3.3.2 SLL reduction using heuristic approach

In order to reduce the SLL at the center frequency pattern using optimization technique, a cost function is required. A well-defined cost function of any optimization problem is important to obtain satisfactory performance. The cost function measures the distances between the desired and obtained values of the radiation parameters which are to be controlled. During the optimization process, the algorithms compare the obtained values of the radiation parameters with those of their respective desired values. Without considering sideband radiation and FNBW, the cost function to realize the patterns of desired SLLs at f0 is defined as

$$\Psi = \left( \text{SL}L\_d - \text{SL}L\_{\text{max}} \right)^2 \tag{13}$$

beamwidth (FNBW) for different values of SLLmax of the main beam radiation pattern has been noted and is plotted in Figure 6. The maximum two harmonics are normalized with respect to the maximum value of the radiation at f0. For the different values of SLLs, the change in SBLmax at the first and second harmonics is shown Figure 7. Since, the Dolph-Chebyshev (DC) method gives the optimum pattern, i.e., the pattern with minimum FNBW for a specific value of SLL or vice versa. For the DC patterns of different SLLs, the corresponding FNBW, SBL1(max), and SBL2(max) are also given in Table 1. The plot SLL vs. FNBW is shown in Figure 6, and that for SLL vs. SBLmax is shown in Figure 7. Figure 6 depicts that for the DC method, FNBW is linearly increased when |SLLmax| is enhanced, whereas Figure 7 shows that SBL1max initially decreases from 3.35 dB and obtained its minimum value of 13.36 dB at 25 dB SLL pattern. Thereafter, it gradually

GA- and Dolph-Chebyshev-based pattern of SLL of 55.6 dB at f0.

The patterns at f0 by DC The patterns at f0 by GA

15 7.2 3.35 8.30 15.06 8.4 10.01 17.91 20 8.4 7.2 17.83 20.03 9.6 9.95 17.06 25 9.8 13.36 20.25 25.78 10.4 8.51 14.27 30 11.2 12.28 19.31 30.28 12.0 9.98 14.75 35 12.4 12.42 17.45 35.66 17.2 12.19 17.39 40 13.8 12.42 17.45 40.04 19.0 6.058 13.13 45 15.2 12.59 17.40 43.76 20.2 10.12 14.572 50 16.6 12.58 17.40 50.52 22.4 7.301 12.7870 55 18.0 12.55 17.37 55.6 23.6 7.3 12.9

Radiations maximum at the first two sidebands and FNBW for the fundamental patterns of different values

SLLmax (dB)

FNBW (deg)

SBL1(max) (dB)

SBL2(max) (dB)

SBL2(max) dB)

SLLmax (dB)

Table 1.

of SLL.

Figure 5.

11

FNBW (deg)

DOI: http://dx.doi.org/10.5772/intechopen.89479

SBL1(max) (dB)

Pattern Synthesis in Time-Modulated Arrays Using Heuristic Approach

where SLLmax is the actual value of the SLL as obtained during each trial of the optimization process and SLLd is its desired value. Any heuristic search global optimization method can be employed to reduce the SLL of the power pattern at f0. Here, one of the useful stochastic search global optimization methods, namely, genetic algorithm (GA), is used to synthesize the power pattern of different values of SLL of the array under consideration [17].

#### 3.3.3 Results and discussion

It can be seen from Eqs. (5)–(10) that the Fourier coefficients and hence amplitudes of the harmonic signals are decreasing gradually with increasing harmonic order. Thus, the radiation energy at the first few harmonics (called sidebands) is most significant. So, the influence on the maximum radiation at the first two harmonics of TMAA is observed by reducing the SLL of the center frequency pattern. Firstly, the SLL of the power pattern at f0 is reduced by using the Dolph-Chebyshev (DC) method [1]. Then a global optimization method is used to synthesize the same pattern as obtained via DC. In order to observe the effects of reducing SLL on SBL and FNBW, these values are noted for different power patterns. Table 1 shows the simulation results of the maximum sideband level (SBLmax) at the first and second harmonics for the fundamental pattern with different values of maximum SLL (SLLmax) ranging from �15 dB to �55 dB. The radiation pattern at f0 as obtained by GA and DC with SLL of �55 dB is shown in Figure 5. The first null

Pattern Synthesis in Time-Modulated Arrays Using Heuristic Approach DOI: http://dx.doi.org/10.5772/intechopen.89479

#### Table 1.

harmonics of the modulation frequency. When the desired power pattern is synthesized at the center frequency, the sideband power is wasted. In this section, the influences on the first null beamwidth (FNBW) and sideband radiation by reducing SLL of the center frequency pattern are observed. The SLL of the power pattern at f0 is reduced by using the conventional amplitude tapering technique, namely, Dolph-Chebyshev (DC) [1], and a heuristic search global optimization method,

The conventional antenna array synthesis technique such as Dolph-Chebyshev (DC) method [1] can be directly used to realize power pattern of the desired value of SLL at the center frequency. For a 30-element uniformly excited (UE) TMAA, the equivalent excitation coefficient of the DC pattern of desired SLL is made equal to the normalized on-time duration of the array elements. Following the DC method, the power pattern of different values of SLL is obtained at the center

