# 6.20 Source coding theorem

 Page 1 / 1
The Source Coding Theorem states that the entropy of an alphabet of symbols specifies to within one bit how many bits on the average need to be used to send the alphabet.

The significance of an alphabet's entropy rests in how we can represent it with a sequence of bits . Bit sequences form the "coin of the realm" in digitalcommunications: they are the universal way of representing symbolic-valued signals. We convert back and forth betweensymbols to bit-sequences with what is known as a codebook : a table that associates symbols to bit sequences. In creating this table, we must be able to assign a unique bit sequence to each symbol so that we can go between symbol and bit sequences without error.

You may be conjuring the notion of hiding information from others when we use the name codebook for thesymbol-to-bit-sequence table. There is no relation to cryptology, which comprises mathematically provable methods ofsecuring information. The codebook terminology was developed during the beginnings of information theory just after WorldWar II.

As we shall explore in some detail elsewhere, digital communication is the transmission of symbolic-valued signals from one place toanother. When faced with the problem, for example, of sending a file across the Internet, we must first represent eachcharacter by a bit sequence. Because we want to send the file quickly, we want to use as few bits as possible. However, wedon't want to use so few bits that the receiver cannot determine what each character was from the bit sequence. Forexample, we could use one bit for every character: File transmission would be fast but useless because the codebookcreates errors. Shannon proved in his monumental work what we call today the Source Coding Theorem . Let $B({a}_{k})$ denote the number of bits used to represent the symbol ${a}_{k}$ . The average number of bits $\langle B(A)\rangle$ required to represent the entire alphabet equals $\sum_{k=1}^{K} B({a}_{k})({a}_{k})$ . The Source Coding Theorem states that the average number of bits needed to accurately represent the alphabet need only to satisfy

$H(A)\le \langle B(A)\rangle < H(A)+1$
Thus, the alphabet's entropy specifies to within one bit how many bits on the average need to be used to send the alphabet.The smaller an alphabet's entropy, the fewer bits required for digital transmission of files expressed in that alphabet.

A four-symbol alphabet has the following probabilities. $({a}_{0})=\frac{1}{2}$ $({a}_{1})=\frac{1}{4}$ $({a}_{2})=\frac{1}{8}$ $({a}_{3})=\frac{1}{8}$ and an entropy of 1.75 bits . 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.

$↔({a}_{0}, \mathrm{00})\text{}↔({a}_{1}, \mathrm{01})\text{}↔({a}_{2}, \mathrm{10})\text{}↔({a}_{3}, \mathrm{11})$
Whenever the number of symbols in the alphabet is a power oftwo (as in this case), the average number of bits $\langle B(A)\rangle$ equals $\log_{2}K$ , which equals $2$ in this case. Because the entropy equals $1.75$ bits, the simple binary code indeed satisfies the Source Coding Theorem—we arewithin one bit of the entropy limit—but you might wonder if you can do better. If we choose a codebook with differingnumber of bits for the symbols, a smaller average number of bits can indeed be obtained. The idea is to use shorter bitsequences for the symbols that occur more often. One codebook like this is
$↔({a}_{0}, 0)\text{}↔({a}_{1}, \mathrm{10})\text{}↔({a}_{2}, \mathrm{110})\text{}↔({a}_{3}, \mathrm{111})$
Now $\langle B(A)\rangle =1·\frac{1}{2}+2·\frac{1}{4}+3·\frac{1}{8}+3·\frac{1}{8}=1.75$ . We can reach the entropy limit! 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 Theorem.

what is variations in raman spectra for nanomaterials
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
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
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
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
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 did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with        By By By