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

χ = {τp} is taken as the optimization parameter vector. The control parameters of ABC such as EN = 30, limit = 900 (limit = EN\*D), and MNC = 700 are selected as per the guidelines given in [38]. W1, W2, and W3 are selected as 2, 1, and 1, respectively. In Eq. (27), δ1d, δ2d, and δ3d are set as 20, 30, and 7 dB, respectively. The ABC optimized far-field power pattern with side lobe level (SLL) of 20.15 dB, FNBW of 6.86°, and sideband levels (SBLs) at the first two sidebands as SBL1 = 30.78 dB and SBL2 = 31.63 dB, respectively, is shown in Figure 15. Table 2 contains the ABC optimized values of τ<sup>p</sup> of the elements used to obtain Figure 15. As compared to [6], SLLmax and SBLmax are improved by a factor of 0.1 and 0.7 dB, respectively, in the proposed work. The total sideband power is calculated by using either of the expressions derived in [14] or [39] and found to be 4.83% of the total power which is quite higher than 3.89% and 3.57% as reported in

Figure 15. ABC optimized power pattern obtained by using the discrete value of τ<sup>p</sup> of Table 2.


Table 2. Optimum discrete values of τ<sup>p</sup> for the power pattern of Figure 15.

f. Employed bees' stage: The greedy nature of the employed bees (EBs) is incorporated, and the new sources (si) surrounding its neighborhood are

sij ¼ χij þ ℜij χij � χzj

row indexes of the position matrix and ij is any randomly generated number through [�1, 1]. When any parameter of the new solution crosses its lower limit, it

<sup>ξ</sup><sup>i</sup> <sup>¼</sup> <sup>0</sup>:<sup>9</sup> <sup>μ</sup><sup>i</sup>

where j∈ {1, 2, … , D} and z∈{1, 2, … , FN} are randomly selected column and

g. Onlooker bees' stage: The quality of the food source is represented by the fitness value, μi, of the cost function, and onlooker bees select the new source by means of the probability, ξi, in terms of the fitness value, determined by

> μmax

where μmax is the maximum fitness value among the current possible solutions. Like employed bees (EBs), the greedy selection is also applicable to onlooker bees

h. Scout bees' stage: In this stage, the abandonment of a food source by the employed bees is simulated. If the fitness value of the cost function is not improved during a specified number of steps called "limit = FN\*D" [25], it is

min <sup>þ</sup> randð Þ 0, 1 <sup>χ</sup><sup>j</sup>

i. Remembering the best solution: The overall new best solution as mentioned in the steps "e–h" replaces the previous best, and the value is then stored.

j. Stopping criterion: Steps "(e)" to "(i)" are repeated until the cost function converges to the desired value or a predetermined value of maximizing the

The VAS-based synthesis problems that have been reported in [6, 33] are considered at first, and the QAS-based time-modulation approach is applied to realize the patterns. Here, the modulation period Tm is quantized in 10 equal discrete levels, i.e., Q = 10. Hence, the discrete search space for the optimization problem

Example 1: A 30-element UE TMLAA is placed along the x-axis with one element at the origin, and a uniform inter-element spacing of 0.7λ is considered. It is desirable in practice for such an array to feed with {Ap} = 1 and {ϕp} = 0. Here,

j i j

max). If, for a new solution, the value of ψ is less than the

(28)

(30)

min) and for the upper limit by

þ 0:1 (29)

, for the new solution is provided randomly

max � χ j min 

generated as follows:

Advances in Array Optimization

its maximum value (χ

(OBs).

24

is replaced by its predetermined minimum value (χ

corresponding old solution, the old is replaced by the new one.

j

ignored, and the parameter, q

number of cycles (MNC).

8. Design examples and discussions

(τp) becomes {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1}.

through the whole search space by Eq. (30):

q j <sup>i</sup> ¼ χ j the work of [6, 40], respectively. However, the method as proposed in [40] may be utilized to reduce the waste of power in the form of sideband radiations.

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

Pattern Synthesis in Time-Modulated Arrays Using Heuristic Approach

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

ABC optimized difference pattern as obtained by time modulating the same percentage (26.7%) of elements as

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

in [33]. SLL and SBLmax of the pattern are obtained as 16.05 and 31.44 dB, respectively.

Figure 17.

Figure 18.

27

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


Table 3.

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

#### Figure 16.

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.