In order to reduce the SLL at the center frequency pattern using optimization technique, a cost function is required. A well-defined cost function of any optimization problem is important to obtain satisfactory performance. The cost function measures the distances between the desired and obtained values of the radiation parameters which are to be controlled. During the optimization process, the algorithms compare the obtained values of the radiation parameters with those of their respective desired values. Without considering sideband radiation and FNBW, the

where SLLmax is the actual value of the SLL as obtained during each trial of the

It can be seen from Eqs. (5)–(10) that the Fourier coefficients and hence amplitudes of the harmonic signals are decreasing gradually with increasing harmonic order. Thus, the radiation energy at the first few harmonics (called sidebands) is most significant. So, the influence on the maximum radiation at the first two harmonics of TMAA is observed by reducing the SLL of the center frequency pattern. Firstly, the SLL of the power pattern at f0 is reduced by using the Dolph-Chebyshev (DC) method [1]. Then a global optimization method is used to synthesize the same pattern as obtained via DC. In order to observe the effects of reducing SLL on SBL and FNBW, these values are noted for different power patterns. Table 1 shows the simulation results of the maximum sideband level (SBLmax) at the first and second harmonics for the fundamental pattern with different values of maximum SLL (SLLmax) ranging from �15 dB to �55 dB. The radiation pattern at f0 as obtained by GA and DC with SLL of �55 dB is shown in Figure 5. The first null

optimization process and SLLd is its desired value. Any heuristic search global optimization method can be employed to reduce the SLL of the power pattern at f0. Here, one of the useful stochastic search global optimization methods, namely, genetic algorithm (GA), is used to synthesize the power pattern of different values

Ψ ¼ ð Þ SLLd � SLLmax 2 (13)

cost function to realize the patterns of desired SLLs at f0 is defined as

namely, genetic algorithm (GA) [16].

Advances in Array Optimization

3.3.2 SLL reduction using heuristic approach

of SLL of the array under consideration [17].

3.3.3 Results and discussion

10

frequency.

3.3.1 SLL reduction using Dolph-Chebyshev technique

Radiations maximum at the first two sidebands and FNBW for the fundamental patterns of different values of SLL.

Figure 5. GA- and Dolph-Chebyshev-based pattern of SLL of 55.6 dB at f0.

beamwidth (FNBW) for different values of SLLmax of the main beam radiation pattern has been noted and is plotted in Figure 6. The maximum two harmonics are normalized with respect to the maximum value of the radiation at f0. For the different values of SLLs, the change in SBLmax at the first and second harmonics is shown Figure 7. Since, the Dolph-Chebyshev (DC) method gives the optimum pattern, i.e., the pattern with minimum FNBW for a specific value of SLL or vice versa. For the DC patterns of different SLLs, the corresponding FNBW, SBL1(max), and SBL2(max) are also given in Table 1. The plot SLL vs. FNBW is shown in Figure 6, and that for SLL vs. SBLmax is shown in Figure 7. Figure 6 depicts that for the DC method, FNBW is linearly increased when |SLLmax| is enhanced, whereas Figure 7 shows that SBL1max initially decreases from 3.35 dB and obtained its minimum value of 13.36 dB at 25 dB SLL pattern. Thereafter, it gradually

Figure 6. FNBW for different values of SLLs of the GA and Dolph-Chebyshev patterns at f0.

shifting; (3) binary optimized time sequence (BOTS); (4) subsectional optimized time steps (SOTS); (5) variable aperture size (VAS) with quantized on-time (VAS-QOT) or quantized aperture size (QAS); and (6) nonuniform period modulation (NPM). From the array factor expression as given in Eq. (6), it can be observed that for TMAA, the array factor at different harmonic can be obtained if the Fourier coefficients of different time-modulated elements are known. Therefore, in the following sections, along with the brief description of different timemodulation approaches, the Fourier coefficients of time switching elements under

This is the first type of time-modulation strategy as reported in [2] where the

In VAS time-modulation scheme, only the switch-"on" time duration is considered for deriving the array factor expression. However, when the RF switches are used to commutate the antenna elements in TMAAs, the radiation patterns at center frequency as well as at different harmonics depend not only on the switch-on time duration but also on the switch-"on" and switch-"off" time instants of the array elements [18, 19]. Thus along with the switch-on time durations as considered in VAS scheme, switch-on and switch-off time instants are also taken as another degree of freedom to control the power pattern in TMAA. For the pulse shifting strategy, periodic switching instants of the pth element over the modulation period

≤ Tm. Thus, two situations may occur. The first case is shown in Figure 8(a) where

p ≤ Tm. Therefore, the switching function as expressed in Eq. (2)

1

1 pand t on p ¼ t 2 p � t 1

p and off-time instant t

2 p

p should be

aperture size of the antenna array is varied with time. The time-modulation

the respective time-modulation scheme are presented.

Switching instants defining pulse shifting strategy under two cases.

Pattern Synthesis in Time-Modulated Arrays Using Heuristic Approach

DOI: http://dx.doi.org/10.5772/intechopen.89479

principle as discussed in Section 2 falls under this category.

are shown in Figure 8. In this case, both on-time instant t

for VAS will be modified and is represented as in Eq. (14):

can be controlled independently such that individually t

4.2 Time modulation through pulse shifting

4.1 Variable aperture size (VAS)

Figure 8.

t 1 p <t 2 p and t 1 p þ t on

13

Figure 7. The plot of SBR1(max) and SBR2(max) for the different values of SLLs of the patterns at f0.

increases and becomes almost steady at 12.5 dB after 30 dB SLL. From Figures 6 and 7, it can be seen that for the GA-based patterns of different SLLs, SBLmax and FNBW vary randomly as in the cost function, only SLL is considered without controlling FNBW and SBL.