2.3.1 The effect of quantization levels on the generated nulls

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 null depths.

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 excitations as compared to amplitude excitations.

#### 2.3.2 The effect of frequency fluctuation on the generated nulls

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

Figure 14. Sensitivity of the proposed multiple null steering method to various phase quantization levels.

frequency. It is assumed that the element positions can be accurately fixed at the design frequency f <sup>o</sup>, and element separation is fixed at d ¼ λo=2, where λ<sup>o</sup> is the free-space wavelength at the frequency fo. In this case, the array factor of the uniformly excited equally spaced linear array can be found from [30] as:

$$AF(f, \theta) = \frac{\sin\left[\frac{N}{2}\frac{f}{f\_s}\pi\sin(\theta)\right]}{\sin\left[\frac{1}{2f\_s}\pi\sin(\theta)\right]}\tag{5}$$

nulls. Generally, it is noticed that the nulls positions are sensitive to frequency changes. The sensitivity of the null angle θ<sup>n</sup> to frequency can be found from Eq. (6)

The effect of the frequency changes on the null positions for N = 10 elements, and design frequency fo ¼ 3GHz

df <sup>¼</sup> <sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>2</sup>nf <sup>o</sup>

The above relation shows that the sensitivity is a nonlinear function of the frequency deviation. This nonlinearity can be obviously noticed by comparing Figure 15(left) and (right), where a 0.5 GHz change in frequency produces different shifts in the null positions depending if the change is positive or negative. It has been found that a �16.7% changes in the frequency result in a shift of 2.26 and 1.67°, respectively, for the first null position. In this example, a relatively large

As we have shown in the previous section, the practical implementation of the feeding network in the electronically null steering methods is a real challenging issue, especially when dealing with large arrays. To solve this problem, many researchers, for example, see [22–26], proposed to mechanically control the spacing between the array elements instead of electronically controlling the amplitude and/ or phase excitations to achieve the required null steering. However, in practice, these fully nonuniform spaced arrays have also some disadvantages and difficulties to build. These difficulties may especially arise when dealing with movable or unknown interfering directions where in such a case it is required to continuously readjust the element positions to achieve the desired null steering. This means that

Nf � �<sup>2</sup> <sup>r</sup> (7)

dθ<sup>n</sup>

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

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

frequency span of 1 GHz has produced null movement of only 3.89°.

the mechanical position of all elements in an array needs repeatedly to be

recalculated and accordingly the whole array elements need to be removed for each specific interfering direction. In such methods of mechanically nonuniform spaced arrays, the simplest way to change the position of the array elements is to use a set of servo-motors connected to each element. For large arrays with fully nonuniform spaced elements, i.e., large number of the movable elements, the computational time (i.e., the number of iterations that are required by the optimization algorithm to converge) becomes a real challenging issue. In addition, an extra time is needed

3. Mechanical null steering methods

as [30]:

23

Figure 15.

[30].

where f is instantaneous frequency, and f/fo is the fluctuation or deviation ratio. From (5), the angle of the nth null θ<sup>n</sup> as a function of the frequency can be written as [30]:

$$\theta\_n(f) = \sin^{-1}\left[\frac{f\_o}{f}\left(\pm\frac{2n}{N}\right)\right] \tag{6}$$

A sample array of N = 10 elements working at an instantaneous frequency f and design frequency of f <sup>o</sup> ¼ 3 GHz is investigated here. Figure 15(left) shows the radiation patterns of the uniform array, plotted for frequencies higher than the design value f <sup>o</sup> ¼ 3 GHz. It can be seen that the angular location of the first null, in the uniform pattern is 11.54°, whereas this null is shifted to 9.871° when f is changed from 3 to 3:5 GHz. The figure also shows that, as the frequency departs from the design value f <sup>o</sup> ¼ 3 GHz, the nulls move toward main beam resulting in anincreased magnitude at the original directions of the nulls. Figure 15(right) shows the radiation patterns of the same array plotted for frequencies lower than the design value fo. It can be seen that the angular location of the first null is shifted from 11.54 to 13.8° when f is changed from 3 to 2:5 GHz; whereas, the forth null is shifted from 53.33 to 74.0° when f is changed from 3 to 2:5 GHz. The figure shows that, for frequencies lower than the design value f <sup>o</sup> ¼ 3 GHz, the nulls move far from the main beam resulting in an increased magnitude at the original directions of the

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

Figure 15.

frequency. It is assumed that the element positions can be accurately fixed at the design frequency f <sup>o</sup>, and element separation is fixed at d ¼ λo=2, where λ<sup>o</sup> is the free-space wavelength at the frequency fo. In this case, the array factor of the uniformly excited equally spaced linear array can be found from [30] as:

> sin <sup>N</sup> 2 f f o

sin <sup>1</sup> 2 f f o

where f is instantaneous frequency, and f/fo is the fluctuation or deviation ratio. From (5), the angle of the nth null θ<sup>n</sup> as a function of the frequency can be written as

A sample array of N = 10 elements working at an instantaneous frequency f and

f �

<sup>π</sup>sinð Þ<sup>θ</sup> h i

h i (5)

(6)

<sup>π</sup>sinð Þ<sup>θ</sup>

2n N � � � �

AF f ð Þ¼ ; θ

Sensitivity of the proposed multiple null steering method to various phase quantization levels.

<sup>θ</sup>nð Þ¼ <sup>f</sup> sin �<sup>1</sup> <sup>f</sup> <sup>o</sup>

design frequency of f <sup>o</sup> ¼ 3 GHz is investigated here. Figure 15(left) shows the radiation patterns of the uniform array, plotted for frequencies higher than the design value f <sup>o</sup> ¼ 3 GHz. It can be seen that the angular location of the first null, in the uniform pattern is 11.54°, whereas this null is shifted to 9.871° when f is changed from 3 to 3:5 GHz. The figure also shows that, as the frequency departs from the design value f <sup>o</sup> ¼ 3 GHz, the nulls move toward main beam resulting in anincreased magnitude at the original directions of the nulls. Figure 15(right) shows the radiation patterns of the same array plotted for frequencies lower than the design value fo. It can be seen that the angular location of the first null is shifted from 11.54 to 13.8° when f is changed from 3 to 2:5 GHz; whereas, the forth null is shifted from 53.33 to 74.0° when f is changed from 3 to 2:5 GHz. The figure shows that, for frequencies lower than the design value f <sup>o</sup> ¼ 3 GHz, the nulls move far from the main beam resulting in an increased magnitude at the original directions of the

[30]:

22

Figure 14.

Array Pattern Optimization

The effect of the frequency changes on the null positions for N = 10 elements, and design frequency fo ¼ 3GHz [30].

nulls. Generally, it is noticed that the nulls positions are sensitive to frequency changes. The sensitivity of the null angle θ<sup>n</sup> to frequency can be found from Eq. (6) as [30]:

$$\frac{d\theta\_n}{df} = \frac{1}{\sqrt{1 - \left(\frac{2uf\_s}{Nf}\right)^2}}\tag{7}$$

The above relation shows that the sensitivity is a nonlinear function of the frequency deviation. This nonlinearity can be obviously noticed by comparing Figure 15(left) and (right), where a 0.5 GHz change in frequency produces different shifts in the null positions depending if the change is positive or negative. It has been found that a �16.7% changes in the frequency result in a shift of 2.26 and 1.67°, respectively, for the first null position. In this example, a relatively large frequency span of 1 GHz has produced null movement of only 3.89°.
