Error Probability for L-branch Coherent BPSK Equal Gain Combiners over Generalized Rayleigh Fading Channels

Abstract

In this paper, a closed-form expression for the probability of error in a coherent BPSK system over Generalized Rayleigh fading channels is derived. An L-branch equal gain combining diversity scheme is used. Theoretical results for the probability of error are plotted for various values of the number of degrees of freedom (n) and diversity order (L). A simulation is performed and the simulated results are found to match very well with the theoretical results.

Appendix

It is interesting to see that Eq. 6 of this paper converges to Eq. 11 of [17] when n = 2. Here is the proof:

For n = 2, (6) can be written as

$$ P_e = 2^L \int\limits_0^{\infty} \ldots \int\limits_0^{\infty} F_{\rm {NORM}(0,\delta^2)}\left( -\sum\limits_{k=1}^L r_k \right) \prod_{k=1}^L {\frac{r_k}{\gamma_k}}. \exp\left( - \sum\limits_{k=1}^L {\frac{r_k^2}{\gamma_k}} \right) dr_1 dr_2 \ldots dr_L, $$

(12)

where ${\gamma_k}=2\sigma_k^2$ Making change of variables in the integrals of (12), where $r^{\prime}_k = {\frac{r_k}{\sqrt{\gamma_k}}}\hbox{ and }dr^{\prime}_k = {\frac{dr_k}{\sqrt{\gamma_k}}}\forall k = 1, 2, \ldots, L$, we have

$$ P_e = 2^L \int\limits_0^{\infty} \ldots \int\limits_0^{\infty} F_{\rm {NORM}(0,\delta^2)}\left( -\sum\limits_{k=1}^L r^{\prime}_k \sqrt{\gamma_k} \right) \prod_{k=1}^L r^{\prime}_k. \exp\left( -\sum\limits_{k=1}^L {r^{\prime}_k}^2 \right) dr^{\prime}_1 dr^{\prime}_2 \ldots dr^{\prime}_L. $$

(13)

Now, (13) reduces to Eq. 11 of [17] when

• the average received SNR at the kth branch, γ _k in this paper is related to the ratio of the average bit energy to average noise density in the kth branch, ρ _k of [17] by $\gamma_k = {\frac{\rho_k}{L}},$ and

• the variance, δ², of the Gaussian random variable, W in this paper is equal to ${\frac{1}{2}}$.