2.1.4 Multiple nulls

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

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 excita-

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 conver-

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

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

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

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

The results for fully electronic null steering method with amplitude and phase excitations for N = 30,

, respectively; whereas, these value were

All of the above results show clearly that the required shape of the array pattern

tions of the full elements are optimized are shown in Figure 4.

gence rate of the optimizer under these two different algorithms.

optimized values of A1, AN, P1, and PN using GA were found to be

effectiveness of the proposed single null steering method.

<sup>0</sup>:8181, <sup>0</sup>:7313, <sup>52</sup>:2824 and <sup>47</sup>:6185o

original and optimized arrays.

Array Pattern Optimization

in the previous example.

SLL = 40 dB, and a single wide null.

Figure 4.

12

applications with multi interference environment, it is desirable to generate multiple wide nulls in the radiation pattern; thus, a set of element excitations have to be modified. In this subsection, a subset of a small number of adjustable elements on

Results for the proposed single null steering method for N = 30 and Dolph excitation.

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

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

Figure 7.

15

Figure 6. The results for the proposed single null steering method for N = 30.

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

Figure 7. Results for the proposed single null steering method for N = 30 and Dolph excitation.

applications with multi interference environment, it is desirable to generate multiple wide nulls in the radiation pattern; thus, a set of element excitations have to be modified. In this subsection, a subset of a small number of adjustable elements on

Figure 6.

Array Pattern Optimization

14

The results for the proposed single null steering method for N = 30.

array elements; whereas, the second array subset contains only a small number of the array elements that will be adjusted adaptively. The GA is used to optimize the

Here in this subsection, the solution proposed in Section 2.1.2 is further extended

amplitude and phase excitations of the second array subset elements [21].

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

to include which elements are to be optimized and to what extent. The small number of the selected elements should have an impact on the controllable nulls. Therefore, we examine three different selection strategies. The first strategy selects the most effective elements that are located on the extremes of the array; while in the second and third strategies, the selections consider the elements that are located at the center of the array or randomly chosen from the whole array elements [21]. The idea of the second strategy was employed in a sidelobe adaptive canceller system [27]. By adapting few elements at the center of the array, the system is capable to produce multiple nulls toward a number of interfering signals [27]. Experience with these three selection strategies showed that the first strategy provides best performance for interference suppression [21]. To apply the first strategy, first, consider an array of an even number of elements 2N, with uniform amplitude excitations and mechanically fixed locations with uniform inter-element spacing d, symmetrically positioned about the origin (i.e., N elements are placed on each side of the origin). Assuming a subset of only 2M elements (out of the 2Nelement array) is optimized to generate the required nulls at unwanted directions (i.e., M outer elements on each end of the array). The remaining 2N-2M elements are kept unchanged, i.e., having uniform amplitude and equal-phase excitations. The overall far-field pattern due to the 2N-2M uniformly excited array elements

2.1.4.1 The electronic multiple null steering method

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

and the 2M adaptive array elements can be written as [21]:

þ ∑ N m¼N�Mþ1

of the outer M elements on each side of the array is shown in Figure 9 [21].

where k ¼ 2π=λ. From (2), it can be noted that the amplitudes (Amr and Aml for right and left subset elements) and phases (Pmr and Pml for right and left subset elements) of 2M adjustable elements array can be considered either symmetric or nonsymmetric (i.e., Aml 6¼ Amr and Pml ¼6 Pmr) like the earlier method. Also note that the 2M adjustable elements are selected from the extremes of the array and they play an important role in generating the required nulls. The structure of the interior 2N-2M uniformly excited array elements with adjustable amplitude and phase excitations

An original uniform linear array with 2N = 100 elements located at fixed positions and having element separation equals to half the wavelength is considered. Figure 10 shows the results obtained from the original uniform array and the optimized array patterns with four required wide nulls each of width u = 0.05 and depth = �60 dB. Five elements at each side of the linear array are used here as the elements to be controlled. To show the effectiveness of the proposed array with respect to the fully optimized array, the radiation pattern of the fully phase-only optimized array and its convergence speed are also included in Figure 10. It can be

<sup>A</sup>mrej ð Þ 2m�<sup>1</sup>

<sup>2</sup> ð Þ kduþPmr <sup>þ</sup> <sup>A</sup>ml


<sup>2</sup> kduþPml ð Þ n o

e�<sup>j</sup> ð Þ 2m�<sup>1</sup>

(2)

cos ð Þ 2n � <sup>1</sup> <sup>2</sup> kdu � �


AF uð Þ¼ 2 ∑

2.1.4.2 The results

17

ð Þ N�M n¼1

Figure 8. Results for the proposed single null steering method for N = 30 and Taylor excitation.

both sides of the array is considered. Therefore, the far-field equation of the overall array is formulated as the summation of two independent array subsets. The first array subset is referred to as a uniform array, which contains the majority of the

array elements; whereas, the second array subset contains only a small number of the array elements that will be adjusted adaptively. The GA is used to optimize the amplitude and phase excitations of the second array subset elements [21].
