3.2 The results

for mechanical movement of the element positions. Thus, these methods of fully

Generally, the optimization parameters of the mechanically nonuniformly spaced arrays can be chosen by either in terms of inter-element spacing between successive elements or in terms of absolute positions of the elements from the center of the array. These two structures are illustrated in Figure 16. Choosing the second structure in the optimization process may cause the element positions to overlap. The overlapping between any two or more elements may help to remove (or turn it off) some redundant elements for the thinning arrays. Thus, the second

The far-field radiation pattern of an array consisting of N isotropic mechanically movable elements that are arranged in nonuniform locations xn according to the

an cosð Þ kxnu (8)

AF uiupperbound&uilowerbound 

 

(9)

N=2

n¼1

where N is assumed an even number, and an is the electronic weighting of the array elements which is chosen to be constant or uniform in this method. In order to introduce the required nulls and at the same time reducing the sidelobe level in the

> þ ∑ I i¼1

where λ=Nd≤u ≤1, and i ¼ 1, 2, …I. Note that λ=Nd represents the angular location of the first null in the array pattern and I represents the total number of the steered nulls. To control the width of the produced nulls, some constraints on the upper and lower bounds are imposed in (9). Note that the first term in (9) corresponds to the peak sidelobe level and the second term corresponds to the required

Array structures in terms of Inter-element spacing and absolute locations from the array center [26].

To overcome these problems and make them more amendable in practice, some researchers, for example, see [22–25], have found that the required nulls can be introduced by controlling the positions of only selected elements rather than con-

nonuniform spaced arrays were not widely used in practice.

trolling the positions of all elements.

Array Pattern Optimization

3.1 Fully nonuniform spaced array

structure is considered in the present work.

second structure (see Figure 16), can be written by [26]

array pattern of (8), the following cost function is used [26]

CostFunction <sup>¼</sup> <sup>10</sup> log<sup>10</sup> max AF u j j ð Þ <sup>2</sup>

nulls with pre-specified width.

Figure 16.

24

AF uð Þ¼ 2 ∑

Each wide null at the desired direction is generated by forming two adjacent nulls with a small spacing equal to u = 0.02 around the interfering directions. The effectiveness of the simplified null steering array [26] compared to the fully nonuniform spaced array has been illustrated by the design of 30 elements linear array with the main beam directed toward the broadside. The smoothing, elite sample selection and population parameters of the optimization algorithm are chosen to be 0.7, 0.1, and 100, respectively [26].

In the first example, the fully nonuniform spaced array where all of its elements are made movable is considered. It is assumed that the width of the required nulls in the optimized array are from 0.42 to 0.44 and from 0.61 to 0.63 in u-space, while the depth of these two nulls is chosen to be �40 dB. Also, it is assumed that the electronic amplitude and phase excitations for all elements in the considered array are uniform, i.e., an ¼ 1. Note, to maintain the overall array length unchanged, the first and the last array elements' locations are fixed. Moreover, in all cases, the cost function represented in (9) is chosen such that it minimizes the output power at the intended null direction(s), i.e., it contains only the second term while the first term which is responsible for sidelobe reduction is omitted. Figure 17 shows the radiation pattern of the fully nonuniform spaced array. For comparison, the radiation pattern of the fully uniform spaced array is also shown in Figure 17(right). From this figure, it can be seen that the capability of the fully nonuniform spaced array for accomplishing the required nulls is more than satisfactory. This is mainly due to the availability of many degrees of freedom. On the other hand, the sidelobe structure has generally increased by few dBs. This is mainly due to the considered cost function as mentioned earlier. The optimized location of all elements with respect to that of the uniformly spaced array is shown in Figure 17(left). Note that, as

Figure 17. Results of fully uniform and nonuniform spaced arrays for N = 30 elements, and two wide nulls.
