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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 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 2 logbase --> x , where x 0 . Then 2 logbase --> x y implies that 2 y x . Taking the natural logarithm on both sides we obtain

2 logbase --> x x 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 i 1 6 p X x i p X x i 6 bits/symbol

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If a soure produces binary information 0 1 with probabilities p and 1 p . The entropy of the source is

H X p 2 logbase --> p 1 p 2 logbase --> 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 .

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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 1 2 1 4 1 8 1 16 1 16 , respectively. The entropy of the random variables are

H X i 1 5 p X x i p X x i 1 2 1 2 3 8 1 4 1 4 15 8 bits/sample
There are 8000 samples per second. Therefore, the source produces 8000 15 8 15000 bits/sec of information.

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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

Do somebody tell me a best nano engineering book for beginners?
s. Reply
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Devang Reply
are you nano engineer ?
s.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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
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for screen printed electrodes ?
SUYASH
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s. Reply
of graphene you mean?
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Ebrahim
in general
s.
Graphene has a hexagonal structure
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what's the easiest and fastest way to the synthesize AgNP?
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Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
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what is the function of carbon nanotubes?
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I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
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what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
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what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
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anybody can imagine what will be happen after 100 years from now in nano tech world
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after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
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name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
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silver nanoparticles could handle the job?
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this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
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Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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how did you get the value of 2000N.What calculations are needed to arrive at it
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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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