5.3 The need of evolutionary algorithm

The first summation indicates that the signals radiated at the center frequency ω<sup>0</sup> are accumulated in the space, whereas the second summation is due to the signals radiated at different harmonics. Now, if the modulation frequencies of the antenna elements are selected in such a way that f1 = f<sup>2</sup> = … = fN = fm, then the scenario becomes UTMAA, and the term kfp in the second summation becomes kfm that means that the kth-order harmonics of all the elements appeared at the same frequency. The scenario is the same for all other order of harmonics. As a result, radiated signals at the same frequency are accumulated in space, which in turn

But in the case of NTMAA, the modulation frequencies are selected in such a way that f1 6¼ f<sup>2</sup> 6¼ … 6¼ fN. So, due to different modulation frequencies of different antenna elements, the signals radiated from different harmonics appeared at different frequencies, and the term kfp in the second summation of [25] becomes different for different elements. That means the kth-order harmonics of different elements appear at different frequencies and the scenario is the same for all the other order harmonics. So, unlike UTMAA, the harmonic signals appeared at different frequencies and are distributed in space, which in turn decreases the resultant SBL [23]. Recently, some research works have reported the calculation of the sideband power of NTMAA [24–25], and also the reduction of the sideband power

In Section 3.3, it is observed that, though the conventional amplitude tapering methods such as Dolph-Chebyshev and Taylor series can be used to obtain the power pattern of the desired SLL with minimum beamwidth at the operating frequency of time-modulated antenna arrays, these methods are not useful to control the undesired power radiated at different sidebands. Similarly, it is also

observed that application of the stochastic computational technique, such as GA, for suppressing side lobe level of the center frequency pattern without taking into account the sideband radiation, cannot reduce sideband signal power. Also, the beamwidth of such patterns is unpredictable. The power pattern with low SLL and

Therefore, the parameters to be considered to synthesize pencil beam pattern in TMAAs as shown in Figure 5 are SLL, FNBW, and SBL. However for the shaped beam pattern such as flattop and cosec squared, in addition to these three parameters, ripple level in the desired shaped region is another parameter to be taken into account. Further, it can be observed that while SLL is reduced, FNBW is increased and SBL is significantly large. In this regard, SLL, SBL, and FNBW for pencil beam pattern and SLL, SBL, FNBW, and ripple level for synthesizing shaped beam

In Eq. (13), the cost function is defined to synthesize the power pattern with a single objective that is to achieve the desired value of SLL in the synthesized power pattern. Conversely, the synthesized pencil beam patterns at the operating frequency should have reduced SLL along with sufficiently suppressed SBL and narrow beamwidth. Thus, TMAA synthesis problems are multi-objective optimization problems where the multiple objectives are low SLL and narrow beamwidth (BW)

suppressed sideband is preferred for the different communication systems.

increases the resultant SBL.

Advances in Array Optimization

losses using NTMAA is investigated [26].

5.1 Pattern synthesis parameters

patterns are the conflicting parameters.

5.2 Multiple objectives

18

5. Synthesis of time-modulated antenna arrays

TMAA synthesis problem is non-convex and nonlinear in nature. A number of numerical techniques as already mentioned—Dolph-Chebyshev and Taylor series [1]—are available to synthesize pencil beam power pattern in conventional antenna arrays (CAAs). Also, some analytical methods are reported to generate shaped beam patterns and phase-only controlled multiple power patterns in CAAs [27, 28, 29]. Durr et al. described a modified Woodward-Lawson technique to design phasedifferentiated multiple pattern antenna arrays with prefixed amplitude distributions [27]. The analytical technique reported in [28] is used to determine the nonlinear phase distribution of linear arrays. A method based on projection approach [29] is proposed to synthesize reconfigurable array antennas of a cosecant<sup>2</sup> beam and a flattop beam (FTB) by using a common amplitude with phase-only control of analog phase shifters. Though these numerical and analytical techniques can also be applied to determine the nonlinear distributions of dynamic excitation coefficient and phase to synthesize power pattern at operating frequency of TMAAs, such methods have no control on sideband power level. Therefore, the powerful global stochastic optimization tools such as genetic algorithm (GA) [30], differential evolution (DE) [4–5, 31, 32], particle swarm optimization (PSO) [7], simulated annealing (SA) [6, 33], and artificial bee colony (ABC) [22, 34] are essentially required to solve such multi-objective TMAA synthesis problems.
