1. Linawati Electrical Engineering Department Udayana University
Information Theory
Linawati
Electrical Engineering Department
Udayana University

2. Information Source
Measuring Information
Entropy
Source Coding
Designing Codes

3. Information Source 4 characteristics of information source
The no. of symbols, n
The symbols, S1, S2, …, Sn
The probability of occurrence of each symbol, P(S1), P(S2), …, P(Sn)
The correlation between successive symbols
Memoryless source: if each symbol is independent
A message: a stream of symbols from the senders to the receiver

4. Examples …
Ex. 1.: A source that sends binary information (streams of 0s and 1s) with each symbol having equal probability and no correlation can be modeled as a memoryless source
n = 2
Symbols: 0 and 1
Probabilities: p(0) = ½ and P(1) = ½

5. Measuring Information
To measure the information contained in a message
How much information does a message carry from the sender to the receiver?
Examples
Ex.2.: Imagine a person sitting in a room. Looking out the window, she can clearly see that the sun is shining. If at this moment she receives a call from a neighbor saying “It is now daytime”, does this message contain any information?
Ex. 3. : A person has bought a lottery ticket. A friend calls to tell her that she has won first prize. Does this message contain any information?

6. Examples …
Ex.3. It does not, the message contains no information. Why? Because she is already certain that is daytime.
Ex. 4. It does. The message contains a lot of information, because the probability of winning first prize is very small
Conclusion
The information content of a message is inversely proportional to the probability of the occurrence of that message.
If a message is very probable, it does not contain any information. If it is very improbable, it contains a lot of information

7. Symbol Information
To measure the information contained in a message, it is needed to measure the information contained in each symbol
I(s) = log2 1/P(s) bits
Bits is different from the bit, binary digit, used to define a 0 or 1
Examples
Ex.5. Find the information content of each symbol when the source is binary (sending only 0 or 1 with equal probability)
Ex. 6. Find the information content of each symbol when the source is sending four symbols with prob. P(S1) = 1/8, P(S2) = 1/8, P(S3) = ¼ ; and P(S4) = 1/2

8. Examples …
Ex. 5.
P(0) = P(1) = ½ , the information content of each symbol is
Ex.6.

9. Examples … Definition the relationships Ex.6.
The symbols S1 and S2 are least probable. At the receiver each carries more information (3 bits) than S3 or S4. The symbol S3 is less probable than S4, so S3 carries more information than S4
Definition the relationships
If P(Si) = P(Sj), then I(Si) = I(Sj)
If P(Si) < P(Sj), then I(Si) > I(Sj)
If P(Si) = 1, then I(Si) = 0

10. Message Information
If the message comes from a memoryless source, each symbol is independent and the probability of receiving a message with symbols Si, Sj, Sk, … (where i, j, and k can be the same) is:
P(message) = P(Si)P(Sj)P(Sk) …
Then the information content carried by the message is

11. Example …
Ex.7.
An equal – probability binary source sends an 8-bit message. What is the amount of information received?
The information content of the message is
I(message) = I(first bit) + I(second bit) + … + I(eight bit) = 8 bits

12. Entropy Entropy (H) of the source Example
The average amount of information contained in the symbols
H(Source) = P(S1)xI(S1) + P(S2)xI(S2) + … + P(Sn)xI(Sn)
Example
What is the entropy of an equal-probability binary source?
H(Source) = P(0)xI(0) + P(1)xI(1) = 0.5x x1 = 1 bit
1 bit per symbol

13. Maximum Entropy
For a particular source with n symbols, maximum entropy can be achieved only if all the probabilities are the same. The value of this max is
In othe words, the entropy of every source has an upper limit defined by
H(Source)≤log2n

14. Example … What is the maximum entropy of a binary source?
Hmax = log22 = 1 bit

15. Source Coding
To send a message from a source to a destination, a symbol is normally coded into a sequence of binary digits.
The result is called code word
A code is a mapping from a set of symbols into a set of code words.
Example, ASCII code is a mapping of a set of 128 symbols into a set of 7-bit code words
A ………………………..>
B …………………………>
Set of symbols ….> Set of binary streams

