## 5 Antenna Synthesis Via an Innovative HYBRID Genetic Algorithm  
When the antenna subset H\({}_{s}\), which is the \(M_{r}\times M_{t}\) channel coefficient matrix [46], was selected via the minimum posterior SNR criterion mentioned in part 4 above, we could think about a linear array displayed in the Figure 4, where the M non-uniform spaced elements were all located alongside an exactly straight line L in the positions \(x_{k}\), \(k=0\), 1, 2, 3,..., \(M-1\). Figure 4: The geometry of the linear array. Figure 3: Performance of LDC-TAS vs. Bell Labs layered space time (BLAST)-TAS. The parameter \(M\) can be calculated by the Equations (6) and (14), which are derived in part 4, and the beam pattern (BP) function of the matrix, \(p(u)\), being achieved as below:  
\[p(u)=\sum_{k=0}^{M-1}w_{k}e^{j\frac{2\pi}{\lambda}x_{k}u}. \tag{15}\]  
In Equation (15), \(w_{k}\) being the weighting coefficients of the \(k\)th element, \(\lambda\) means the wavelength of the propagation channel, \(u=\sin\theta-\sin\theta_{0}\), and \(\theta_{0}\) and \(\theta\) are respectively the steering angles of these arrays and the incident angle of the impinging plane wave. It is very essential to combine an array configuring to produce a BP that meets some requirements (e.g., side lobes level (SLL)) or rebuilds a special shape (\(p^{\text{ref}}\) (\(u\))). First of all, we defined the fitness function \(f\) that symbolizes a measurement of the difference among desired and combined BP [47; 48], as follows:  
\[f(\overline{\zeta})=\frac{1}{k_{1}f_{\text{SLL}}(\overline{\zeta})+k_{2}f_{ \text{BP}}(\overline{\zeta})+k_{3}f_{N}(\overline{\zeta})+k_{4}f_{\text{D}}( \overline{\zeta})}, \tag{16}\]  
where \(\overline{\zeta}\) = [M; x\({}_{0}\),..., x\({}_{k}\),..., x\({}_{\text{M-1}}\); w\({}_{0}\),..., w\({}_{\text{K}}\),..., w\({}_{\text{M-1}}\)]t being the uncertain quantity arrays and where:  
\[f_{\text{SLL}}(\overline{\zeta})=\frac{Q}{\max\limits_{u_{start}\leq t\leq 1 }[PdB(u)]},\]  
\[f_{\text{BP}}(\overline{\zeta})=\int\limits_{u\in S}\bigg{(}\frac{PdB(u)}{Q}- p_{dB}^{ref}(u)\bigg{)}du,\]  
\[f_{N}(\overline{\zeta})=M,\]  
\[f_{D}(\overline{\zeta})=D\]  
In the above equations, \(\mu_{start}\) is a threshold, which allows removing the main-lobe, which could be derived from the computation of the SLL (side-lobes level) [49; 50; 51]. Moreover, D is an aperture of the array, Q is a normalized constant parameter, S is a region of these values for \(\{p_{\text{dB}}\) (u)/Q\(\}>p_{dB}^{ref}(u)\), but \(p_{dB}^{ref}(u)\) is a desired BP shape. Finally, based on the optimization method, \(k_{1}\), \(k_{2}\), \(k_{3}\), and \(k_{4}\) were all selected as the normalized parameters. A fitness function \(f\) that has been assumed in Expression (2) is extremely non-linear with a lot of the local peak values, where the deterministic processes can be captured. Accordingly, in order to address the maximization issue at hand, we applied a stochastic algorithm to remove these local maxima and effectively explore highly nonlinear search spaces. The GAs already displayed their correctness and pretty effectiveness for optimizing those antenna arrays and seem to be the most possible, feasible, and reasonable option. GAs are optimization algorithms that are basically built on [PERSON]'s Theory of Evolution. The method assumes an evolution process for a population of these individuals (i.e., a set of experimental ways for the solved problem dealt with) in time that are improving some individual characteristics (i.e., the fitness with several constraints assessed using fitness function \(f\)).