# Huffman source coder

 Page 1 / 1
This example shows how a Huffman coder allocates variable length codewords to the transmitted symbols depending on their probability of occurence.

## Source coding

Huffman coding deploys variable length coding and then allocates the longer codewords to less frequently occurring symbols and shorter codewords to more regularly occurring symbols. By using this technique it can minimize the overall transmission rate as the regularly occurring symbols are allocated the shorter codewords.

## Simple source coding

 Symbol Probability A 0.10 B 0.18 C 0.40 D 0.05 E 0.06 F 0.10 G 0.07 H 0.04

We have to start with knowledge of the probabilities of occurrence of all the symbols in the alphabet. The table above shows an example of an 8-symbol alphabet, A…H, with the associated probabilities for each of the eight individual symbols.

[link] shows that the entropy of this source data is 2.5524 bits/symbol.

 Symbol Code A 000 B 001 C 010 D 011 E 100 F 101 G 110 H 111

This shows the application of very simple coding where, as there are 8 symbols, we adopt a 3-bit code. [link] shows that the entropy of such a source is 2.5524 bit/symbol and, with the fixed 3 bit/symbol length allocated codewords, the efficiency of this simple coder would be only 2.5524/3.0 = 85.08%, which is a rather poor result.

## Huffman coding

This is a variable length coding technique which involves two processes, reduction and splitting.

## Reduction

We start by listing the symbols in descending order of probability, with the most probable symbol, C, at the top and the least probable symbol, H, at the foot, see left hand side of [link] . Next we reduce the two least probable symbols into a single symbol which has the combined probability of these two symbols summed together. Thus symbols H and D are combined into a single (i.e. reduced) symbol with probability 0.04 + 0.05 = 0.09.

Now the symbols have to be reordered again in descending order of probability. As the probability of the new H+D combined symbol (0.09) is no longer the smallest value it then moves up the reordered list as shown in the second left column in [link] .

This process is progressively repeated as shown in [link] until all symbols are combined into a single symbol whose probability must equal 1.00.

## Splitting

The variable length codewords for each transmitted symbol are now derived by working backwards (from the right) through the tree structure created in [link] , by assigning a 0 to the upper branch of each combining operation and a 1 to the lower branch.

The final “combined symbol” of probability 1.00 is thus split into two parts of probability 0.60 with assigned digit of 0 and another part with probability 0.40 with assigned digit of 1. This latter part with probability 0.40 and assigned digit of 1 actually represents symbol C, [link] .

The “combined symbol” with probability 0.60 (and allocated first digit of 0) is now split into two further parts with probability 0.37 with an additional or second assigned digit of 0 (i.e. its code is now 00) and another part with the remaining probability 0.23 where the additional assigned digit is 1 and associated code will now be 01. Huffman coder splitting process to generate the variable length codewords and allocate these depending on symbol probabilities.

This process is repeated by adding each new digit after the splitting operation to the right of the previous one. Note how this allocates short codes to the more probable symbols and longer codes to the less probable symbols, which are transmitted less often.

 Symbol Code A 011 B 001 C 1 D 00010 E 0101 F 0000 G 0100 H 00011

## Code efficiency

[link] summarises the codewords now allocated to each of the transmitted symbols A…H and also calculates the average length of this source coder as 2.61 bits/symbol. Note the considerable reduction from the fixed length of 3 in the simple 3-bit coder in earlier table. Summary of allocated codewords for each symbol, A ...H, and calculation of average length of transmitted codeword.

Now recall from [link] that the entropy of the source data was 2.5524 bits/symbol and the simple fixed length 3-bit code in the earlier table, with a length of 3.00 which gave an efficiency of only 85.08%.

The efficiency of the Huffman coded data with its variable length codewords is therefore 2.5524/2.62 = 97.7% which is a much more acceptable result.

If the symbol probabilities all have values 1/( ${2}^{n}$ ) which are integer powers of 2 then Huffmann coding will result in 100% efficiency.

This module has been created from lecture notes originated by P M Grant and D G M Cruickshank which are published in I A Glover and P M Grant, "Digital Communications", Pearson Education, 2009, ISBN 978-0-273-71830-7. Powerpoint slides plus end of chapter problem examples/solutions are available for instructor use via password access at http://www.see.ed.ac.uk/~pmg/DIGICOMMS/

are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
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
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
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
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
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?
what is a peer
What is meant by 'nano scale'?
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 ?
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?
what king of growth are you checking .?
Renato
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!

#### Get Jobilize Job Search Mobile App in your pocket Now! By OpenStax By John Gabrieli By OpenStax By OpenStax By Jazzycazz Jackson By OpenStax By Yasser Ibrahim By OpenStax By David Martin By Madison Christian