2.2.2 The results

In this subsection, a uniform planar array with N = 6 and M = 6 isotropic elements spaced by half a wavelength is considered. The required half power beamwidth (HPBW) of the proposed planar array pattern is chosen to be 17o, i.e., <sup>Ω</sup>BW <sup>¼</sup> <sup>8</sup>:5o. Note that, for a uniformly excited planar array with size 6�6 elements the HPBW is also 17o. This means that the HPBW of the optimized array is constrained to be the same as that of the uniformly excited planar array.

Assume the direction of the desired signal is known, which is set to be 90o. The weights of the perimeter elements in the proposed planar array are optimized such that the corresponding radiation pattern has equal sidelobe level at �20 dB and two nulls at <sup>ϕ</sup> ¼ �30o, <sup>θ</sup> <sup>¼</sup> <sup>80</sup><sup>o</sup> (i.e., ux <sup>¼</sup> <sup>0</sup>:852, uy ¼ �0:492<sup>Þ</sup> and <sup>ϕ</sup> <sup>¼</sup> <sup>45</sup><sup>o</sup>, <sup>θ</sup> ¼ �70<sup>o</sup> (i.e., ux ¼ �0:664, uy ¼ �0:664Þ. Figure 12 shows the radiation patterns of the original uniform planar array and the proposed array. It can be seen that the required sidelobe level and the desired null are efficiently accomplished by optimizing only the perimeter elements.

The corresponding complex weights (i.e., the magnitudes and the phases) of all elements in the proposed planar array are shown in Figure 13. It can be seen that only the magnitudes and the phases of the 20 perimeter elements are adjusted, whereas the 16 interior elements remain holding their uniform excitations.

Figure 12. Optimized radiation pattern of the proposed planar array (right) and the uniformly excited planar array (left).

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

Figure 13. Amplitude and phase excitations of the proposed planar array pattern that is shown in Figure 12.

## 2.3 The sensitivity analysis

AFð Þ¼ θ; ϕ ∑

Array Pattern Optimization

M-1 [28].

Figure 12.

(left).

20

2.2.2 The results

N�1 n¼2 ∑ M�1 m¼2 e j 2πd

mizing only the perimeter elements.

þ ∑ M m¼1

<sup>λ</sup> <sup>½</sup> <sup>n</sup>�Nþ<sup>1</sup> ð Þ <sup>2</sup> sin ð Þ<sup>θ</sup> cosð Þ� <sup>ϕ</sup> <sup>β</sup><sup>x</sup> <sup>ð</sup> Þþ <sup>m</sup>�Mþ<sup>1</sup> ð Þ <sup>2</sup> ð Þ sin ð Þ<sup>θ</sup> sin ð Þ� <sup>ϕ</sup> <sup>β</sup><sup>y</sup>

N�1 n¼2


ð Þ wn1f g: þ wnMf g:

(4)


ðw1mf g: þ wNmf g: Þ þ ∑

where f g: represents the exponential term as expressed in the first term of Eq. (3). The perimeter elements in the lower term of (4) are expressed as the sum of 2 rows and 2 columns. In the two rows, the value of n is set to n ¼ 1 and n ¼ N, while the value of m is allowed to change from 1 to M. In the two columns, the value of m is set to m ¼ 1 and m ¼ M, while the value of n is allowed to change from 2 to

In this subsection, a uniform planar array with N = 6 and M = 6 isotropic elements spaced by half a wavelength is considered. The required half power beamwidth (HPBW) of the proposed planar array pattern is chosen to be 17o, i.e., <sup>Ω</sup>BW <sup>¼</sup> <sup>8</sup>:5o. Note that, for a uniformly excited planar array with size 6�6 elements

Assume the direction of the desired signal is known, which is set to be 90o. The weights of the perimeter elements in the proposed planar array are optimized such that the corresponding radiation pattern has equal sidelobe level at �20 dB and two nulls at <sup>ϕ</sup> ¼ �30<sup>o</sup>, <sup>θ</sup> <sup>¼</sup> <sup>80</sup><sup>o</sup> (i.e., ux <sup>¼</sup> <sup>0</sup>:852, uy ¼ �0:492<sup>Þ</sup> and <sup>ϕ</sup> <sup>¼</sup> <sup>45</sup><sup>o</sup>, <sup>θ</sup> ¼ �70<sup>o</sup> (i.e., ux ¼ �0:664, uy ¼ �0:664Þ. Figure 12 shows the radiation patterns of the original uniform planar array and the proposed array. It can be seen that the required sidelobe level and the desired null are efficiently accomplished by opti-

The corresponding complex weights (i.e., the magnitudes and the phases) of all elements in the proposed planar array are shown in Figure 13. It can be seen that only the magnitudes and the phases of the 20 perimeter elements are adjusted, whereas the 16 interior elements remain holding their uniform excitations.

Optimized radiation pattern of the proposed planar array (right) and the uniformly excited planar array

the HPBW is also 17o. This means that the HPBW of the optimized array is constrained to be the same as that of the uniformly excited planar array.

In this section, a realistic situation is investigated where the required element excitations in amplitude and phase cannot be realized exactly in practice, or when there is some fluctuation in the frequency of operation. The performance of the proposed optimization techniques and the null positions are investigated when there are some errors in the excitation of the array elements.
