Closed-Form Analysis of Various Diversity Techniques for Multiband OFDM UWB System Over Log-Normal Fading Channels

Abstract

In this paper, we investigate the effect of multipath fading on the combined signal-to-noise ratio (SNR) in conjunction with multiband orthogonal frequency division multiplexing ultra-wideband system with different diversity schemes such as maximal ratio combining, equal gain combining and selection combining. In particular, we derived the probability density function of the combiner output SNR followed by performance measures, average combined SNR, outage probability and average bit error rate, in the case of L-fold log-normal fading channels. The derived mathematical expressions are illustrated by applying the method to some selected numerical examples of interest showing the impact of fading variance as-well-as a number of diversity paths, which is combined, on the performance metrics of combiner output for above diversity techniques. The results show that the number of diversity paths significantly improve the effect of multipath fading on derived performance measures.

Appendix

1.1 Mean and Variance Calculations

Consider $\gamma$ as a log-normal random variable with PDF given in (4), i.e. $10~{\text {log}}~\gamma \approx {\mathcal {N}}\left( \mu ,\Omega \right)$. The expectation value of $\sqrt{\gamma }$ is

$$\begin{aligned} E\left[ \sqrt{\gamma }\right] =&\int \limits _{-\infty }^{\infty }\left( 10^{\frac{x}{10}}\right) ^{\frac{1}{2}} \frac{1}{\sqrt{2\pi \Omega }}{\text {exp}}\left( -\frac{\left( x-\mu \right) ^{2}}{2\Omega }\right) dx \\ =&\,{\text {exp}}\left( \frac{\Omega }{8\zeta ^{2}}+\frac{\mu }{2\zeta }\right) \end{aligned}$$

(58)

The expectation value of $\gamma$ is

$$\begin{aligned} E\left[ \gamma \right] =&\int \limits _{-\infty }^{\infty }10^{\frac{x}{10}}\frac{1}{\sqrt{2\pi \Omega }} {\text {exp}}\left( -\frac{\left( x-\mu \right) ^{2}}{2\Omega }\right) dx \\ =&\,{\text {exp}}\left( \frac{\Omega }{2\zeta ^{2}}+\frac{\mu }{\zeta }\right) \end{aligned}$$

(59)

and the expectation value of $\gamma ^{2}$ is

$$\begin{aligned} E\left[ \gamma ^{2}\right] =&\int \limits _{-\infty }^{\infty }\left( 10^{\frac{x}{10}}\right) ^{2}\frac{1}{\sqrt{2\pi \Omega }} {\text {exp}}\left( -\frac{\left( x-\mu \right) ^{2}}{2\Omega }\right) dx \\ =&\,{\text {exp}}\left( \frac{2\Omega }{\zeta ^{2}}+\frac{2\mu }{\zeta }\right) \end{aligned}$$

(60)

Thus variance of $\sqrt{\gamma }$ is

$$\begin{aligned} {\text {var}}\left( \sqrt{\gamma }\right) =&E\left[ \gamma \right] -E^{2}\left[ \sqrt{\gamma }\right] \\ =&\,{\text {exp}}\left( \frac{\Omega }{4\zeta ^{2}}+\frac{\mu }{\zeta }\right) \left( {\text {exp}}\left( \frac{\Omega }{4\zeta ^{2}}\right) -1\right) \end{aligned}$$

(61)

and variance of $\gamma$ is

$$\begin{aligned} {\text {var}}\left( \gamma \right) =&E\left[ \gamma ^{2}\right] -E^{2}\left[ \gamma \right] \\ =&{\text {exp}}\left( \frac{\Omega }{\zeta ^{2}}+\frac{2\mu }{\zeta }\right) \left( {\text {exp}}\left( \frac{\Omega }{\zeta ^{2}}\right) -1\right) \end{aligned}$$

(62)