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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:
$\overline{{P}_{B}}=\int {P}_{B}(x)p(x)\text{dx}$
where ${P}_{B}(x)$ 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 $p(x)$ 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 :
$p(x)=\frac{1}{\Gamma}\text{exp}(-\frac{x}{\Gamma})$ $x\ge 0$
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, $i=\mathrm{1,}\text{2,}\text{.}\text{.}\text{.}\text{,}M$ , has an instantaneous $\text{SNR}={\gamma}_{i}$ , and we assume that each branch has the same average $\text{SNR}$ given by $\Gamma $ , then
$p({\gamma}_{i})=\frac{1}{\Gamma}\text{exp}(-\frac{{\gamma}_{i}}{\Gamma})$ ${\gamma}_{i}\ge 0$
The probability that a single branch has $\text{SNR}$ less than some threshold $\gamma $ is:
$P({\gamma}_{i}\le \gamma )={\int}_{0}^{\gamma}p({\gamma}_{i}){\mathrm{d\gamma}}_{i}={\int}_{0}^{\gamma}\frac{1}{\Gamma}\text{exp}(-\frac{{\gamma}_{i}}{\Gamma}){\mathrm{d\gamma}}_{i}$
$=1-\text{exp}(-\frac{\gamma}{\Gamma})$
The probability that all $M$ independent signal diversity branches are received simultaneously with an $\text{SNR}$ less than some threshold value $\gamma $ is:
$P({\gamma}_{1},\text{.}\text{.}\text{.},{\gamma}_{M}\le \gamma )={\left[1-\text{exp}(-\frac{\gamma}{\Gamma})\right]}^{M}$
The probability that any single branch achieves $\text{SNR}>\gamma $ is:
$P({\gamma}_{i}>\gamma )=1-{\left[1-\text{exp}(-\frac{\gamma}{\Gamma})\right]}^{M}$
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:
$P({\gamma}_{1},{\gamma}_{2},{\gamma}_{3},{\gamma}_{4}\le \text{10}\text{dB})={\left[1-\text{exp}(-0\text{.}1)\right]}^{4}=8\text{.}2\times {\text{10}}^{-5}$
or, using selection diversity, we can say that
$P({\gamma}_{i}>\text{10}\text{dB})=1-8\text{.}2\times {\text{10}}^{-5}=0\text{.}\text{9999}$
Without diversity,
$P({\gamma}_{1}\le \text{10}\text{dB})={\left[1-\text{exp}(-0\text{.}1)\right]}^{1}=0\text{.}\text{095}$
$P({\gamma}_{1}>\text{10}\text{dB})=1-0\text{.}\text{095}=0\text{.}\text{905}$
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