# Optimization Design for Sparse Planar Array in Satellite Communications
## 2 Sparse Planar Array Optimization Design Model
### Sparse Area Array Pattern Mathematical Model  
The core of the array antenna is the beamforming technology. The output of each array element is weighted according to specific criteria. This weighting process ensures that the array output is adjusted to align with the desired receiving direction of the array, leading to the concentration of gain in a single direction and the generation of directional beams directed toward the intended user.  
The antenna pattern is a critical indicator for evaluating the array's performance and is generally used to indicate the change in the relative field strength of the radiation field with direction. At the mathematical level, in accordance with the principle of antenna pattern multiplication, it is expressed as the product of the unit factor \(T_{mn}(\theta,\varphi)\) (single element pattern) and the array factor \(AF(\theta,\varphi)\):  
\[F(\theta,\varphi)=T_{mn}(\theta,\varphi)\cdot AF(\theta,\varphi). \tag{5}\]  
Assuming that the single-element pattern is wide enough to form an array with isotropic elements, the single-element pattern \(T_{mn}(\theta,\varphi)=1\) in Equation (5) can be rewritten as  
\[F(\theta,\varphi)=AF(\theta,\varphi). \tag{6}\]  
Assuming that the current distribution of the antenna element can be separated, the planar array with factorable positions is distributed on the _xoy_ plane and contains \(M*N\) elements within the array aperture; its antenna pattern can be represented as follows [11]:  
\[\begin{array}{l}F(\theta,\varphi)=\sum\limits_{m=0}^{M-1}\sum\limits_{n=0}^{ N-1}w_{mn}e^{jk[d_{m}(\sin\theta\cos\varphi-\sin\theta_{0}\cos\varphi_{0})+d_{a}( \sin\theta\sin\varphi-\sin\theta_{0}\sin\varphi_{0})]}\\ =|F_{1}(\theta,\varphi)|\cdot|F_{2}(\theta,\varphi)|\end{array}, \tag{7}\]  
where \(w_{mn}\) represents the complex excitation (including amplitude and phase) of the \((m,n)\) element, \(k=2\pi/\lambda\) is the number of spatial wavelengths, \(\theta\) represents the pitch angle of the observation direction relative to the origin, \(\varphi\) denotes the azimuth angle of the observation direction relative to the origin, \((\theta_{0},\varphi_{0})\) is the maximum value of the main beam of the array, \(d_{m}\) and \(d_{n}\) are the coordinates of the \((m,n)\) element, and \(|F_{1}(\theta,\varphi)|\) and \(|F_{2}(\theta,\varphi)|\) denote the antenna radiation pattern in the x-direction and y-direction, respectively. The above expression is for the general case of planar arrays.  
In order to compare the differences in antenna patterns between a uniform planar array and a sparse planar array, Figure 4a,b show the array layout of a uniform planar array with 169 elements and a sparse planar array with 73 elements, respectively. Figure 5a,b compare the gain patterns of the two arrays on two orthogonal sections (\(\varphi=0^{\circ}\) and \(\varphi=90^{\circ}\)), which are both HFSS full-wave simulation results. It can be seen that, although the two arrays have different numbers of elements, they can still obtain similar antenna pattern shapes and gains. These results are also in line with the theory of array antennas. On the one hand, the similarity in lobe widths between the two arrays can be attributed to their identical aperture sizes. On the other hand, the gain of the distribution of the aperture field is dependent on various factors such as the number of elements, the position of the elements, and the excitation of the elements. Consequently, sparse arrays can achieve the same gain as a full array at the same caliber by optimizing the position and excitation of the elements.