# A Study on the Amplitude Comparison Monopulse Algorithm
## 6 Simulations  
Simulations were performed to evaluate the proposed least-squares-based monopulse algorithms and their performance prediction with various combinations of measurement noise level, beam width, squint angle, and target direction angles. In the simulations, a track axis gain of \(g_{0}=1\) and a received signal gain of \(A=1\)\(V\) were used. The estimates of the proposed least-squares-based algorithms are given after four iterations. Figures 4 and 5 and Tables 1-3 show the simulation results for the conventional and the proposed methods. They show RMS (root-mean-square) errors of the target location estimate in terms of azimuth and elevation angle deviations from the center of track axis. The simulation results are all based on \(N=100,000\) Monte Carlo runs. The experimental RMS error in the azimuth estimate is defined as  
\[\sigma_{az}=\sqrt{\frac{1}{N}\ \sum_{i=1}^{N}\Big{(}\theta_{az}-\theta_{az}^{(i )}\Big{)}^{2}} \tag{47}\]  
where superscript \((i)\) denotes the results from run \(i\). A similar expression yields the RMS error in the elevation estimate. Figures 4 and 5 show RMS errors with respect to the various standard deviations (\(\sigma\)) of measurement noise in the case of \(\theta_{3 dB}=3^{\circ},\ \theta_{s}=1^{\circ},\ \theta_{az}=0^{\circ},\ \theta_{el}=0^{\circ}\). RMS errors increase linearly with the increase in measurement noise until \(\sigma\approx 0.3^{\circ}\) is reached, which corresponds approximately to an SNR (signal-to-noise ratio) of 5.13 dB. However, for larger noise levels than \(\sigma\approx 0.3^{\circ}\), the performances of all three algorithms become severely worse, although the proposed least-squares-based algorithms are better than the conventional one. Note that all three algorithms provide almost the same estimate accuracy, and the proposed covariance prediction Equations (44) and (46) determine very accurate RMS errors with low noise levels. Therefore, the covariance prediction equation can be used to analytically calculate the RMS errors even  
Figure 4: Three position vectors and true anomalies. Figure 5: Three position vectors and true anomalies. for the conventional monopulse algorithm, and this equation could be considered a measure for the performance analysis of amplitude comparison monopulse radar algorithms. Table 1 shows the RMS errors in Figures 4 and 5 in tabular form. In Table 1, the values in parenthesis in the 'Full' and 'Sum-Difference' columns are analytically computed RMS errors with Equations (43) and (45) respectively, and the values in the 'Prediction' column are standard deviations computed with Equation (44). Table 2 shows RMS errors of target location estimates with respect to unknown target locations (\(\theta_{az},~{}\theta_{el}\)) in the case of \(\theta_{3 dB}=3^{\circ},~{}\theta_{s}=1^{\circ},\sigma=0.05\). It is shown that the target location itself affects the accuracy of the target location estimate, and that the proposed least-squares-based algorithms provide better estimates than the conventional one for the situation where the unknown target location is far from the center of the track axis. Table 3 shows RMS errors with respect to beam width and squint angle (\(\theta_{3 dB},~{}\theta_{s}\)) in the case of \(\theta_{az}=0^{\circ},~{}\theta_{el}=0^{\circ},~{}\sigma=0.1\). As expected from Equation (44), it is clear that the accuracy of the estimates gets better where \(\theta_{s}\) is larger and beam width is smaller.