# Optimization Design for Sparse Planar Array in Satellite Communications
## 4 Sparse Planar Array Optimization Design Method
### Intelligent Optimization Methods
#### 4.2.3 Local Optimization Methods
By introducing beamforming, a suitable SPA with a lower lobe and grating lobe can be reconstructed during the scanning process based on the JCO method, and a more accurate reconstruction map can be obtained. In addition, the alternating projection method mentioned in [77], the semi-fixed relaxation method mentioned in [78], and the heavy weighted \(l_{1}\) mode minimization method mentioned in [79] can all be used to deal with antenna pattern reconstruction problems. Some methods can also be classified as convex optimizations. In 2004, [PERSON], [PERSON], [PERSON], and [PERSON] proposed compressed sensing (CS) [80; 81], providing new ideas for the design of sparse planar arrays, such as the basis purity (BP) reconstruction algorithm [82], orthogonal matching tracking method (OMP) [83], underdetermined system local method (FOCUSS) algorithm [84], and Bayesian compressed sensing (BCS) reconstruction algorithm [85], which have become popular in recent years. Compressed sensing theory can reconstruct the original signal with a sampling rate far lower than the Nyquist sampling theorem, and its basic idea can be summarized as follows: if the signal is sparse in a transformation domain, a linear measurement matrix can be selected to project the high-dimensional signal in the sparse domain into the low-dimensional space, and a small number of projections on the low-dimensional space can be used to solve an optimization problem. Finally, the original signal is reconstructed with a high probability. The sparse array construction process is shown in Figure 15.  
The construction of sparse area arrays according to compressed sensing theory is mainly divided into three basic steps: sparse representation of signal, linear measurement of signal, and signal reconstruction. Sparse representation of signal: when there is at most one nonzero value in the signal, the signal is said to be sparse; signals can also be considered sparse if they are not sparse themselves but are expressed sparingly in some basis matrix. Linear measurement of the signal: After determining that the original signal is sparse in a transformation domain, a sublinear measurement of the signal is performed to obtain a measured value. Figure 15: A sparse array process constructed according to compressed sensing theory. Signal reconstruction: Signal reconstruction, also known as signal reconstruction or signal recovery, relies on the measurement matrix and low-dimensional measurement vectors to reconstruct the high-dimensional original signal. When the signal satisfies the restricted isometry property (RIP) condition, the signal satisfies the uniqueness of signal reconstruction. According to the principle of compressed sensing, the antenna pattern reconstruction problem of sparse planar arrays can be transformed into a solution:  
\[\underset{w}{\min}\Big{|}\Big{|}w\Big{|}\Big{|}_{0} \tag{16}\] \[s.t.||f-f_{REF}||_{2}\leq\tau\]  
where \(f_{REF}\) represents the target antenna pattern, and \(\tau\) is the upper limit of the reconstruction error set by the user. According to the convex optimization technique, the pattern reconstruction problem established according to the principle of compressed sensing can be transformed into  
\[\underset{w}{\min}\Big{|}\Big{|}w\Big{|}\Big{|}_{1} \tag{17}\] \[s.t.||f-f_{REF}||_{2}\leq\tau\]  
Aiming at the sparse area array synthesis problem, the authors of [86] combined compressed sensing theory to propose an iterative convex optimization scannable sparse array synthesis method, which optimizes the excitation weight and element position, significantly reduces the number of elements, and obtains a low peak sidelobe level. Compressed sensing and low-rank recovery techniques were applied to sparse array pattern synthesis [87], and an improved compressed sensing reconstruction algorithm orthogonal multi-match tracking (OMMP) algorithm was proposed, which has better robustness, faster atomic selection speed, and lower sidelobes while saving 20-30% of the array elements.