# 7.9 Diversity techniques

 Page 1 / 1

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,

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

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

#### Questions & Answers

how can chip be made from sand
Eke Reply
are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

### Read also:

#### Get Jobilize Job Search Mobile App in your pocket Now!

Source:  OpenStax, Principles of digital communications. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10805/1.1
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Principles of digital communications' conversation and receive update notifications?

 By OpenStax By Richley Crapo By Ryan Lowe By Zarina Chocolate By Katy Keilers By Brooke Delaney By OpenStax By OpenStax By Rachel Woolard By David Martin