# 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

anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
how did you get the value of 2000N.What calculations are needed to arrive at it
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