7.9 Diversity techniques

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:

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

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

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:

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,

$P({\gamma }_1>\text{10}\text{dB})=1-0\text{.}\text{095}=0\text{.}\text{905}$