5.4 Cost function with multiple objectives

Most of the TMAA synthesis problems are solved by applying single-objective optimization method where all the objectives are added with different weighting factors to form a single cost function and the cost function is minimized by employing heuristic evolutionary algorithms. The different stochastic optimization techniques are used with the objective to synthesize desired patterns at the operating frequency by reducing SLL and SBL. One of the commonly used techniques to define the cost function of such conflicting multi-objective TMAA synthesis problem is as expressed in Eq. (26):

$$\Psi(\boldsymbol{\chi}) = \sum\_{h=0}^{h=V} \mathcal{W}\_h \mathbf{H}(\delta\_{hd} - \delta\_h) . (\delta\_{hd} - \delta\_h)^{\rho} \tag{26}$$

where χis the set of unknown parameters, termed as optimization parameter vector which is to be determined by the used evolutionary algorithm; δ<sup>h</sup> with h = 0, 1, 2, … .V are the different parameters of the desired patterns; and δhd are the desired values of the specific parameters. For example, δ<sup>0</sup> is the maximum SLL (SLLmax) of the pattern at f0, δ<sup>1</sup> is the maximum of sideband radiations (SBRmax) among the first five sidebands, and δ<sup>2</sup> represents FNBW. "Wh" is the weighting factors for the corresponding terms. Hð Þ: is the Heaviside step function. "ρ" is any natural number. It can be seen from Eqs. (13) and (26) that when the obtained values of δ<sup>h</sup> are close to their desired values, the cost function value is moving toward zero. Thus, reaching zero value of the cost function confirms that the synthesized pattern satisfies the requirements in terms of the desired values of the intended synthesizing parameters. To illustrate the effectiveness of the cost

function as defined in Eq. (26), three multi-objective TMAA synthesis problems have been solved in Section 8. It is to be noted that, for different switching techniques, there is a trade-off between sideband level (SBL) and radiated total sideband power (SRp). Reducing the sideband level usually does not guarantee the power reduction. Hence, in that case a power term should be added into cost function.

6.1 Quantized time modulator (QTM)

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

Pattern Synthesis in Time-Modulated Arrays Using Heuristic Approach

width of t

Figure 13.

Figure 14.

21

Oq∀q∈ð Þ 1, … , Q in Figure 13.

The proposed quantized time modulator (QTM).

on

For appropriate switching operation at pth element, a current pulse with a pulse

antenna array elements is shown in Figure 13. The QTM has two parts, namely, quantized pulse generator (QPG) and pulse width selector (PWS). In QPG, the consecutive tap delay output line, TAP<sup>i</sup> with i = 1, 2, ..Q, introduces an equal delay of "Tm/Q." The pulse output from the pulse generator (PG) is used to set the

The wave form of the input and output pulses of different pulse widths that can be obtained at the outputs

<sup>p</sup> is required [2]. The proposed scheme for the periodical switching of the
