## 2 Multibeam Pattern Subtraction (MPS) Beamformer Modelling
Therefore, we assume that the significant set of sidelobes for multibeam subtraction are \(\alpha\) times the number of major sidelobes in Equation (8) or:  
\[P=\alpha(M+N-4) \tag{9}\]  
The resulted SLL of the uniform feeding case can be improved by using weighted multibeam pattern subtraction where multi-mainlobes are formed and weighted at the same directions of sidelobes and subtracted from the main pattern with proper amplitude weighting. The process can be repeated to achieve the required SLL if the convergence of the process is guaranteed. If the number of multibeams used for SLL reduction is \(P\), then the resulting array factor after SLL reduction iterations (\(i\)) can be written as:  
\[AF_{i+1}(\theta,\phi)=AF_{i}(\theta,\phi)-\beta\sum_{p=1}^{P}SUM(AF_{i}^{H}( \theta_{p},\phi_{p})|_{N}S(\theta_{p},\phi_{p})\circ S(\theta,\phi)) \tag{10}\]  
where \(AF_{i+1}(\theta,\phi)\) is the resulted array factor after SLL reduction for the previous cycle array factor \(AF_{i}(\theta,\phi)\), \(\beta\) is a control factor for maintaining convergence, \(AF_{i}^{H}(\theta_{p},\phi_{p})|_{N}\) is the normalized array factor for a secondary mainlobe formed at the same direction of the pth sidelobe (\(\theta_{p},\phi_{p}\)) and \(S(\theta_{p},\phi_{p})\) is the array steering matrix at the pth sidelobe direction. The presence of sidelobes in the uniform feeding case helps SLL reduction where each sidelobe can be reduced by generating a secondary mainlobe in the same direction and with the same amplitude assuming that the two lobes have approximately the same widths. The  
Figure 2: Normalized power gain in dB of a broadside beam originating from \(16\times 16\) planar array with uniform feeding. algorithm in [32] used pattern subtraction also but with the adoption of only one sidelobe at a time and the process was repeated to sequentially reduce the next highest sidelobe in the array power pattern. The analysis in [32] was mainly conducted for one-dimensional array and extended to two-dimensional arrays, however, very long processing time was required to reduce the SLL to deep levels. Therefore, in the current paper, the adoption of multibeam pattern subtraction is expected to reduce the processing time greatly for planar arrays. The proposed MPS beamformer in this paper is demonstrated for SLL reduction after three numbers of iterations as shown in Figure 3 for a planar rectangular array formed by \(16\times 16\) isotropic elements using 84 beams for subtraction at \(\beta=0.1\). The overall normalized multibeam subtracted patten after one reduction cycle is displayed in Figure 3a while Figure 3b shows the resulting array pattern where the maximum SLL is reduced to approximately \(-15\) dB. Proceeding with more reduction iterations or cycles leads to more reduced SLL where Figure 3c displays the normalized multibeam cumulative pattern after 16 iterations to reduce the SLL to \(-30\) dB which is shown in Figure 3d, and finally Figure 3e depicts the normalized multibeam pattern required for reducing the SLL to \(-50\)dB as shown in Figure 3f after 68 subtraction cycles. Figure 3: _Cont._