Performance Analysis of Rayleigh Fading Channels with Hybrid Correlation Effects with 2-Branch Maximal Ratio Combining

Abstract

This paper deals with the impact of channel estimation errors on the capacity of correlated fading MIMO channels. In this paper, closed-form expressions for cumulative density function and probability density function for Rayleigh fading channels with combined effects of impairments due to combining errors and branch correlation errors for 2-branch case are derived. We also derive closed-form expressions for capacities per unit bandwidth for fading channels with effects of hybrid correlation for two branch maximal ratio combining diversity reception case for different adaptation policies. We show that hybrid correlation leads to increased performance degradation when combining correlation is low and branch correlation is high and vice versa. Even though hybrid correlation exists between branches, capacity gains are significantly more for adaptation policies especially for OPRA policy as compared to other policies for both cases.

Appendix 1

Figure 1a, b, through Fig. 5a, b show the system simulation through the steps discussed below.

Step 1:

The base for the simulation is to find out numerical instantaneous SNR values (γ). This is obtained from the CDF of the received instantaneous SNR γ for (1) combining errors at the output of a M-branch MRC output (when the pilot and message signals are not perfectly correlated, i.e., 0 < ρ < 1) which is given in (3), (2) branch correlation at the output of two branch system given in (1), (3) combined CE & BC hybrid correlation of two branch system given in (4) as

$$ \mathop F\nolimits_{\text{M}}^{{ ( {\text{CE)}}}} (\gamma ) = 1 - \exp \left( { - \frac{\gamma }{\varGamma }} \right)\sum\limits_{n = 0}^{M - 1} {\left( \begin{gathered} M - 1 \hfill \\ \quad n \hfill \\ \end{gathered} \right)\rho^{2n} \left( {1 - \rho^{2} } \right)^{M - 1 - n} \sum\limits_{k = 0}^{n} {\frac{1}{k!}} \left( {\frac{\gamma }{\varGamma }} \right)}^{k} , $$

(17)

$$ \mathop {\text{F}}\nolimits_{ 2}^{{ ( {\text{BC)}}}} (\gamma ) = 1 - \frac{1}{2\varepsilon }\left[ {(1 + \varepsilon )\exp \left( { - \frac{\gamma }{\varGamma (1 + \varepsilon )}} \right) - (1 - \varepsilon )\exp \left( { - \frac{\gamma }{\varGamma (1 - \varepsilon )}} \right)} \right] , $$

(18)

and

$$ \begin{aligned} F_{2}^{{({\text{CE}}\& {\text{BC}})}} (\gamma ) =\, & \frac{{(1 - \rho^{2} )(1 + \varepsilon )}}{{2\varepsilon^{2} }}\left[ {\exp \left( {\frac{ - \gamma }{\varGamma }} \right) - (1 + \varepsilon )\exp \left( {\frac{ - \gamma }{\varGamma (1 + \varepsilon )}} \right)} \right] - \frac{{(1 - \rho^{2} )(1 - \varepsilon )}}{{2\varepsilon^{2} }}\left[ {(1 - \varepsilon )\exp \left( {\frac{ - \gamma }{\varGamma (1 - \varepsilon )}} \right) - \exp \left( {\frac{ - \gamma }{\varGamma }} \right)} \right] \\ & + \frac{{\rho^{2} (1 + \varepsilon )^{2} \varGamma }}{{2\varepsilon^{3} }}\left[ {\exp \left( {\frac{ - \gamma }{\varGamma }} \right) - (1 + \varepsilon )\exp \left( {\frac{ - \gamma }{\varGamma (1 + \varepsilon )}} \right)} \right] + \frac{{\rho^{2} (1 + \varepsilon )}}{{2\varepsilon^{2} }}\left[ {\varGamma \exp \left( {\frac{ - \gamma }{\varGamma }} \right) + \gamma \exp \left( {\frac{ - \gamma }{\varGamma }} \right)} \right] \\ & + \frac{{\rho^{2} (1 - \varepsilon )^{2} }}{{2\varepsilon^{3} }}\left[ {(1 - \varepsilon )\exp \left( {\frac{ - \gamma }{\varGamma (1 - \varepsilon )}} \right) - \exp \left( {\frac{ - \gamma }{\varGamma }} \right)} \right] + \frac{{\rho^{2} (1 - \varepsilon )}}{{2\varepsilon^{2} \varGamma }}\left[ {\varGamma \exp \left( {\frac{ - \gamma }{\varGamma }} \right) + \gamma \exp \left( {\frac{ - \gamma }{\varGamma }} \right)} \right] \\ \end{aligned} $$

(19)

Step 2:

The CDF is equated to a uniform random variable, $U \in (0,1)$. Then 10⁶ Uniform random numbers are generated using rand command in MATLAB.

Step 3:

Using the above values of U, the maximum and minimum values of U are found out from the array in which 10⁶ Uniform random numbers (U) are stored.

Step 4:

The maximum value of received SNR (γ_max), and the minimum value of received SNR (γ_min) are found out for different values of Individual Branch SNRs($\bar{\gamma }$), different values of Correlation (ρ) and for different number of diversity order (M) which is tabulated in Table 1.

Table 1 Impairments due to hybrid correlation errors

Step 5:

Using the maximum and minimum values of instantaneous SNR (γ), the capacity for all four policies is obtained for different values of Individual Branch SNRs ($\bar{\gamma }$), different values of Correlation (ρ), and for different number of diversity orders (M) with step size 0.001.

Step 6:

The simulation results for all four policies for M = 2 and M = 3, M = 4 and M = 5 at ρ = 0.5 are plotted for (1) different values of Individual Branch SNRs $\bar{\gamma }$, and for (2) different values of correlation ρ at $\bar{\gamma }$ = 5 dB and compared with the analytical results for the above four cases. Figures 1a,1b through 5a,5b show the analytical vesus simulation results for impairments due to combining errors.