Envelope correlation coefficient

ECC is one of the diversity parameters to investigate the performance of the MIMO antenna system for wireless applications which shows how many antennas are correlated to each other. It is calculated by equation 2(a) using S-parameters of antennas but this equation is only valid when power is uniformly distributed in the system elements, therefore, currently, the researcher has not preferred this equation for the calculation of ECC. Calculated ECC using S-parameters are shown in Fig. 9(b).

(2a)$$\rho _{ecc, ij} = \displaystyle{{{\left[{\sum\nolimits_{n = 1}^N {S_{in}^\ast S_{nj}} } \right]}^2} \over {\left({1-\sum\nolimits_{n = 1}^N {{\vert {S_{ni}} \vert }^2} } \right)\left({1-\sum\nolimits_{n = 1}^N {{\vert {S_{nj}} \vert }^2} } \right)}}$$

Fig. 9.

Measured and simulated results of the proposed MIMO antenna: (a) envelope correlation coefficient from radiated fields, (b) envelope correlation coefficient from S-parameters, and (c) multiplexing efficiency.

As equation 2(a) is not preferred by the researcher, the accurate ECC to be calculated with radiated fields by equation 2(b), and the results are shown in Fig. 9(a).

(2b)$$\rho _{ecc,ij} = \left| {\displaystyle{{\int\!\!\!\int {XPR.E_{\theta i}.E_{\theta j}^{\ast} .P_\theta + E_{\phi i}.E_{\phi j}^{\ast} .P_\phi d\Omega } } \over {\sqrt {\int\!\!\!\int {XPR.G_{\theta i}.P_\theta + G_{\phi i}.P_\phi d\Omega } } \sqrt {\int\!\!\!\int {XPR.G_{\theta j}.P_\theta + G_{\phi j}.P_\phi d\Omega } } }}} \right|^2$$

In equation 2(b), XPR is the power ratio of the cross-polarization of the incident wave. E_θ(Ω) and E _Φ(Ω) are the θ and Φ components of an active electric field where Ω is the beam solid angle. P_θ and P _Φ are the θ and Φ components of the angular power density of incoming waves. The unity cross-polarization is achieved when P_θ = P _Φ = 1/4π. Polarized signal active gain patterns are denoted by G_θ and G _Φ for any propagation environment. G_θi = E_i × E_i*, G_θj = E_j × E_j*, G _Φi = E_i × E_i*, G _Φj = E_j × E_j*are the relations between active gain patterns and electric field components. Using these relations, equation 2(b) becomes equation 2(c) [Reference Loyka28].

(2c)$$\rho _{ecc,ij} = \displaystyle{{{\left| {\int\!\!\!\int\limits_{4\pi } {\left[ {E_i\left( {\theta ,\phi } \right) \times E_j\left( {\theta ,\phi } \right)} \right]d\Omega } } \right|}^2} \over {\int\!\!\!\int\limits_{4\pi } {{\left| {E_i\left( {\theta ,\phi } \right)} \right|}^2d\Omega \; \times \int\!\!\!\int\limits_{4\pi } {{\left| {E_j\left( {\theta ,\phi } \right)} \right|}^2d\Omega } } }}$$

The electric field of the i ^th and the j ^th element is denoted by E_i and E_j. Ideally, ECC has zero value for an un-correlated MIMO antenna system but the practically acceptable value for the MIMO system is less than or equal to 0.5. The calculated and measured values of ECC between two antenna elements of a designed four-element antenna by S-parameters and radiated fields in all prescribed bands are observed <0.1 as exhibited in Figs 9(a) and 9(b), respectively. Hence, this MIMO antenna shows less correlation between the two antennas in the MIMO system and gets excellent diversity performance.

Diversity gain

Diversity gain (DG) is also an important parameter of the MIMO antenna system. The DG is an increment of signal-to-interference ratio due to some diversity scheme or to reduce the transmission power by using this scheme without compromising the performance of an antenna. Diversity gain is depending on cross-correlation between the transmitted signals from the adjacent antennas and the relative mean power level. The DG of the MIMO antenna is calculated by $\;DG = 10 \times \sqrt {( {1-ECC^2} ) }$. For the effective operation of the MIMO antenna system, the value of DG should be close to 10 dB. The calculated values of DG in all intended bands are more than 9.8 dB.

Multiplexing efficiency

The main feature of multiplexing efficiency is to show the correlation and efficiency imbalance between MIMO antennas used in a wireless system. The ratio of radiated power of an i ^th antenna and reference (isotropic) antenna is multiplexing efficiency (η_MUX). The η_MUX of the MIMO antenna is calculated by $ME = \left[{\left({\prod\limits_{i = 1}^k {\eta_i} } \right)\bullet \det ( {\psi}^{\kern1ptp} ) } \right]^{1/k}$, where η_i stands for the total efficiency of the i ^th antenna, ψ ^p is the normalized antenna correlation matrix [Reference Li, Xu, Ban and Yu29].

