2.1.4.2 The results

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

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

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

Figure 8.

Array Pattern Optimization

16

#### Figure 9.

Configuration of the multiple null steering method [21].

number of adjustable elements, where the searching spaces are restricted to include only the elements on the array's perimeter [28]. The optimization was performed under some constraints to obtain the desired radiation characteristics such as narrower beamwidth, asymmetric low sidelobes, and controlled nulls in some prespecified directions. Unlike the existing methods, in which all the array elements are changeable, the proposed planar array enjoys a faster convergence of the optimizer as its interior elements are fixed and the whole array keeps maintaining a

Amplitude and phase excitations of the proposed array and fully phase-only array.

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

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

Consider a rectangular planar array composed of N rows and M columns of isotropic elements with mechanically fixed inter-element spacing d ¼ λ=2 in both x and y directions. The radiation pattern of such rectangular planar array can be

where θ is the elevation angle, ϕ is the azimuth angle, and β<sup>x</sup> ¼ sin ð Þ θ<sup>0</sup> cos ϕ<sup>0</sup> ð Þ,

In this work, the weights of the interior elements of the array (i.e., the central

wnm ¼ 1, while the elements on the perimeter are only considered to be adjustable (or controllable) subject to some required constraints. Thus, the array factor of (3)

part having dimensions ð Þ N � 2 X Mð Þ � 2 ) are assumed to be constant, i.e.,

β<sup>y</sup> ¼ sin ð Þ θ<sup>0</sup> sin ϕ<sup>0</sup> ð Þ are progressive phase shifts in x and y directions that are necessary to direct the mainbeam to the angle(θ0, ϕ0), and wnm is the complex weight of the (n,m)th element. Clearly, the array factor in (3) represents a fully controllable planar array in which all of its elements are adjustable and the resulting feeding network is a relatively complex system. Furthermore, to meet the required radiation characteristics, it is necessary to impose some constraints on the array weights which lead to an added complexity to the adaptive system. Thus, the necessity of controlling a small number of array elements arises especially with the practical implementation of large planar arrays or when faster adaptation is

<sup>λ</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> ½ � ð Þ sin ð Þ<sup>θ</sup> sin ð Þ� <sup>ϕ</sup> <sup>β</sup><sup>y</sup> (3)

good performance [28].

written as [28]

Figure 11.

desirable.

19

can be rewritten as [28]

AFð Þ¼ θ; ϕ ∑

N n¼1 ∑ M m¼1

2.2.1 The electronic planar null steering method

wnme j 2πd

Figure 10. Results of multiple wide null steering method.

seen that the proposed array with only 10 adjustable outer elements is capable of generating the required multi nulls at pre-specified depths and widths. The same performance is obtained with the fully phase-only optimized array but with an extra requirement of modifying the phases of 100 elements (i.e., more cost and more optimization parameters). The half power beam width of the fully uniform and the proposed arrays are 1:0085 and 1:0772o, respectively. Moreover, the proposed method converges much faster than the method of fully phased-only optimized array. The amplitude and phase excitations of the proposed and the fully phased-only optimized arrays are shown in Figure 11. From this figure, it can be seen that the required percentage of the perturbed element excitation represents only 10% of those needed for the fully phased-only optimized array. This drastically reduces the RF components of the feeding network and consequently the cost while maintaining the same performance of interference suppression.

#### 2.2 Planar array

In this section, the selection process of the controllable elements is extended to the large planar arrays to reach the desired radiation pattern with a minimum

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

Figure 11. Amplitude and phase excitations of the proposed array and fully phase-only array.

number of adjustable elements, where the searching spaces are restricted to include only the elements on the array's perimeter [28]. The optimization was performed under some constraints to obtain the desired radiation characteristics such as narrower beamwidth, asymmetric low sidelobes, and controlled nulls in some prespecified directions. Unlike the existing methods, in which all the array elements are changeable, the proposed planar array enjoys a faster convergence of the optimizer as its interior elements are fixed and the whole array keeps maintaining a good performance [28].

## 2.2.1 The electronic planar null steering method

Consider a rectangular planar array composed of N rows and M columns of isotropic elements with mechanically fixed inter-element spacing d ¼ λ=2 in both x and y directions. The radiation pattern of such rectangular planar array can be written as [28]

$$AF(\theta,\phi) = \sum\_{n=1}^{N} \sum\_{m=1}^{M} w\_{nm} e^{j\frac{2\pi l}{\lambda} \left[ \left( n - \frac{M+1}{2} \right) (\sin \left( \theta \right) \cos \left( \phi \right) - \beta\_x \right] + \left( m - \frac{M+1}{2} \right) \left( \sin \left( \theta \right) \sin \left( \phi \right) - \beta\_y \right)} \tag{3}$$

