# Optimal Design of Aperiodic Reconfigurable Antenna Array Suitable for Broadcasting Applications
## 2 Geometry Definition  
As mentioned before, CST is employed to model and simulate an eight-element AMAA for DVB-T broadcasting at 698 MHz. The feeding network is out of scope for this paper and therefore the array elements are considered to be directly excited at their input ports by sources. The inset feeding method is used in order to achieve better impedance matching to 50 \(\Omega\) sources. The geometry of AMAA is  
Figure 1: Flowchart of the optimization procedure. illustrated in Figure 2 and is defined by the physical dimensions \(W\) and \(L\) of the rectangular array elements, four distances \(d_{0},d_{1},d_{2},d_{3}\) between the array elements, where \(d_{0}<d_{1}<d_{2}<d_{3}\) (aperiodic structure), the transmission line length \(t_{l}\) and width \(t_{w}\) which connects each array element to its respective source, the inset depth \(s\) and the inset width \(g\). PSOvm is employed to find the optimal values for all the above parameters. In order to help the optimization algorithm converge faster, the search space for all the geometry parameters must be confined between an upper and a lower limit. As indicated by Reference [19] the theoretical values of \(W\) and \(L\) are given by:  
\[W_{th}=\frac{c}{2f_{r}}\sqrt{\frac{2}{\epsilon_{r}+1}} \tag{1}\]  
\[L_{th}=\frac{c}{2f_{r}\sqrt{\epsilon_{r,eff}}}-2\Delta L\,, \tag{2}\]  
where \(c\) is the free space wave velocity, \(f_{r}\) is the operation frequency (698 MHz in this paper), \(\epsilon_{r}\) is the dielectric constant of the substrate used here, \(\epsilon_{r,eff}\) is the effective dielectric constant, and \(\Delta L\) is the length reduction applied at both sides of the effective length to get physical length \(L\). It is noted that the subscript \"th\" denotes the theoretical value of a variable. The values of \(\epsilon_{r,eff}\) and \(\Delta L\) are respectively estimated [19] by:  
\[\epsilon_{r,eff}=\frac{\epsilon_{r}+1}{2}+\frac{\epsilon_{r}-1}{2}\left[1+12 \frac{h}{W_{th}}\right]^{-1/2},\frac{W_{th}}{h}>1, \tag{3}\]  
\[\Delta L=0.412h\frac{(\epsilon_{r,eff}+0.3)(W_{th}/h+0.264)}{(\epsilon_{r,eff} -0.258)(W_{th}/h+0.8)}\,, \tag{4}\]  
where \(h\) is the substrate height (or thickness). The substrate used in this paper is Duroid RT5880 LZ with \(\epsilon_{r}=2\) and \(h=2.54\) mm [20]. The thickness of the copper cladding (used in CST modeling) at both sides of the substrate is equal to 35 um [20]. By applying the above equations, we get \(W_{th}=175\) mm and \(L_{th}=151\) mm. It is expected that the optimal values for \(W\) and \(L\) will not deviate more than 40% from the theoretical values given by (1) and (2) and thus, \(W\) is limited between 0.6 \(W_{th}\) and 1.4 \(W_{th}\) while \(L\) is limited between 0.6 \(L_{th}\) and 1.4 \(L_{th}\). The distance \(d_{0}\) is confined between 0.5 \(\lambda\) and 0.8 \(\lambda\) (\(\lambda\) is the free space wavelength at 698 MHz) where the maximum \(FG\) is expected to be found. Since \(d_{0}\), \(d_{1}\), \(d_{2}\) and \(d_{3}\) are in incremental order to create the aperiodical structure of the array, we adopt the rule  
\[d_{i}=d_{i-1}+r_{i}\,\quad i=1,2,3, \tag{5}\]  
where \(r_{i}\), \(i=1,2,3\), are random numbers between 0 and 0.2 \(\lambda\). Therefore, the parameters \(d_{1}\), \(d_{2}\) and \(d_{3}\) are indirectly optimized through \(r_{1}\), \(r_{2}\) and \(r_{3}\), respectively. The values of \(t_{l}\) are confined between 0 and \(\lambda/4\) which helps for impedance matching. As for the width of the transmission line \(t_{w}\), the microwave theory predicts that a width of 8.4 mm will result in a transmission line of characteristic impedance equal to 50 \(\Omega\) however due to the mutual coupling between the array elements a more relaxed constraint needs to be applied in this variable and therefore the lower limit has been set at half the theoretical value (4.2 mm) and double (16.8 mm) for the upper limit.