Hence, the values of total and multiplexing efficiency are more or less identical (70%) and the value of efficiency goes down at the unintended bands as shown in Fig. 9(c).

Channel capacity

Channel capacity of the MIMO antenna is investigated to know the data rate which is dependent on bandwidth and signal-to-noise ratio (SNR). The channel capacity of the four-element pentaband MIMO antenna is verified with the average channel capacity of the standard MIMO system which is calculated by equation (3). The minimum and maximum threshold levels of channel capacity of the four-element antenna are varying between 14.85 and 22.68 bps/Hz at 20 dB SNR in a uniform environment as shown in Fig. 10(a).

(3)$$C_{4 \, MIMO \, Max.} = k\{ {{\log }_2[ {\det ( {[ I] + SNR[ H] [ H^\ast ] } ) } ] } \} $$

Channel capacity directly depends upon the number of antenna elements and the correlation between them. But the CCL also increases with the increase of the number of antennas in the system for fixed S/N value. Therefore, CCL reduces the throughput or channel capacity of the proposed MIMO antenna as shown in Fig. 10(c), where k is the number of antennas used in a MIMO system for transmitting or receiving EM signals. An identity matrix is denoted by [I] and SNR is the ratio of signals of device terminals and distribution channels in a Rayleigh fading environment. The value of SNR for the calculation of channel capacity is 20 dB. H and H* are the channel fading matrix and Hermitian transpose of H, respectively. For an ideal MIMO antenna, the correlation between them is negligible so [H] [H*] = [I]. Maximum channel capacity of the four-element MIMO antenna system is 26.63 bps/Hz but the permissible limit is 65–70% of its maximum value (26.63 × 0.65 = 17.30 bps/Hz) [Reference Saxena, Jain and Awasthi26, Reference Loyka28]. The main factor behind the use of the MIMO antenna is to get an enhanced channel capacity at an appropriate level. However, the channel capacity of a wireless communication system is also dependent on the number of antenna elements used in the system and the correlation between antenna elements. This means channel capacity will be increased with an increase of un-correlation between antenna elements.

Fig. 10.

The proposed four-element antenna: (a) calculated average channel capacity, (b) simulated and measured results of channel capacity loss, and (c) simulated channel capacity versus SNR.

Channel capacity loss

CCL is also related to the amount of correlation between antenna elements used in the MIMO system and it is calculated by using equations (4)–(6) [Reference Saxena, Jain and Awasthi26, Reference Saxena, Jain and Awasthi27, Reference Biswal and Das30]. The values of simulated and measured CCL are below the prescribed (0.5 bps/Hz) limit for intended pentabands as shown in Fig. 10(b). However, the CCL of the MIMO system is linearly increased when we increase the number of antenna elements. If we increase the antenna elements then the correlation factor between antenna elements increases which increases the CCL. Therefore, the diversity performance of the proposed antenna will be affected by increasing the number of antenna elements. CCL of four-element antenna is calculated as follows:

(4)$$CCL = {-}\log _2( \psi ^p) $$

where, $\psi ^p = \left[{\matrix{ {\rho_{11}} & {\rho_{12}} & {\rho_{13}} & {\rho_{14}} \cr {\rho_{21}} & {\rho_{22}} & {\rho_{23}} & {\rho_{24}} \cr {\rho_{31}} & {\rho_{32}} & {\rho_{33}} & {\rho_{34}} \cr {\rho_{41}} & {\rho_{42}} & {\rho_{43}} & {\rho_{44}} \cr } } \right]$

(5)$$\rho _{ii} = 1-\left({\sum\limits_{n = 1}^N {\vert {S_{in}^\ast S_{ni}} \vert } } \right)$$

(6)$$\rho_{ij} = {-}\left({\sum\limits_{n = 1}^N {\vert {S_{in}^\ast S_{nj}} \vert}} \right)$$

where i, j, and n = 1, 2, 3, 4.

Figure 10(b) shows the CCLs and it is <0.35 bps/Hz over the intended bands, which indicate a good diversity performance of the proposed pentaband MIMO antenna.

Total active reflection coefficient

In a four-port antenna system, adjacent antenna elements interrupt each other when they are excited at the same instant. This type of interruption affects the antenna performance parameters like overall gain, efficiency, and bandwidth of multiband. Therefore, the actual performance of the MIMO antenna system will not be extracted by scattering parameters only, but also by the total active reflection coefficient (TARC). It is defined as the square root of the sum of all incident powers at the ports minus radiation power, divided by the sum of all incident powers at the ports. The TARC of the proposed four-element MIMO system is calculated by equation (7)–(9) [Reference Saxena, Jain and Awasthi26, Reference Saxena, Jain and Awasthi27, Reference Biswal and Das30].

