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.