# An ADS-Based Sparse Optimization Method for Sonar Imaging Sensor Arrays
## 4 Analysis of ADS Thinned Array
### Azimuth Angle Performance  
In order to evaluate the beam performance of the obtained array in different directions, the 3D power patterns were simulated in different directions. In this paper, the beam power diagrams are given in Figure 7 when \(\theta\) and \(\varphi\) are \(0^{\circ}\), \(30^{\circ}\), \(45^{\circ}\), \(60^{\circ}\), and \(75^{\circ}\), respectively. The main lobe width is about \(2.6^{\circ}\) in the case of \(-3\) dB, and increases slightly as the azimuth angle increases. The slight change of the main lobe width is probably a result of the size of the array aperture being small when compared to the detection range, only about 30 cm under a carrier frequency of 100 KHz. In addition, the amplitude of the main lobe remains consistent when the azimuth angle is less than \(45^{\circ}\) and drops significantly when the azimuth angle exceeds \(45^{\circ}\). Nevertheless, the _PSLL_ also remains relatively steady from 11.83 to 11.85 when the azimuth angle changes. It can also be seen from these simulation results that the width and amplitude of the main lobe and the _PSLL_ change slightly within the scope of azimuth angular variation.  
Figure 6: Power pattern when \(K=20\), \(\phi=0\), and \(\varphi=0\); \(\sigma=0\), \(15\), and \(28\), respectively.  
Figure 7: _Cont._  
In order to further identify the performance of the array thinned by ADS, we compare the beam pattern of the target array with those of the original Mills Cross array with \(40\times 40\) elements and the sparse array by the CDS method. When using the CDS method, the parameters (\(N\), \(K\), \(\Lambda\)) are set as (40, 21, 1) to make the element count of the thinned array approximately the same as that after ADS. The _PSLL_ values of the beam patterns at various steering angles are given in Table 2. Meanwhile, the beam pattern graphs of the Mills Cross array and the thinned array by CDS are also presented here. The beam patterns are given in the case of \(\theta\) and \(\varphi\) at 45\({}^{\circ}\) and are shown in Figures 8 and 9.  
These results indicate that the original Mills array has the best beam performance compared to the other thinned arrays with roughly equal beam width. The major difference is the _PSLL_ values of the beam patterns. This difference is due to a larger thinned ratio in CDS and ADS sequence results in the spatial undersampling of the array. Comparison of the two arrays thinned by CDS and ADS demonstrates this point. In our investigation, the thinned ratios of the CDS and ADS methods are 21 and 20, respectively, and the element counts of the two virtual arrays are 441 and 400, respectively. Therefore, the _PSLL_ values of the beam patterns following CDS method are slightly lower than those using ADS. In addition, ADS is able to generate more sequence sets and arrays compared to CDS, and thus it is easier to obtain an estimated array. Unfortunately, we cannot deduce other sets due to our weaker mathematical ability, but we will contact [PERSON]'s research group and invite them to guide us in creating a new set. At that time, we will provide the further investigation results. Here, the results are just to demonstrate that the ADS method can be applied to sparse the sensor array.