## 3 Performance Analysis of the MPS Beamformer
### Performance of MPS Beamformer with the Number of Multibeams \(P\)  
Adopting larger number of multibeams in Equation (9) is expected to reduce the processing time significantly in the MPS beamformer, however the divergence of the learning curve becomes faster also due to the cumulative addition of multibeams sidelobes especially at higher values of \(\beta\) as discussed in the previous section. To investigate this learning behavior, we examine the learning curves at different values of \(P\) as shown in Figure 5 for an array of \(16\times 16\) elements with \(\beta=0.1\) for the multibeam case while we use \(\beta=0.7\) for the single sidelobe sequential suppression as in [32] to speed up the convergence for that algorithm and for comparison purposes. The iteration cycles continue to achieve a \(-40\) dB SLL at several values of \(P\) as shown in Figure 5 where there is a significant reduction in the required number of iterations in the MPS beamformer if we include larger number of multibeams in the reduction process. The single beam sidelobe sequential damping (SSD) algorithm [33] provides a stable convergence profile, however, it requires very large number of iterations to achieve deep SLL which is more than five times of that required in the MPS beamformer case although a higher value of \(\beta\) is used.  
In terms of the absolute processing time in seconds, the comparison in Figure 5 has been conducted by a platform using i7-8550U CPU @ 1.80 GHz with 16 GB RAM. In addition, MATLAB 2021b is used in the simulation of the proposed MPS beamformer and the SSD algorithm as well. The average absolute processing time in seconds is shown Table 1 which is consistent with the data presented in Figure 5.  
The performance of MPS beamformer at different number of multiple major sidelobes is almost the same at SLL values of \(-35\) dB or higher, while there is a significant difference at  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Number of Multibeams** & **Average Processing Time in Seconds** \\ \hline Single beam (SSD Algorithm in [32]) & 469.59 \\ \hline \(M+N-4\) & 105.1 \\ \hline \(2(M+N-4)\) & 78.0 \\ \hline \(3(M+N-4)\) & 86.5 \\ \hline \(4(M+N-4)\) & 88.0 \\ \hline \(5(M+N-4)\) & 86.6 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Average processing time required for achieving \(-40\) dB SLL using the MPS beamformer at different number of multibeams and SSD algorithm in [32].  
Figure 5: Convergence profiles at different number of subtracted multibeams.  
lower SLL values as shown in Figure 6. Utilizing the first major sidelobes can provide deep SLL of \(-50\) dB after 143 iterations while incorporating the second set of major sidelobes quickly achieve this level after approximately 95 iterations. Increasing the number of multibeams to include the third and fourth major sidelobes decreases the processing time where at \(P=4\) (\(M+N-4\)) the required number of iterations is reduced to only 64 cycles. Any increase in \(P\) will not add any tangible performance improvement and it may requires decreasing the value of \(\beta\) to maintain convergence which in turn requires larger number of iterations. Therefore, including the first two to four major sidelobe groups is very sufficient for achieving deep SLL at lower processing time.