Error Probability Distribution and Density Functions for Weibull Fading Channels With and Without Diversity Combining

Abstract

In this letter, a detailed theoretical analysis of probability distribution and density functions of probability of error in a wireless system is considered. Closed form expressions for distribution and density functions of the probability of error are derived for Weibull fading channels for the cases of (i) No Diversity (ND), (ii) Selection Combining (SC) diversity, and (iii) Switch and Stay Combining (SSC) diversity. Numerical results are plotted and discussed in detail for the various cases.

Appendix

The expression for the error rate of a BPSK system as a function of the received SNR incorporating fading, $\gamma_b = \frac{\alpha^2 E_b}{N_0},$ is given by [21]

$$ P_2\left( \gamma_b \right) = Q\left( \sqrt{2\gamma_b} \right), $$

(12)

where α is the “faded random variable”, $Q(x) = \frac{1} {\sqrt{2\pi}} \int\limits_x^{\infty} \exp\left( -\frac{y^2}{2} \right) dy$ is the Q(.) function in probability theory [19]. The unconditional BER for BPSK signaling, P ₂, is obtained by averaging P ₂(γ _b ) over the PDF of γ _b , and is given by [21]

$$ \begin{aligned} P_2 &= \int\limits_0^{\infty} P_2(\gamma_b) f(\gamma_b) d\gamma_b \\ &= \frac{p}{2\sigma^2} \int\limits_0^{\infty} \gamma_b^{p-1} \exp\left( -\frac{\gamma_b^p}{2\sigma^2} \right) Q\left( \sqrt{2\gamma_b} \right) d\gamma_b \\ &= \frac{p}{4\sigma^2} \int\limits_0^{\infty} \gamma_b^{p-1} \exp\left( -\frac{\gamma_b^p}{2\sigma^2} \right) d\gamma_b - \frac{p}{4\sigma^2} \int\limits_0^{\infty} \gamma_b^{p-1} \exp\left( -\frac{\gamma_b^p}{2\sigma^2} \right) \hbox {erf}(\sqrt{\gamma_b}) d\gamma_b, \\ \end{aligned} $$

(13)

where $\hbox {erf}(.)$ is the error function in probability theory and $Q(x) = \frac{1}{2} \left( 1 - \hbox {erf}\left( \frac{x} {\sqrt{2}} \right) \right)$ [19].

Making change of variables in the integral of (13) with $t = \frac{\gamma_b^p}{2\sigma^2}$ and $dt = \frac{p \gamma_b^{p-1}} {2\sigma^2}d\gamma_b,$ we have

$$ P_2 = \frac{1}{2} \left( 1 - \int\limits_0^{\infty} \hbox {erf}\left( \left(2t\sigma^2\right)^{\frac{1}{2p}} \right) e^{-t} dt \right). $$

(14)

Now, consider the infinite series expansion for $\hbox{erf}(x)$ from [22] given by

$$ \hbox {erf}(x) = \frac{2}{\sqrt{\pi}} \sum\limits_{i=1}^{\infty} (-1)^{i+1} \frac{x^{2i-1}}{(2i-1)(i-1)!} $$

(15)

Substituting (15) into (14), rearranging and simplifying, we have

$$ P_2 = y = g(\gamma_c) = \frac{1}{2} - \frac{\gamma_c^{1/k}} {k\sqrt{\pi}} \left( \Upgamma\left( \frac{1}{k} \right) - \gamma_c^{2/k} \Upgamma\left( \frac{3}{k} \right) + \frac{1}{2} \gamma_c^{4/k} \Upgamma\left( \frac{5} {k} \right) - \frac{1} {6} \gamma_c^{6/k} \Upgamma\left( \frac{7} {k} \right) + \frac{1}{24} \gamma_c^{8/k} \Upgamma\left( \frac{9} {k} \right) + \cdots \right), $$

(16)

where ${\gamma_c}=2\sigma^2$ is the SNR without fading. In our case, we can consider ${\gamma_c}=\hbox{SNR}.$ So, $g^{-1}(y) \equiv \hbox {SNR}.$

Let x ≡ γ _c . Then, $\frac{\partial g^{-1}(y)}{\partial y} = \frac{\partial x}{\partial y}.$ From (16), we have

$$ \frac{\partial y}{\partial \gamma_c} = -\frac{1}{k^2 \sqrt{\pi}} \left[ \Upgamma\left( \frac{1}{k} \right) \gamma_c^{\frac{1}{k}-1} - \Upgamma\left( \frac{3}{k} \right) \gamma_c^{\frac{3} {k}-1} + \frac{5}{2} \Upgamma\left( \frac{5}{k} \right) \gamma_c^{\frac{5} {k}}-1 - \frac{7}{6} \Upgamma\left( \frac{7}{k} \right) \gamma_c^{\frac{7}{k}-1} + \frac{3}{8}\Upgamma\left( \frac{9}{k} \right) \gamma_c^{\frac{9}{k}-1} \right]. $$

(17)

Thus, $c_1(y) = -\frac{1}{\frac{\partial y}{\partial \gamma_c}},$ and it is substituted in Eqs. 7, 9 and 11 to compute the PDF of the probability of error in Weibull fading channels in the cases of no diversity, SC diversity, and SSC diversity, respectively.