# Optimization Design for Sparse Planar Array in Satellite Communications
## 4 Sparse Planar Array Optimization Design Method
### Sparse Area Array Optimization Based on Analysis Methods  
The primary way to solve the problem of sparse array optimization design in the early stage is to use analysis methods, which are mainly used for array synthesis under known array layout conditions; most of these methods are aimed at uniform arrays. For example, the Dolph-Chebyshev synthesis method first proposed in 1946 obtained an equal secondary lobe level. It achieved the narrowest main lobe for a uniform array [18], and the Taylor synthesis method obtained a lower sidelobe level by changing the zero position of the sidelobe. The pattern decreased with the secondary lobe interval [19]. Both of these methods are optimized for single targeting. Bayliss synthesis is a two-parameter differential beam synthesis method that minimizes peak sidelobe levels and maximizes angular accuracy [20]. The Elliott synthesis method is based on the Taylor and Bayliss distribution to achieve different sidelobe structures on the left and right [21]. The above analytic methods only optimize the peak sidelobe level and main lobe width, but do not optimize the position of the array elements. The almost difference set (ADS) synthesis method was proposed in [22; 23], which can determine the position of the array elements, predict the upper bound of the sidelobes a priori, and reduce the peak sidelobe level. The above sparse planar array optimization method based on the analysis method has simple ideas, but strong pertinence and a small scope of application. The Woodward-Lawson method is aimed at the beamforming problem and is suitable for the antenna pattern synthesis problem of arbitrary array beamforming [24], which can achieve a good fit of the expected pattern, but the peak sidelobe level cannot be controlled in the non-fitting area. The analysis methods based on statistics mainly include the dynamic programming method (DP) [25], density cone method (DT) [26], and fractional order Legendre transform method (FLT) [27]. Statistical-based optimization methods are feasible when dealing with the design of small-scale sparse arrays; however, as the scale of the array increases, the processing level of these algorithms decreases significantly. In order to improve the calculation speed, some scholars proposed a matrix decomposition-based method, which can solve the problem of sparse area array pattern reconstruction and beamforming; the most commonly used are the matrix pencil method (MPM) [28; 29] and the augmented matrix beam (MEMP) method [30], which can minimize the number of array elements and do not need iteration, thus exhibiting low computational complexity. The authors of [31] proposed a 0-1 integer programming method (ILP), which can optimize the element excitation, reduce the peak sidelobe level, and improve the computational efficiency without iteration. The representative analysis methods for large-scale sparse area matrix synthesis problems are based on Fourier transform methods, such as fast Fourier transform technology [32] and iterative Fourier transform (IFT) [33]. The IFT method is an analysis method that has been used more frequently in sparse area array optimization designs in recent years. IFT was proposed by Dutch scholar [PERSON] and applied to large sparse planar matrix synthesis [34]. IFT requires a known sparsity rate and uses the Fourier variation relationship between the excitation and array functions to iterate the excitation coefficient. Its advantage is that the expected peak sidelobe level can be obtained, and the excitation can be optimized. It has a fast convergence speed in processing large planar arrays, but the disadvantage of this method is that it is easy to fall into the local optimum. Assume \(M\times N\) array elements are arranged in a square grid. Let \(u=sin\theta\cos\varphi\), \(v=\sin\theta\sin\varphi\), and \(\Delta d_{x}\) and \(\Delta d_{y}\) represent the element space on the \(x\)- and \(y\)-axes.