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

Table 3. It can be observed that the sum and difference pattern is obtained by time modulating the same number of elements as found in [33]. As compared to [33], SLLmax and SBLmax of the sum pattern are improved by 2.03 and 1.5 dB, respectively. In case of difference pattern, the SBLmax is reduced by 2.37 dB with only 0.37 dB rise in SLL. Also, for both the sum and difference patterns, the amount of sideband power is found to be 3.35% and 4.69% of the total power which are 4.30% and 5.45% in the respective patterns of [33]. The FNBW of ABC optimized sum

Figure 17.

the work of [6, 40], respectively. However, the method as proposed in [40] may be

Example 2: In the second example, the synthesis problem as discussed in [33] is considered. From the list of static and dynamic excitations of one-half of the linear arrays as presented in Table 3, Ref. [33], it was found that out of the five edge elements, only three are time-modulated to synthesize the sum pattern, whereas, for the difference pattern, time modulation is applied only on four center elements. In this work, to synthesize the sum and difference pattern, the proposed method is applied in the following way. For the UE TMLAA, the sum pattern is synthesized by taking the discrete τ<sup>p</sup> values of five edge elements (in one-half of the array) as "χ." In order to compare the ABC optimized results with those of SA, during optimization, the three lower values of τ<sup>p</sup> are rounded off to their nearest quantization levels, whereas the higher two τ<sup>p</sup> values are kept to 1 so that the ABC optimized pattern is obtained by time modulating the same number of (i.e., three) elements as observed in SA. However, to synthesize the difference pattern, perturbation of discrete τ<sup>p</sup> values of four center elements are considered. In Eq. (27), the same values of δhd's as used in Example 1 are set. Figures 16 and 17 show the ABC optimized sum and difference patterns, respectively. For optimizing the sum and difference pattern with NE = 30 and limit = 450, the ABC takes only 23 and 5 iterations, respectively (refer to Figure 18). The corresponding optimum discrete values of τ<sup>p</sup> are shown in

utilized to reduce the waste of power in the form of sideband radiations.

Element numbers 1 &

Advances in Array Optimization

Difference pattern

Table 3.

Figure 16.

26

30

2 & 29

3 & 28

4 & 27

5 & 26

τ<sup>p</sup> Sum pattern 1 1 0.2 0.9 0.1 1 1 1 1 1

Optimum discrete values of τ<sup>p</sup> of ABC optimized sum and difference pattern, as shown in Figures 12 and 13.

ABC optimized sum pattern as obtained by time modulating the same percentage (20%) of elements as in [33].

SLL and SBLmax of the pattern are obtained as 17.87 and 31.44 dB, respectively.

6–11 & 25–20

1 1 1 1 1 1 0.1 0.9 0.3 0.1

12 & 19

13 & 18

14 & 17

15 & 16

ABC optimized difference pattern as obtained by time modulating the same percentage (26.7%) of elements as in [33]. SLL and SBLmax of the pattern are obtained as 16.05 and 31.44 dB, respectively.

Figure 18. Convergence characteristics of ABC for the synthesized sum and difference patterns of Figures 5 and 6.

Figure 19. Sideband levels of the first 30 sidebands for the different patterns in examples 1 and 2.

pattern and difference pattern was found as 6.12 and 4.56°, respectively, which are quite comparable to 5.88 and 4.59° as for the patterns in [33].

Figure 19 shows SBLs of the first 30 sidebands for the synthesized patterns as considered in Example 1 and Example 2. It can be observed that at the higher sidebands also, the SBLs are below SBLmax. Further observation shows that the no radiation is produced at 10th, 20th, and 30th sideband with quantized values of τ<sup>p</sup> as at these harmonics the array factor expression becomes zero for all elements.

Example 3: In this example, it is shown that the same time modulator can also be used to synthesize a flattop pattern. Accordingly, a symmetrical TMLA with element number N = 20 and inter-element spacing d0 = 0.5λ is considered. Here, the objective is to synthesize a flattop pattern in the broadside direction with digitally controlled static excitation amplitudes and phases by using five digital attenuators and phase shifters. A flattop pattern with a beamwidth of 30°, maximum ripple level (Rmax) at the flat region of less than 1 dB, and transition width of 8° is selected as the target pattern. Although such pattern with more stringent design specification is reported in [6], analog attenuators and phase shifters are required. Due to symmetry, the dimension of the parameter vector χ = {Ap, ϕp, τp} becomes 30. During optimization, Ap and ϕp∀p ∈ð Þ 1, : … , N are perturbed within the search range of (0.2–1) and (�180 to +180) with step sizes of 0.5/2<sup>5</sup> and 360/2<sup>5</sup> , respectively. The number of quantization states for τ<sup>p</sup> is selected as 20. In Eq. (27), both δ1d and δ2d are selected as �30 dB, while δ3d is set to 1 dB. Setting FN = 150, the ABC parameters are obtained as in [38]. ABC converges after 2000 iterations, while the weighting factors are selected as W1 = 2; W2 = 1; and W3 = 5. The ABC optimized 3D space pattern at fundamental frequency along with the first 30 sidebands is shown in Figure 20. Table 4 contains the corresponding discrete values of Ap, ϕp, andτp. The flattop pattern in Figure 20 is obtained with SLL, SBLmax, and Rmax of �29.31, �29.9, and 1.22 dB, respectively. The absolute value of Rmax is measured in the region of 75 ≤ θ ≤ 105°. Hence, only 0.22 dB higher values of Rmax are obtained by satisfying other design specification of the pattern. Also, it is observed that no such improvement in the pattern is obtained when the continuous value of τ<sup>p</sup> is used to

synthesize the pattern. However, with Q = 10, almost the same pattern is obtained

Optimum discrete values of Ap, ϕp, and τ<sup>p</sup> for the flattop power pattern of Figure 20.

ABC optimized space pattern at f0 and the first 30 sidebands. At f0, the flattop pattern is obtained with SLL,

Element numbers (p) Normalized on-time, τ<sup>p</sup> Discrete values of excitation

1 & 20 0.65 0.200 33.75 2 & 19 0.95 0.325 22.50 3 & 18 0.95 0.475 0 4 & 17 1 0.525 33.75 5 & 16 1 0.675 67.50 6 & 15 1 0.975 90 7 & 14 1 1 112.50 8 & 13 1 0.800 135 9 & 12 1 0.600 180 10 & 11 1 0.700 146.25

Amplitude, Ap Phase, ϕ<sup>p</sup>

individually controlled switching circuits. Other time-modulation schemes such as BOTS [30] and SOTS [21] need a complex programmable logic device (CPLD) for controlling the "on–off" timing of the connected switches. The continuous values of

In the continuous search space of VAS time-modulation method [2, 3–6], the ontime duration of array elements can be of any value between 0 and Tm. In [2], for each time-modulated elements, the current pulse required with pulse width over

<sup>p</sup> < 0.9Tm) is obtained by using the RF switches with

with Rmax of 1.80 dB.

Table 4.

29

Figure 20.

the range of (0.1Tm < t

on

SBLmax, and Rmax of 29.31, 29.9, and 1.22 dB, respectively.

Pattern Synthesis in Time-Modulated Arrays Using Heuristic Approach

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