(7)$$( {TARC} ) \Gamma _A^t = \displaystyle{{\sum\nolimits_{n = 1}^N {{\vert {b_i} \vert }^2} } \over {\sum\nolimits_{n = 1}^N {{\vert {a_i} \vert }^2} }} = {-}\sqrt {\displaystyle{{{( S_{ii} + S_{ij}) }^2 + {( S_{\,jj} + S_{\,ji}) }^2} \over n}} $$

where a_i and b_i are an incident and reflected EM wave, respectively, and n denotes the number of antenna elements used in the MIMO system at transmitting or receiving ends.

(8)$$[ b ] = [ S ] [ a ] $$

where [S], [a], and [b] are scattering, incident, and reflected matrix of four-port multiband MIMO antenna, respectively.

(9)$$\left[{\matrix{ {b_1} \cr {b_2} \cr {b_3} \cr {b_4} \cr } } \right] = \left[{\matrix{ {S_{11}} & {S_{12}} & {S_{13}} & {S_{14}} \cr {S_{21}} & {S_{22}} & {S_{23}} & {S_{24}} \cr {S_{31}} & {S_{32}} & {S_{33}} & {S_{34}} \cr {S_{41}} & {S_{42}} & {S_{43}} & {S_{44}} \cr } } \right]\left[{\matrix{ {a_1} \cr {a_2} \cr {a_3} \cr {a_4} \cr } } \right]$$

In general, TARC should be less than −10 dB for better MIMO antenna performance, and TARC of the fabricated and simulated proposed MIMO antenna is less than −40 dB in the intended bands. Hence, the designed antenna is depicted as a good diversity criterion for a MIMO wireless communication system.

Mean effective gain

MEG is an important parameter to investigate the diversity performance of the MIMO antenna and it gives information about how much mean power is received when input power is fed along the same route. MEG of the proposed pentaband MIMO antenna can be calculated by equation (10) [Reference Saxena, Jain and Awasthi26, Reference Saxena, Jain and Awasthi27, Reference Biswal and Das30].

(10)$$\hskip-1pc \eqalign{ MEG_i = \displaystyle{{P_{rec}} \over {P_{inc}}} = \oint {\left[{\displaystyle{{XPR\,\times \,G_{\theta i}( \Omega ) + G_{\phi i}( \Omega ) \,\times \,P_\phi ( \Omega ) } \over {1 + XPR}}} \right]d\Omega }} $$

where power gain [G_θi (Ω), G _Φi (Ω),] and density [P _Φ (Ω)] are the functions of the beam solid angle of the incident wave. The acceptable MEG of i ^th and j ^th antenna elements is ≤ −3 dB.

The MEG is calculated for the proposed antenna based on three propagation models where XPR has different values as 0dB, 1.0dB, and 6.0 dB for isotropic, outdoor, and indoor with different mediums like isotropic, Gaussian, and Laplacian as shown in Table 4 and Fig. 11. The MEG of the proposed antenna falls within the permissible limit (≤ ± 3 dB) for all bands. Hence, the proposed antenna has high-quality channel performance which applies to wireless communication.

Fig. 11.

Simulated results of mean effective gain of the proposed antenna in isotropic, Laplacian, and Gaussian medium for indoor and outdoor values.

Table 4.

Simulated MEG results of pentaband MIMO antenna at various XPR and frequencies

The various power levels in terms of power loss, power accepted, power outgoing to all ports and power radiated by the proposed antenna have been examined for SAR calculation as shown in Fig. 12. Here, 0.5 W stimulated power is used for the calculation of all powers and diversity parameters. In the proposed antenna, most of the power is radiated because this antenna has minimum power loss in intended bands. Power accepted is almost equal to stimulated power except unintended bands due to poor return loss means power returns back to the same ports. The radiated power is less than stimulated power because dielectric, conductor, and surface wave losses are associated with the proposed antenna. Powers going to all ports highly interfere at other than the intended bands because at unintended bands most of the power is correlated between ports and power at the ports will not reach the antenna for radiation. Table 5 shows the comparison of the performance of the pentaband MIMO antenna with existing multiband MIMO antennas at diverse parameters. So, it is observed from Table 5 that the proposed antenna is having good radiation efficiency, ECC, and CCL including dual-band circular polarization characteristics. A lower value of ECC and CCL provides better channel capacity and less outage probability and circular polarization in two-band is applicable to immune the noise, produced due to Faraday rotation when the radiated field passes through the ionosphere layer. Hence, the proposed MIMO antenna provides better performance in comparison with the existing references.

Fig. 12.

Power levels of the designed MIMO antenna in various states as power loss, power stimulated, power accepted, and power radiated, power absorbed, and power outgoing.

Table 5.

Comparison of the performance of pentaband MIMO antenna with existing multiband MIMO antennas at diverse parameters

[P], Proposed.