7.2 Implementation of ABC

flip-flop (FFs) outputs to logic level 1, whereas the delayed pulses from the

version at the S and R inputs of i

Advances in Array Optimization

current pulses with quantized values of t

modulate antenna element using the quantized values of t

multiplexers, it is very easy to reconfigure different patterns.

of the algorithm have been briefly discussed in the following section.

7. Artificial bee colony (ABC) algorithm

7.1 Food foraging behavior of real bees

22

corresponding tap outputs of the delay line are applied to reset the flip-flop outputs to logic level 0. To avoid the simultaneous appearance of PG output and its delayed

Tm/Q can be used. The current waveforms of the input pulse applied to the set (S) inputs of the flip-flops and output pulses appeared at the outputs Oq with q = 1, 2, … Q of different flip-flops as shown in Figure 14. Therefore, the QPG, consisting of a pulse generator, simple tapped delay line, and flip-flops, provides the required

on p . One of the most important features in TMAAs is to reconfigure different antenna patterns just by changing the on-time sequence across each element. Such a feature can easily be obtained in the proposed QTM employing PWS. The PWS consists of N number of (Q 1) multiplexers and their outputs that are used to

combination at the select inputs I0, I1, ... IB of the multiplexers, one of the quantized pulses at the output of QPG is selected to time modulate the corresponding antenna element. Thus, just by using the appropriate combination of the select lines of

Karaboga [37] introduced the artificial bee colony (ABC) algorithm to simulate intelligent food foraging behavior of the honeybee swarm. The ABC algorithm shows excellent performance for optimizing multivariable functions as compared to other similar algorithms like genetic algorithm (GA), differential evolution (DE), and particle swarm optimization (PSO). ABC is a robust search and optimization algorithm with relatively fewer control parameters [38]. Although GA is extensively used due to its efficiency to solve the optimization problems with binary/discrete variables, it requires high computational time as well as high memory consumption to store unnecessary binary data during the conversion of a real number to binary and vice versa. The decoding method as applied in ABC algorithm requires one-line MATLAB code which directly quantizes continuous values of the variables by rounding off them. The food foraging behavior of real bees and the implementation

The constituents of the food foraging systems are the unemployed bees (UBs) and the employed bees (EBs) in a beehive and food sources (FSs) in their surroundings. Initially, all the bees are unemployed, and after they find a rich food source, they become employed. UBs are categorized into scout bees (SBs) and onlooker bees (OBs). The food foraging process is initiated when the SBs start to explore the rich food source randomly from any location by moving toward any direction of the search space. When SBs find a rich food source, it becomes an EB and returns to the hive to attract other bees by performing a special dance known as the waggle dance. Depending on the quality of the food source, the EBs recruit some bees to extract nectar from the source. The EBs abandon the current food source when the nectar of the source is finished and becomes scout bees (SBs). However, in the dancing area, OBs examine the quality and quantity of the food sources with the information provided by the EBs, and after examinations EBs select a food source. Thus during the food foraging process, exploration is carried out by SBs, and

th flip-flop, respectively, the pulse width less than

on

<sup>p</sup> . With appropriate bit

In the following steps, the real bee colony behavior into the problem space is implemented:


$$\boldsymbol{\Psi}(\boldsymbol{\chi}) = \sum\_{h=0}^{h=2} \boldsymbol{W}\_h \cdot \mathbf{H}(|\delta\_{hd}| - |\delta\_h|) \cdot \left(\delta\_{hd} - \delta\_h\right)^2 \tag{27}$$

where δ<sup>h</sup> with h = 0, 1, and 2 are the instantaneous values of different parameters of the desired patterns, while δhd is the desired values of the specific parameters. For all examples as considered in Section 8, δ<sup>0</sup> is the maximum SLL (SLLmax) of the pattern at f0 and δ<sup>1</sup> is the value of SBLmax among the first five sidebands. But, for the first two examples, δ<sup>2</sup> represents FNBW, and, for the third case, it is the ripple level of the flattop pattern for which the positions of δhd and δ<sup>h</sup> are interchanged in the Heaviside step function Hð Þ� . "Wh" is the weighting factor for the corresponding terms. The cost function ψ in Eq. (27) depends on "D," the independent parameters of optimization parameter vector χ. A possible set of the parameter values may be considered as a point in the search space of D dimensional coordinate system. In ABC, the cost function ψ of the optimization problem has resembled with the food sources of the bees and each possible point as its location. The solutions of the optimization problem represent locations of the food sources, whereas the corresponding value of cost function ψ due to each point in its solution set is considered as the quality of the food source:


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

$$\mathbf{x}\_{\vec{\text{ij}}} = \boldsymbol{\chi}\_{\vec{\text{ij}}} + \mathfrak{R}\_{\vec{\text{ij}}} \left( \boldsymbol{\chi}\_{\vec{\text{ij}}} - \boldsymbol{\chi}\_{\vec{\text{adj}}} \right) \tag{28}$$

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

Pattern Synthesis in Time-Modulated Arrays Using Heuristic Approach

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

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

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

Element numbers (p) τ<sup>p</sup> 1 1 2 0.30 3 0.10 4–22 1 23 0.90 24 0.90 25 0.10 26 0.10 27 0.10 28 0.90 29 0.10

Figure 15.

Table 2.

25

where j∈ {1, 2, … , D} and z∈{1, 2, … , FN} are randomly selected column and 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 is replaced by its predetermined minimum value (χ j min) and for the upper limit by its maximum value (χ j max). If, for a new solution, the value of ψ is less than the corresponding old solution, the old is replaced by the new one.

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

$$\xi\_i = 0.9 \left( \frac{\mu\_i}{\mu\_{\text{max}}} \right) + 0.1 \tag{29}$$

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 (OBs).

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 ignored, and the parameter, q j i , for the new solution is provided randomly through the whole search space by Eq. (30):

$$q\_i^j = \chi\_{\text{min}}^j + rand(\mathbf{0}, \mathbf{1}) \left( \chi\_{\text{max}}^j - \chi\_{\text{min}}^j \right) \tag{30}$$

