## 2 Multibeam Pattern Subtraction (MPS) Beamformer Modelling  
In this section, the MPS beamformer is modelled and investigated. Assuming a rectangular array formed by \(M\times N\) elements in the XY-plane as shown in Figure 1 where the antenna elements are interseparated by a half-wavelength distance (\(\lambda/2\)) as in most array designs. Figure 1: Planar two-dimensional array formed by \(M\times N\) isotropic elements in the XY-plane. A popular planar array design has rounded corners which can be simply inserted in the array model by including an array masking matrix \(\mathbf{\mu}\). This structure gains the benefits of low sidelobe levels as in the concentric circular arrays but with suitable element distribution for more easier feeding of the antenna elements. The array center is located by a distance \(\delta_{m,n}\) (normalized to \(\lambda/2\)) from \(m\),\(n\)-th element which can be written as:  
\[\delta_{m,n}=\sqrt{\left(m-\left(M-1\right)/2\right)^{2}+\left(n-\left(N-1 \right)/2\right)^{2}} \tag{1}\]  
where \(m=0\), \(1\), \(2\),..., \(M-1\), \(n=0\), \(1\), \(2\),..., \(N-1\) are the element location indices on the array plane. The maximum normalized distance from the array center is given by:  
\[\delta_{MAX}=\sqrt{\left(\left(M-1\right)/2\right)^{2}+\left(\left(N-1\right) /2\right)^{2}} \tag{2}\]  
The array center has been chosen as a reference point to form the rounded-corners array structure by using the array masking matrix, \(\mathbf{\mu}\), whose elements are given by:  
\[\mathbf{\mu}_{m,n}=\left\{ \begin{array}{ll}1,&\delta_{m,n}\leq\delta_{T}\\ 0,&otherwise\end{array} \right. \tag{3}\]  
where \(\delta_{T}\) is a normalized threshold distance that defines the array corners trim profile where \(\delta_{T}<\delta_{MAX}\) for circularly trimmed corners and \(\delta_{T}=\delta_{MAX}\) for the full rectangular array structure (no trimmed elements at the four array corners). The array steering matrix which defines the relative phase shifts between elements is defined as follows:  
\[S(\theta,\phi)=\mathbf{\mu}\circ\mathbf{A}(\theta,\phi) \tag{4}\]  
where \(\circ\) is the Hadamard product (element-wise product) and \(\mathbf{A}(\theta,\phi)\) is the rectangular array steering matrix whose \(m\),\(n\)-th element is given by:  
\[a_{m,n}(\theta,\phi)=e^{j\pi\sqrt{m^{2}+n^{2}}\cos\left(\phi-\phi_{m,n}\right) \sin\left(\theta\right)} \tag{5}\]  
where \(\phi_{m,n}=\tan^{-1}\left(\frac{n}{m}\right)\). The array factor can be written as:  
\[AF(\theta,\phi)=SUM\Big{(}\mathbf{W}^{H}\mathbf{S}(\theta,\phi)\Big{)} \tag{6}\]  
where \(SUM()\) is an operator in MATLAB that finds the sum of all elements in a matrix, \(H\) is the Hermitian operator, and \(\mathbf{W}\) is the array weighting matrix. For uniform feeding case, \(\mathbf{W}\) is given by:  
\[\mathbf{W}=\mathbf{S}(\theta_{o},\phi_{o}) \tag{7}\]  
which results in a mainlobe directed towards \((\theta_{o},\phi_{o})\). For full rectangular arrays (all elements in \(\mathbf{\mu}\) are equal to 1), the sidelobes have identified peaks and the two major sets of them are located on two perpendicular planes centered around the mainlobe as shown in Figure 2 for uniformly feeding of \(16\times 16\) antenna array. We consider only the front side array pattern which is in the range \(-90^{\circ}\leq\theta\leq 90^{\circ}\) where one mainlobe is generated while the back image pattern is always attenuated or damped by the practical antenna structures. The number of sidelobes in these two major sidelobe planes in this visible angular range is given by:  
\[n_{M,N}=M+N-4 \tag{8}\]  
These major sidelobes are lined in the two perpendicular planes at \(\phi=0^{\circ}\) and \(\phi=90^{\circ}\) with highest SLL of approximately \(-13.26\) dB. On the other hand, there are also many minor sidelobes located between these two major planes with highest SLL of approximately \(-26.43\) dB.