where θ is the elevation angle, ϕ is the azimuth angle, and β<sup>x</sup> ¼ sin ð Þ θ<sup>0</sup> cos ϕ<sup>0</sup> ð Þ, β<sup>y</sup> ¼ sin ð Þ θ<sup>0</sup> sin ϕ<sup>0</sup> ð Þ are progressive phase shifts in x and y directions that are necessary to direct the mainbeam to the angle(θ0, ϕ0), and wnm is the complex weight of the (n,m)th element. Clearly, the array factor in (3) represents a fully controllable planar array in which all of its elements are adjustable and the resulting feeding network is a relatively complex system. Furthermore, to meet the required radiation characteristics, it is necessary to impose some constraints on the array weights which lead to an added complexity to the adaptive system. Thus, the necessity of controlling a small number of array elements arises especially with the practical implementation of large planar arrays or when faster adaptation is desirable.

In this work, the weights of the interior elements of the array (i.e., the central part having dimensions ð Þ N � 2 X Mð Þ � 2 ) are assumed to be constant, i.e., wnm ¼ 1, while the elements on the perimeter are only considered to be adjustable (or controllable) subject to some required constraints. Thus, the array factor of (3) can be rewritten as [28]

seen that the proposed array with only 10 adjustable outer elements is capable of generating the required multi nulls at pre-specified depths and widths. The same performance is obtained with the fully phase-only optimized array but with an extra requirement of modifying the phases of 100 elements (i.e., more cost and more optimization parameters). The half power beam width of the fully uniform and the proposed arrays are 1:0085 and 1:0772o, respectively. Moreover, the proposed method converges much faster than the method of fully phased-only optimized array. The amplitude and phase excitations of the proposed and the fully phased-only optimized arrays are shown in Figure 11. From this figure, it can be seen that the required percentage of the perturbed element excitation represents only 10% of those needed for the fully phased-only optimized array. This drastically reduces the RF components of the feeding network and consequently the cost while

In this section, the selection process of the controllable elements is extended to

the large planar arrays to reach the desired radiation pattern with a minimum

maintaining the same performance of interference suppression.

2.2 Planar array

18

Figure 9.

Array Pattern Optimization

Figure 10.

Configuration of the multiple null steering method [21].

Results of multiple wide null steering method.

$$AF(\theta,\phi) = \underbrace{\sum\_{n=2}^{N-1} \sum\_{m=2}^{M-1} e^{j\frac{2\pi\theta}{\lambda} \left(n - \frac{N+1}{2}\right) \left(\sin\left(\theta\right)\cos\left(\phi\right) - \beta\_x\right) + \left(m - \frac{M+1}{2}\right) \left(\sin\left(\theta\right)\sin\left(\phi\right) - \beta\_y\right)}\_{\text{interior elements}}}\_{\text{interior elements}}$$

$$+ \underbrace{\sum\_{m=1}^{M} \left(w\_{1m}\{.\cdot\} + w\_{Nm}\{.\cdot\}\right) + \sum\_{n=2}^{N-1} \left(w\_{n1}\{.\cdot\} + w\_{nM}\{.\cdot\}\right)}\_{\text{Perimeter elements}}\tag{4}$$

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 M-1 [28].

2.3 The sensitivity analysis

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

Figure 13.

null depths.

21

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

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

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

In this section, the sensitivity analysis is concerned with how much the generated nulls are robust with respect to unavoidable variations of the reconfigured amplitude and phase weights due to the quantization errors that are associated with the used digital attenuators and/or digital phase shifters. The electronic null steering methods including the above-mentioned methods require phase shifters and attenuators that are digitally controlled. With such digital components, it is well-known that only a finite number of quantized values are available. For example, a one-bit digital phase shifter produces only two phase values of 0 and π, while a two-bit digital shifter can realize four phases of 0, π=2, π, and 3π=2. Accordingly, with the use of discrete phase shifters and/or discrete attenuators, precise control over both amplitudes and phases of the adjustable elements is not possible. Therefore, unavoidable quantization error in the phase and/or amplitude excitations causes some modifications of the radiation pattern from the desired one. However, such performance degradations may be lesser in the proposed approaches than that in the fully optimized array elements where the number of the adjustable elements is small. Figure 14 shows the sensitivity of the proposed multiple null steering method to various phase quantization levels. The effect is obvious on the sidelobe level and

The degradation in the optimized array pattern due to random errors in the phase and amplitude of the element excitations was investigated in [29]. Such errors can cause an elevation in the sidelobe level and changing the angular locations of the desired nulls. The simulation results showed that the nulls and the sidelobe level in the adaptive arrays are more sensitive to random errors in the element phase

In this subsection, we assume that there is a fluctuation in the frequency of operation, or the system works on a certain band of frequencies around the center

there are some errors in the excitation of the array elements.

2.3.1 The effect of quantization levels on the generated nulls

excitations as compared to amplitude excitations.

2.3.2 The effect of frequency fluctuation on the generated nulls
