<< Chapter < Page Chapter >> Page >
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 B A required to represent the entire alphabet equals 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 B A 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 1 2 a 1 1 4 a 2 1 8 a 3 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 00 a 1 01 a 2 10 a 3 11
Whenever the number of symbols in the alphabet is a power oftwo (as in this case), the average number of bits B A equals 2 logbase --> 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 a 1 10 a 2 110 a 3 111
Now B A 1 · 1 2 2 · 1 4 3 · 1 8 3 · 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.

Got questions? Get instant answers now!

Questions & Answers

what does nano mean?
Anassong Reply
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?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
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?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
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
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
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
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
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
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
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.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
how to synthesize TiO2 nanoparticles by chemical methods
Zubear
what's the program
Jordan
?
Jordan
what chemical
Jordan
Answers please
Nikki Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Fundamentals of electrical engineering i. OpenStax CNX. Aug 06, 2008 Download for free at http://legacy.cnx.org/content/col10040/1.9
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Fundamentals of electrical engineering i' conversation and receive update notifications?

Ask