+\left[ \begin{array}{c}n_{1} \\ n_{2}\\ n_{3}\\ n_{4}\end{array} \right] \tag{4}\]  
where \(s_{i}\) is the measurement signal of \(i\)-th quadrant, which is contaminated by the measurement noise \(n_{i}\). Here, \(s_{i}\) can be considered as pseudorange since it is a function of geometrical distance weighted by the antenna gain. The conventional amplitude comparison monopulse algorithm for estimating azimuth and elevation angle deviations is briefly explained in the next few paragraphs [4; 5; 12]. Figure 2: Target location in the track axis. Assuming that \(\theta_{az}^{2}+\theta_{d}^{2}\approx 0\), the quadrant 1 voltage gain is simplified as follows:  
\[g_{1}(\theta)=g_{1}(\theta_{az},\theta_{el})\ \approx\ g(\theta_{s})\ \exp\left[2\sqrt{2}\left(\frac{ \theta_{s}}{\theta_{3 dB}}\right)\left(\frac{\theta_{az}+\theta_{el}}{\theta_{3 dB }}\right)\log 2\right] \tag{5}\]  
If a monopulse error slope coefficient is defined as  
\[k_{m}=2\sqrt{2}\left(\frac{\theta_{s}}{\theta_{3 dB}}\right)\log 2 \tag{6}\]  
then, voltage gain for the beam in quadrant 1 should be  
\[g_{1}(\theta_{az},\theta_{d})=g(\theta_{s})\ \exp\left[k_{m}\left(\frac{\theta_{ az}+\theta_{el}}{\theta_{3 dB}}\right)\right]\approx g_{0}\left[1+\frac{k_{m}}{ \theta_{3 dB}}\ (\theta_{az}+\theta_{el})\right] \tag{7}\]  
assuming \(k_{m}\big{(}\frac{\theta_{az}+\theta_{d}}{\theta_{3 dB}}\big{)}\ll 1\).