# Shape and Weighting Optimization of a Subarray for an mm-Wave Phased Array Antenna
## 2 Weighting Optimization for Individual Subarrays by Quadratic Programming
### Weighting Optimized by Quadratic Programming  
QP is a mathematical program that determines the global optimal value given by a mathematical model defined as a quadratic function composed by a real number. The standard objective function of QP is defined as shown in Equations (6) and (7).  
\[minimize\ \ \ \frac{1}{2}x^{T}Qx+c^{T}x \tag{6}\]  
Figure 2: Comparison of the subarray shape: (**a**) first quadrant array structure before optimizing the subarray shape (regular shape), and (**b**) first quadrant array structure after optimizing the subarray shape (aperiodic shape).  
\[\text{{subject}}\text{ {to}}\begin{cases}Ax\leq b\\ Cx=f\end{cases} \tag{7}\]  
Here, \(T\) is the vector transpose operator, with lowercase letters used to express vectors and capital letters used to express matrices. QP is an optimization method that finds the minimum or maximum value of a quadratic objective function when linear equality and inequality limits are given. Therefore, the weighting of the feed signal that minimizes the SLL can be obtained by QP [19]. In the proposed phased array antenna, the weighting of the radiating elements in each subarray should be identical; thus, the objective function (\(Obj_{\textit{frac}}\)) is defined by Equation (8).  
\[Obj_{\textit{frac}}\)) can be immediately utilized as the objective function for QP because it has both real and imaginary parts according to [PERSON]'s formula and does not have a quadratic form. Therefore, in order to formulate the \(Obj_{\textit{frac}}\) quadratic function, it was divided into real and imaginary parts. The real part and the imaginary part were substituted with \(s\) and \(t\), as shown in Equations (9) and (10).  
\[Real(Obj_{\textit{frac}})=\sum_{k=1}^{K}w_{k}\sum_{n=1}^{N_{k}}cos(\frac{2\pi }{\lambda}(x_{k,n}u+y_{k,n}v))=s \tag{9}\]  
\[Imag(Obj_{\textit{frac}})=\sum_{k=1}^{K}w_{k}\sum_{n=1}^{N_{k}}sin(\frac{2\pi }{\lambda}(x_{k,n}u+y_{k,n}v))=t \tag{10}\]  
As the objective function should be a quadratic function composed of real numbers for optimization by QP, it was expressed as the sum of the square of the maximum value \(s\) of the real part of \(Obj_{\textit{frac}}\) and the square of the maximum value \(t\) of the imaginary part of \(Obj_{\textit{frac}}\) as in Equation (11). In addition, when the beam is in the boresight direction, \(Obj_{\textit{frac}}\) should reach its maximum value. Accordingly, the optimization condition was set as shown in Equation (12) such that the SLL in a direction other than the front is the minimum (where \(u_{SLL}\) and \(v_{SLL}\) are sample regions of the SLL) [25].  
\[minimize\quad s^{2}+t^{2} \tag{11}\]  
\[subject\text{ {to}}\begin{cases}Obj_{\textit{frac}}(u=0,v=0)=1\\ Real(Obj_{\textit{frac}}(u=u_{SLL},v=v_{SLL}))\Big{|}\leq s\\ Imag(Obj_{\textit{frac}}(u=u_{SLL},v=v_{SLL}))\Big{|}\leq t\end{cases} \tag{12}\]  
In order to substitute Equations (11) and (12) into the standard objective functions (6) and (7), QP parameters were derived, as shown in Equations (14)-(21). In these equations, the QP parameters make the objective function composed of the quadratic function and the real number identical to the standard objective function of QP.  
\[s^{2}+t^{2}=\frac{1}{2}x^{T}Qx+c^{T}x \tag{13}\]  
\[x=[w_{1},w_{2},w_{3},\cdot\cdot\cdot,w_{K},s,t]^{T} \tag{14}\]