# Optimization Design for Sparse Planar Array in Satellite Communications
## 4 Sparse Planar Array Optimization Design Method
### Sparse Area Array Optimization Based on Analysis Methods
Under the conditions of an ideal omnidirectional element, the antenna pattern expression for a planar array can be expressed as follows [35]:  
\[F(u,v)=\sum_{m=0}^{M-1}\sum_{n=0}^{N-1}w_{mn}\mathrm{e}^{jk(n\Delta d_{x}u+m \Delta d_{y}v)}. \tag{11}\]  
Figure 11: Classification of sparse planar array optimization design methods. Applying the inverse Fourier transform on Equation (11) yields  
\[f(x,y)=\frac{\sum\limits_{m=0}^{M-1}\sum\limits_{n=0}^{N-1}F(m,n)\mathrm{e}^{j2 \pi(mx/M+my/N)}}{MN}. \tag{12}\]  
It is not difficult to find that there is a two-dimensional Fourier transform relationship between the array element excitation \(w_{mn}\) and the antenna pattern, which can be understood as the antenna pattern representing the time-domain signal and the array element excitation denoting the spectrum. The flowchart of the iterative Fourier algorithm is shown in Figure 12.  
Following the proposal of the IFT algorithm, some scholars also proposed many improved algorithms. For example, for the large sparse planar array, the IFT method easily falls into the local optimum. The authors of [36] proposed an improved iterative Fourier transform (MIFT), which introduces a variable filling factor to obtain a lower peak sidelobe level than the IFT algorithm. In [36], circular arrays with diameters ranging from 25\(\lambda\) to 100\(\lambda\) were thinned. Results indicated that, when the array element number was 828, and the fill factor was 42.95%, an MSLL value 1.05 dB lower than that obtained using the IFT method and a directivity 0.12 dB higher than that obtained using the IFT method were achieved. Similarly, when the array element number was 3127 and the fill factor was 40.15%, an MSLL value 1.60 dB lower than that obtained using the IFT method and a directivity of 0.13 dB higher than that obtained using the IFT method were observed. Lastly, for an array element number of 12,514 and a fill factor of 40.00%, the MSLL value was 6.18 dB lower than that obtained using the IFT method, while the directivity was 0.06 dB higher than that obtained using the IFT method. Figure 12: Flowchart of IFT-based sparse array optimization design. The authors of [37] proposed a sparse area matrix synthesis algorithm based on adaptive probability model optimization by constructing a probability estimation model to characterize the distribution of sparse arrays and combined it with the iterative Fourier method; the probability model was used to determine the position of the smallest sidelobe level element in the sparse area matrix, highlighting the global search ability and robustness. In the experimental results presented in [37], a circular array aperture with a diameter of 50\(\lambda\) was utilized with 3116 turned ON elements and a filling factor of 40%. Results indicated that the PLIFT method yielded a _PSLL_ value that was 1.5 dB lower than that obtained using the IFT method. Aiming at the problem of beamforming, the authors of [38] proposes a fast pencil beamforming method for large-scale 4D thin cloth arrays by combining the least squares interpolation method with iterative fast IFT, which greatly improves the calculation speed. In the proposed method for nonuniform array synthesis, the array is first divided into \"conventional arrays\" based on excitation. Subsequently, a uniform virtual array structure is obtained from interpolation of the nonuniform array structure via the least squares interpolation method. Lastly, beam synthesis is accomplished using the IFT method. Aiming at the sparse area array synthesis problem, the authors of [39] used amplitude monotonicity conditions to propose an IFT method with high gain and smaller amplitude dynamics. This study presented nine experimental simulation results, covering array element numbers ranging from nine to 9999. The directivity of the proposed method was shown to improve by 0.42-0.51 dB, while the beam width was reduced by 0.1\({}^{\circ}\)-3.2\({}^{\circ}\), and the amplitude was dynamically decreased by 2.56-7.72, when compared to the IFT method.