# 7.9 Diversity techniques

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This section shows the error-performance improvements that can be obtained with the use of diversity techniques.

The bit-error-probability, $\overline{{P}_{B}}$ ­­­­, averaged through all the “ups and downs” of the fading experience in a slow-fading channel is as follows:

$\overline{{P}_{B}}=\int {P}_{B}\left(x\right)p\left(x\right)\text{dx}$

where ${P}_{B}\left(x\right)$ is the bit-error probability for a given modulation scheme at a specific value of $\text{SNR}=x$ , where $x={\alpha }^{2}{E}_{b}/{N}_{0}$ , and $p\left(x\right)$ is the $\text{pdf}$ of $x$ due to the fading conditions. With ${E}_{b}$ and ${N}_{0}$ constant, $\alpha$ is used to represent the amplitude variations due to fading.

For Rayleigh fading , $\alpha$ has a Rayleigh distribution so that ${\alpha }^{2}$ , and consequently $x$ , have a chi-squared distribution :

$p\left(x\right)=\frac{1}{\Gamma }\text{exp}\left(-\frac{x}{\Gamma }\right)$ $x\ge 0$

where $\Gamma =\overline{{\alpha }^{2}}{E}_{b}/{N}_{0}$ is the $\text{SNR}$ averaged through the “ups and downs” of fading. If each diversity (signal) branch, $i=1,\text{2,}\text{.}\text{.}\text{.}\text{,}M$ , has an instantaneous $\text{SNR}={\gamma }_{i}$ , and we assume that each branch has the same average $\text{SNR}$ given by $\Gamma$ , then

$p\left({\gamma }_{i}\right)=\frac{1}{\Gamma }\text{exp}\left(-\frac{{\gamma }_{i}}{\Gamma }\right)$ ${\gamma }_{i}\ge 0$

The probability that a single branch has $\text{SNR}$ less than some threshold $\gamma$ is:

$P\left({\gamma }_{i}\le \gamma \right)={\int }_{0}^{\gamma }p\left({\gamma }_{i}\right){\mathrm{d\gamma }}_{i}={\int }_{0}^{\gamma }\frac{1}{\Gamma }\text{exp}\left(-\frac{{\gamma }_{i}}{\Gamma }\right){\mathrm{d\gamma }}_{i}$

$=1-\text{exp}\left(-\frac{\gamma }{\Gamma }\right)$

The probability that all $M$ independent signal diversity branches are received simultaneously with an $\text{SNR}$ less than some threshold value $\gamma$ is:

$P\left({\gamma }_{1},\text{.}\text{.}\text{.},{\gamma }_{M}\le \gamma \right)={\left[1-\text{exp}\left(-\frac{\gamma }{\Gamma }\right)\right]}^{M}$

The probability that any single branch achieves $\text{SNR}>\gamma$ is:

$P\left({\gamma }_{i}>\gamma \right)=1-{\left[1-\text{exp}\left(-\frac{\gamma }{\Gamma }\right)\right]}^{M}$

This is the probability of exceeding a threshold when selection diversity is used.

Example: Benefits of Diversity

Assume that four-branch diversity is used, and that each branch receives an independently Rayleigh-fading signal. If the average $\text{SNR}$ is $\Gamma =\text{20}\text{dB}$ , determine the probability that all four branches are received simultaneously with an $\text{SNR}$ less than $\text{10}\text{dB}$ (and also, the probability that this threshold will be exceeded).

Compare the results to the case when no diversity is used.

Solution

With $\gamma =\text{10}\text{dB}$ , and $\gamma /\Gamma =\text{10}\text{dB}-\text{20}\text{dB}=-\text{10}\text{dB}=0\text{.}1$ , we solve for the probability that the

$\text{SNR}$ will drop below $\text{10}\text{dB}$ , as follows:

$P\left({\gamma }_{1},{\gamma }_{2},{\gamma }_{3},{\gamma }_{4}\le \text{10}\text{dB}\right)={\left[1-\text{exp}\left(-0\text{.}1\right)\right]}^{4}=8\text{.}2×{\text{10}}^{-5}$

or, using selection diversity, we can say that

$P\left({\gamma }_{i}>\text{10}\text{dB}\right)=1-8\text{.}2×{\text{10}}^{-5}=0\text{.}\text{9999}$

Without diversity,

$P\left({\gamma }_{1}\le \text{10}\text{dB}\right)={\left[1-\text{exp}\left(-0\text{.}1\right)\right]}^{1}=0\text{.}\text{095}$

$P\left({\gamma }_{1}>\text{10}\text{dB}\right)=1-0\text{.}\text{095}=0\text{.}\text{905}$

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