Sidelobe Nulling by Optimizing Selected Elements in the Linear and Planar Arrays DOI: http://dx.doi.org/10.5772/intechopen.84507

elements are controlled including the two edge elements) is considered. In this example, the total number of array elements are chosen to be N = 30 elements, the amplitude excitation of the original array elements is chosen to be uniform and the phase excitation is set to zero. In this scenario, the genetic algorithm is used to optimize the amplitude excitations of all array elements while the phase excitations are left unchanged. The required sidelobe level was set at 40 dB. Figure 2 shows the optimized array pattern along with the original uniform array pattern. It can be seen that the required sidelobe level is accurately achieved while the HPBW and FNBW have increased. The amplitude excitations of all array elements are greatly changed except a small number of the central elements. Moreover, the optimizer needs at least 250 iterations to converge.

For a fully controlled array, the number of degrees of freedom is quite enough to reduce the sidelobe level and at the same time to place the desired nulls, as shown in Figure 3. Here, as in the previous case, the required sidelobe level is chosen to be

Figure 2. Results for fully electronic null steering method for N = 30 and SLL = 40 dB.

Figure 3.

Results for fully electronic null steering method with amplitude only control for N = 30, SLL = 40 dB, and a single wide null.

where ψ ¼ ð Þ 2πd=λ u þ β. Here, u ¼ sin ð Þθ , and θ is the observation angle from the array normal, d is the element spacing which is selected to be fixed at d ¼ λ=2 for all array elements, and β is the phase shift required to steer the angle of the main beam. Knowing the direction of the interfering signals, ui, i ¼ 1, 2, …I (where I is the total number of interfering signals), and substituting for AF uð Þ¼<sup>i</sup> 0 according to the interference suppression condition, the nulls directions ui can be computed from (1). For asymmetric array, note that the above equation cannot be solved analytically using the method introduced in [17] since it is a function of four unknown parameters, i.e., A1, AN, P1, and PN. On the other hand, the optimal values of these four parameters, subject to some constraints, can be easily found using any global optimization algorithm such as genetic algorithm or particle swarm optimization (PSO), as can be seen in the following subsection [20]. As mentioned earlier, the main constraints that are applied during the optimization process are the depth of the generated nulls and the main beam shape preservation. Moreover, some constraints on the optimization parameters are also considered, where the minimum and maximum values of the optimized amplitudes A1 and AN are set to 0 and 1, respectively, and for optimized phases P1 and PN are set to �π=2andπ=2, respectively [20]. To show the effectiveness of the proposed method, it is applied to uniformly excited linear arrays as well as some nonuniformly excited linear arrays such as Dolph and Tayler arrays as can be seen in the following subsection [20].

In order to show the advantages of the proposed array with controlled two edge elements, first the fully controlled array (i.e., the amplitude excitation of all array

2.1.3 The results

Array Pattern Optimization

Figure 1.

10

Block diagram of the single wide null method [20].

40 dB and a single wide null centered at u = 0.75 (ranged from u = 0.73 to 0.77) is considered. Similar results are obtained when the amplitude and the phase excitations of the full elements are optimized are shown in Figure 4.

All of the above results show clearly that the required shape of the array pattern can be obtained only when precisely choosing the values of the attenuators. In practice, the attenuators are digital and they have a limited number of quantized levels. Thus, these required shapes are far or even impossible to get. Therefore, the arrays that are composed of a few controllable elements are very desirable in practice. In the next example, we consider an array of N = 100 elements and only the edge element are optimized by either the GA or PSO algorithm. Here, only a single wide null is required to be placed from u = 0.4 to 0.5 with depth 60 dB. Moreover, the original excitations of all array elements are assumed to be uniform. Figure 5 shows the radiation patterns of the optimized arrays using GA and PSO along with the original uniform array pattern. This figure also shows the convergence rate of the optimizer under these two different algorithms.

It can be seen that the optimized arrays by GA and PSO are equally capable of achieving the required wide sidelobe nulling. The HPBW of the original uniform array, and the optimized array are 1:0084 and 1:0314o, respectively. For this case, the optimized values of A1, AN, P1, and PN using GA were found to be <sup>0</sup>:8181, <sup>0</sup>:7313, <sup>52</sup>:2824 and <sup>47</sup>:6185o , respectively; whereas, these value were found to be 0:7322, <sup>0</sup>:8179, <sup>47</sup>:7388 and <sup>52</sup>:4027<sup>o</sup> for PSO design. The computational times for GA and PSO were found to be 0.15429 and 0.12667 min, respectively.

In Figure 6, the results of the proposed single null steering method is examined for N = 30 elements and the desired null is from u = 0.7 to 0.75 with a depth equal to 60 dB. This figure also shows the required amplitudes and phases of both the original and optimized arrays.

By comparing the results of Figure 6 with those of Figure 3 or Figure 4, it can be clearly seen that the proposed single null steering method requires only one attenuator and two phase shifters to realize the modified element excitations. However, the fully controlled array requires at least 30 attenuators and 30 phase shifters to realize the reconfigured amplitude and phase weights. This fully confirms the effectiveness of the proposed single null steering method.

Moreover, to show the generality of the proposed method, we extend it to the nonuniformly excited arrays such as Dolph and Taylor arrays. Figures 7 and 8 show the radiation patterns of the Dolph and Taylor arrays (N = 30 elements, and SLL = 40 dB), where the excitations of the edge elements are optimized using GA for the purpose of generating sector sidelobe nulling with same width and depth as in the previous example.

2.1.4 Multiple nulls

The results for the proposed single null steering method for N = 100.

Figure 5.

13

As we have shown in the previous subsection that a single wide null requires at least controlling the excitations of the two end elements in a linear array. In many

Sidelobe Nulling by Optimizing Selected Elements in the Linear and Planar Arrays

DOI: http://dx.doi.org/10.5772/intechopen.84507

#### Figure 4.

The results for fully electronic null steering method with amplitude and phase excitations for N = 30, SLL = 40 dB, and a single wide null.

Sidelobe Nulling by Optimizing Selected Elements in the Linear and Planar Arrays DOI: http://dx.doi.org/10.5772/intechopen.84507

Figure 5. The results for the proposed single null steering method for N = 100.
