<< Chapter < Page Chapter >> Page >

Let us assume that we represent a symbol x n , with probability p n , by l n bits. Then, the average number of bits spent per symbol will be

L n 1 N p n l n
We see that this equation is equal to the entropy if the code words are selected to have the lengths l n p n . Thus, if the source produces stochastically independent outcomes with probabilities p n , such that p n is an integer, then we can easily find an optimal code as we show in the next example.

Finding a minimal representation

A four-symbol alphabet produces stochastically independent outcomes with the following probabilities. x 1 1 2 x 2 1 4 x 3 1 8 x 4 1 8 and an entropy of 1.75 bits/symbol. Let's see if we can find a codebook for this four-letter alphabet that satisfies the Source CodingTheorem. The simplest code to try is known as the simple binary code : convert the symbol's index into a binary number and use the same number of bits for each symbol byincluding leading zeros where necessary.

x 1 00 x 2 01 x 3 10 x 4 11
As all symbols are represented by 2 bits, obviously the average number of bits per symbol is 2.Because the entropy equals 1.75 bits, the simple binary code is not a minimal representation according to the source coding theorem.If we chose a codebook with differing number of bits for the symbols, a smaller average number of bits can indeed be obtained. The idea is to use shorter bit sequences for the symbols that occur more often , i.e., symbols that have a higher probability. One codebook like this is
x 1 0 x 2 10 x 3 110 x 4 111
Now L 1 1 2 2 1 4 3 1 8 3 1 8 1.75 . We can reach the entropy limit! This should come as no surprise, as promised above, when p n is an integer for all n , the optimal code is easily found.

The simple binary code is, in this case, less efficient than theunequal-length code. Using the efficient code, we can transmit the symbolic-valued signal having this alphabet 12.5%faster. Furthermore, we know that no more efficient codebook can be found because of Shannon's source coding theorem.

Got questions? Get instant answers now!

Optimality of the ascii code

Let us return to the ASCII codes presented in . Is the 7-bit ASCII code optimal, i.e., is it a minimal representation? The 7-bit ASCII code assign an equal length (7-bit) to all characters it represents. Thus, it would be optimal if all of the 128 characters wereequiprobable, that is each character should have a probability of 1 128 . To find out whether the characters really are equiprobable an analysis of all English texts would be needed. Such an analysis is difficult to do. However, the letter "E" is more probable than the letter "Z", so the equiprobable assumption does not hold, and the ASCII codeis not optimal.

(A technical note: We should take into account that in English text subsequent outcomes are not stochastically independent. To see this, assume the first letter to be "b", then it is more probable that the next letter is "e", than "z". In the case where the outcomesare not stochastically independent, the formulation we have given of Shannon's source coding theorem is no longer valid, to fix this, we should replace the entropy with the entropy rate, but we will not pursue this here).

Got questions? Get instant answers now!

Generating efficient codes

From Shannon's source coding theorem we know what the minimum average rate needed to represent a source is. But other than in the case when the logarithm of the probabilities gives an integer, we do not get any indications on how to obtain that rate. It is a large area of research to getclose to the Shannon entropy bound. One clever way to do encoding is the Huffman coding scheme.

Questions & Answers

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
I'm not sure why it wrote it the other way
I got X =-6
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
Idrissa Reply
im all ears I need to learn
right! what he said ⤴⤴⤴
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
I'm not good at math so would you help me
what is the problem that i will help you to self with?
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
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
what is system testing
what is the application of nanotechnology?
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
anybody can imagine what will be happen after 100 years from now in nano tech world
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
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
can nanotechnology change the direction of the face of the world
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.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Information and signal theory. OpenStax CNX. Aug 03, 2006 Download for free at http://legacy.cnx.org/content/col10211/1.19
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Information and signal theory' conversation and receive update notifications?