# 5.3 Entropy

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Entropy

The self information gives the information in a single outcome. In most cases, e.g in data compression, it is much moreinteresting to know the average information content of a source. This average is given by the expected value of the self information with respect to the source's probabilitydistribution. This average of self information is called the source entropy.

## Definition of entropy

Entropy
If symbol has zero probability, which means it never occurs, it should not affect the entropy. Letting $0\lg 0=0$ , we have dealt with that.

In texts you will find that the argument to the entropy function may vary. The two most common are $H(X)$ and $H(p)$ . We calculate the entropy of a source X, but the entropy is,strictly speaking, a function of the source's probabilty function p. So both notations are justified.

## Calculating the binary logarithm

Most calculators does not allow you to directly calculate the logarithm with base 2, so we have to use a logarithm base that mostcalculators support. Fortunately it is easy to convert between different bases.

Assume you want to calculate $\log_{2}x$ , where $x> 0$ . Then $\log_{2}x=y$ implies that $2^{y}=x$ . Taking the natural logarithm on both sides we obtain

$\log_{2}x=\frac{\ln x}{\ln 2}$

## Examples

When throwing a dice, one may ask for the average information conveyed in a single throw. Using the formula for entropy we get $H(X)=-\sum_{i=1}^{6} {p}_{X}({x}_{i})\lg {p}_{X}({x}_{i})=\lg 6\mathrm{bits/symbol}$

If a soure produces binary information $\{0, 1\}$ with probabilities $p$ and $1-p$ . The entropy of the source is

$H(X)=-(p\log_{2}p)-(1-p)\log_{2}(1-p)$
If $p=0$ then $H(X)=0$ , if $p=1$ then $H(X)=0$ , if $p=1/2$ then $H(X)=1$ . The source has its largest entropy if $p=1/2$ and the source provides no new information if $p=0$ or $p=1$ .

An analog source is modeled as a continuous-time random process with power spectral density bandlimited to the bandbetween 0 and 4000 Hz. The signal is sampled at the Nyquist rate. The sequence of random variables, as a result ofsampling, are assumed to be independent. The samples are quantized to 5 levels $\{-2, -1, 0, 1, 2\}$ . The probability of the samples taking the quantized values are $\{\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{16}\}$ , respectively. The entropy of the random variables are

$H(X)=-\sum_{i=1}^{5} {p}_{X}({x}_{i})\lg {p}_{X}({x}_{i})=\frac{1}{2}+\frac{1}{2}+\frac{3}{8}+\frac{1}{4}+\frac{1}{4}=\frac{15}{8}\mathrm{bits/sample}$
There are 8000 samples per second. Therefore, the source produces $8000\frac{15}{8}=15000$ bits/sec of information.

Entropy is closely tied to source coding. The extent to which a source can be compressed is related to its entropy.There are many interpretations possible for the entropy of a random variable, including

• (Average)Self information in a random variable
• Minimum number of bits per source symbol required to describe the random variable without loss
• Description complexity
• Measure of uncertainty in a random variable

## References

• ien, G.E. and Lundheim,L. (2003) Information Theory, Coding and Compression , Trondheim: Tapir Akademisk forlag.

#### Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
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
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
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
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Information and signal theory. OpenStax CNX. Aug 03, 2006 Download for free at http://legacy.cnx.org/content/col10211/1.19
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