Figure 12 Surface current distributions for the proposed fractal MPA array at (a) 5.35 GHz (Port-1) (b) 5.35 GHz (Port-2) (c) 7.78 GHz (Port-1) (d) 7.78 GHz (Port-2) (e) 9.47 GHz (Port-1) (f) 9.47 GHz (Port-2)
DIVERSITY CHARACTERISTICS
To characterize the feasibility of the proposed dual-port fractal array for UWB-MIMO systems, various diversity performance metrics such as ECC, DG, MEG, CCL and TARC are analyzed in this section.
For a two-element MPA, ECC is a crucial diversity parameter to quantify the amount of correlation between the signals received from adjacent communication channels. DG is defined as the figure of merit to determine the potency of the applied diversity scheme. For any practical MIMO application, the acceptable limit of ECC and DG to receive uncorrelated signals is less than 0.05 and greater than 9.95 respectively. ECC (\(\rho)\) can be computed from S-parameters using Equation (15) and Equation (16) defines the relationship between ECC and DG [8].
\(\rho=\frac{\left|S_{11}^{*}S_{12}+S_{21}^{*}S_{22}\right|^{2}}{(\left(1-{(\left|S_{11}\right|}^{2}+\left|S_{21}\right|^{2}\right))(1-{(\left|S_{22}\right|}^{2}+\left|S_{12}\right|^{2})))}\)(15)
\(DG=10\sqrt{1-\left|\rho\right|^{2}}\) (16)
Figure 13 (a, b) shows the variation of ECC and DG with respect to frequency for the proposed fractal array. It shows a low level of ECC (≤ 0.0021 (simulated), ≤ 0.0023 (measured)) and a high degree of DG (≥ 9.989 (simulated), ≥ 9.988 (measured)) for a complete operational range which affirms a good diversity performance of the proposed MPA array.
CCL is the third significant diversity parameter that determines the degree of deterioration of array performance as a result of the correlation in the MIMO channel. For high SNR, CCL can be computed from S-parameters using Equation (17) [6].
\(\text{CCL}=-\ \log_{2}\det\left(\Psi^{R}\right)\) (17)
where \(\Psi\)R is a 2 × 2 correlation matrix and given by Equation (18).
\(\Psi^{R}=\ \par \begin{pmatrix}\end{pmatrix}=\par \begin{pmatrix}1-{(\left|S_{11}\right|}^{2}+\left|S_{12}\right|^{2}&{-(S}_{11}^{*}S_{12}+S_{21}^{*}S_{12})\\ {-(S}_{22}^{*}S_{21}+S_{12}^{*}S_{21})&1-{(\left|S_{22}\right|}^{2}+\left|S_{21}\right|^{2})\\ \end{pmatrix}\) (18)
Figure 13 (c) shows that the value of CCL is ≤ 0.4 bits/s/Hz (simulated and measured) for the entire operational range which indicates high port-to-port isolation between the array elements.
MEG is another important diversity metric as it determines the antenna gain performance by taking into account the real fading scenario. Equations (19, 20) are used for calculating MEG (for both the ports) by considering the radiation efficiencies at the two ports. For good diversity performance, the difference in MEG for two ports should be less than 3 dB (Equation (21)) [6].
\(\text{MEG}_{port-1}=0.5\eta_{rad,\ \ port-1}=0.5(1-\left|S_{11}\right|^{2}-\left|S_{12}\right|^{2})\)(19)
\(\text{MEG}_{port-2}=0.5\eta_{rad,\ \ port-2}=0.5(1-\left|S_{21}\right|^{2}-\left|S_{22}\right|^{2})\)(20)
\(\left|\text{MEG}_{port-1}-\text{MEG}_{port-2}\right|<3dB\)(21)
As shown in Figure 13 (d), a maximum MEG of -3.018dB (simulated) at 5.88 GHz frequency and -3.011dB (measured) at 6.2 GHz frequency is observed with an overall MEG ≥ -3.7 dB (simulated) and ≥ -3.85 dB (measured). The difference between the MEG for two ports is 0 dB (simulated) and 0.69 dB (measured). This makes the proposed fractal array feasible for MIMO employment in UWB radio systems.
TARC is also an essential diversity metric to properly identify the antenna array performance in terms of bandwidth and efficiency. TARC consists of a single curve, obtained by concentrating all the details of scattering parameters for a multi-element antenna array. Mathematically, TARC is calculated using Equation (22-27) [15].
\(\Gamma_{a}^{t}=\sqrt{\frac{\text{reflected\ power}}{\text{incident\ power}}}=\sqrt{\frac{\sum_{i=1}^{N}\left|b_{i}\right|^{2}}{\sum_{i=1}^{N}\left|a_{i}\right|^{2}}}\)(22)
\(\left[b\right]=\left[S\right].[a]\)(23)
where b is the reflected signal vector, a is the incident signal vector and S is scattering matrix. Assuming multipath wave propagation and gaussian MIMO channels, the reflected signal for two-port antenna network is given by Equation (24-26).
\(\par \begin{bmatrix}b_{1}\\ b_{2}\\ \end{bmatrix}=\par \begin{bmatrix}S_{11}&S_{12}\\ S_{21}&S_{22}\\ \end{bmatrix}\par \begin{bmatrix}a_{1}\\ a_{2}\\ \end{bmatrix}\) (24)
\(b_{1}=S_{11}a_{1}+S_{12}a_{2}=a_{0}S_{11}e^{i\theta_{1}}+a_{0}S_{12}e^{i\theta_{2}}=a_{1}(S_{11}+e^{\text{iθ}}S_{12})\ \ \ \ \ \ \ \ \ \)(25)
\(b_{2}=S_{21}a_{1}+S_{22}a_{2}=a_{0}S_{21}e^{i\theta_{1}}+a_{0}S_{22}e^{i\theta_{2}}=a_{1}(S_{21}+e^{\text{iθ}}S_{22})\)(26)
By putting values of Equation (25,26) in Equation (22), the final formula for TARC is given in Equation (27).
\(\text{\ \ \ \ \ \ \ \ }\Gamma_{a}^{t}=\frac{\sqrt{\left|\left(S_{11}+S_{12}e^{\text{jθ}}\right)\right|^{2}+\left|\left(S_{11}+S_{12}e^{\text{jθ}}\right)\right|^{2}}}{\sqrt{2}}\)(27)
where ‘θ’ is the input feed phase difference.
As shown in Figure 13 (e, f), TARC (dB) curves are plotted for variation in ‘θ’ (0˚, 30˚, 60˚, 90˚, 120˚, 150˚,180˚) with respect to the frequency\(.\) The simulated and measured TARC values less than -10 dB and -8.9 dB respectively are observed for all variations of ‘\(\theta\)’. The simulated and measured TARC resonance curves (at a feed phase difference of 30˚) show a good resemblance with the simulated and measured S11/S22 (dB) curves respectively. A slight deviation in resonant characteristics is observed due to the presence of some cross-coupling between the actively radiating elements of the array. Table 2 shows that the proposed fractal array exhibits improved performance in terms of compactness, large bandwidth and acceptable diversity characteristics as compared to the previously published 2 × 2 fractal antenna arrays.