# Shape and Weighting Optimization of a Subarray for an mm-Wave Phased Array Antenna
## 2 Weighting Optimization for Individual Subarrays by Quadratic Programming  
### Array Factor Definition of the Subarray  
Figure 1 shows the structure of a phased array antenna with an attenuator and a phase shifter positioned at the subarray stage. Accordingly, the weighting of the radiating elements belonging to the same subarray is identical, and the phases are also in phase with each other. In order to implement the phased array antenna system applied to the subarray as shown in Figure 1, the excitation amplitude and phase of the radiating elements belonging to the same subarray must be equal. The amplitude and phase of the radiating elements in the subarray can be made the same by using feeding coaxial lines with an  
identical electrical length, power divider and various calibration methods for phased array antennas [22]. Figure 1 is expressed as Equations (1)-(3) [1,23]. \[AF(\theta,\phi)\approx\left(\sum_{k=1}^{K}f_{k}(\theta,\phi)\right)f_{super}( \theta,\phi) \tag{1}\]  
\[f_{k}(\theta,\phi)=\sum_{n=1}^{N_{k}}e^{j\frac{2\pi}{\lambda}(x_{k,n}\mu+y_{k }x^{n})} \tag{2}\]  
\[f_{super}(\theta,\phi)=\sum_{k=1}^{K}w_{k}e^{j\frac{2\pi}{\lambda}(\rho_{k,x}( u-u_{0})+\rho_{k,y}(v-v_{0}))} \tag{3a}\]  
\[u=sin(\theta)cos(\phi)\quad(-90^{\circ}\leq\theta\leq 90^{\circ},0^{\circ} \leq\phi\leq 360^{\circ}) \tag{3b}\]  
\[v=sin(\theta)sin(\phi)\quad(-90^{\circ}\leq\theta\leq 90^{\circ},0^{\circ} \leq\phi\leq 360^{\circ}) \tag{3c}\]  
\[u_{0}=sin(\theta_{0})cos(\phi_{0})\quad(-90^{\circ}\leq\theta_{0}\leq 90^{ \circ},0^{\circ}\leq\phi_{0}\leq 360^{\circ}) \tag{3d}\]  
\[v_{0}=sin(\theta_{0})sin(\phi_{0})\quad(-90^{\circ}\leq\theta_{0}\leq 90^{ \circ},0^{\circ}\leq\phi_{0}\leq 360^{\circ}) \tag{3e}\]  
Here, \(AF(\phi,\theta)\) is the average subarray pattern and determined by \(f_{k}(\theta,\phi)\) and \(f_{super}(\theta,\phi)\)[1]. \(f_{k}(\theta,\phi)\) is the array factor of the \(k\)th subarray and is defined by the location at which the radiating elements are positioned. In addition, the weighting and phase value of the radiating element in \(k\)th subarray are identical because each attenuator and phase shifter is located in each subarray stage, as shown in Figure 1. \(f_{super}(\theta,\phi)\) is the array factor when each subarray is assumed to be one radiator, and this is defined as a superarray. The superarray is determined by the position of the phase center \((\rho_{k,x},\rho_{k,y})\) of the subarray, \(u_{0},v_{0}\) applied from the phase shifter located at the subarray and the weighting \((w_{k})\) applied from the attenuator, which control the amplitude of the feed signal, located at the subarray. The phase center of the subarray is determined by the position of the radiating element in the subarray and the applied weighting through Equation (4) [23]. If the same weighting is applied to the radiating elements in the subarray as in the proposed structure, the phase center of the subarray is determined by the location of the radiating elements and the number of radiating elements in the subarray. \[\rho_{k,p}=\frac{\sum_{n=1}^{N_{k}}w_{k,n}p_{k,n}}{\sum_{n=1}^{N_{k}}w_{k,n}} (p=x\,or\,y)=\frac{\sum_{n=1}^{N_{k}}p_{k,n}}{N_{k}}(w_{k}=1) \tag{4}\]  
In this study, before weighting optimization, the subarray shape was optimized into 96 subarrays by GSO of a rectangular lattice array (24 \(\times\) 24) antenna initially with 576 radiating elements. Using Equation (1) as a cost function of GSO, the subarray shape was optimized to avoid grating lobes when scanning on the condition of uniform weighting  
Figure 1: Phased array antenna with a subarray. (\(w_{k}=1\)). Equation (5) shows that the grating lobe (\(\theta_{GL}\)) is not a function of weighting, which is the amplitude of the feed signal [24]. It is a function of the distance (\(d\)) between radiating elements, the number of radiating elements (\(n\)) and beam-scanning angle (\(\theta_{0}\)).