16. Fixed- and Variable-Length Code
A code can be designed with all the code words the same length (fixed-length code) or with different lengths (variable length code)
Examples
A code with fixed-length code words:
S1 -> 00; S2 -> 01; S3 -> 10; S4 -> 11
A code with variable-length code words:
S1 -> 0; S2 -> 10; S3 -> 11; S4 -> 110

17. Distinct Codes Each code words is different from every other code word
Example
S1 -> 0; S2 -> 10; S3 -> 11; S4 -> 110
Uniquely Decodable Codes
A distinct code is uniquely decodable if each code word can be decoded when inserted between other code words.
Not uniquely decodable
S1 -> 0; S2 -> 1; S3 -> 00; S4 -> 10 because
0010 -> S3 S4 or S3S2S1 or S1S1S4

18. Instantaneous Codes
A uniquely decodable
S1 -> 0; S2 -> 01; S3 -> 011; S4 -> 0111
A 0 uniquely defines the beginning of a code word
A uniquely decodable code is instantaneously decodable if no code word is the prefix of any other code word

19. Examples …
A code word and its prefixes (note that each code word is also a prefix of itself)
S -> ; prefixes: 0, 10, 010, 0100, 01001
A uniquely decodable code that is instantaneously decodable
S1 -> 0; s2 -> 10; s3 -> 110; s4 -> 111
When the receiver receives a 0, it immediately knows that it is S1; no other symbol starts with a 0. When the rx receives a 10, it immediately knows that it is S2; no other symbol starts with 10, and so on

20. Relationship between different types of coding
Instantaneous codes
Uniquely decodable codes
Distinct codes
All codes

21. Code … Average code length L=L(S1)xP(S1) + L(S2)xP(S2) + … Example
Find the average length of the following code:
S1 -> 0; S2 -> 10; S3 -> 110; S4 -> 111
P(S1) = ½, P(S2) = ¼; P(S3) = 1/8; P(S4) = 1/8
Solution
L = 1x ½ + 2x ¼ + 3x 1/8 + 3x1/8 = 1 ¾ bits

22. Code …
Code efficiency
 (code efficiency) is defined as the entropy of the source code divided by the average length of the code
Example
Find the efficiency of the following code:
S1 ->0; S2->10; S3 -> 110; S4 -> 111
P(S1) = ½, P(S2) = ¼; P(S3) = 1/8; P(S4) = 1/8
Solution

23. Designing Codes Two examples of instantaneous codes
Shannon – Fano code
Huffman code
An instantaneous variable – length encoding method in which the more probable symbols are given shorter code words and the less probable are given longer code words
Design builds a binary tree top (top to bottom construction) following the steps below:
1. List the symbols in descending order of probability
2. Divide the list into two equal (or nearly equal) probability sublists. Assign 0 to the first sublist and 1 to the second
3. Repeat step 2 for each sublist until no further division is possible

24. Example of Shannon – Fano Encoding
Find the Shannon – Fano code words for the following source
P(S1) = 0.3 ; P(S2) = 0.2 ; P(S3) = 0.15 ; P(S4) = 0.1 ; P(S5) = 0.1 ; P(S6) = 0.05 ; P(S7) = 0.05 ; P(S8) = 0.05
Solution
Because each code word is assigned a leaf of the tree, no code word is the prefix of any other. The code is instantaneous. Calculation of the average length and the efficiency of this code
H(source) = 2.7
L = 2.75
 = 98%

25. Example of Shannon – Fano Encoding
0.30 S2
0.20 S3
0.15 S4
0.10 S5 S6
0.05 S7 S8 1 00 01
100
101
1100
1101
1110
1111

26. Huffman Encoding
An instantaneous variable – length encoding method in which the more probable symbols are given shorter code words and the less probable are given longer code words
Design builds a binary tree (bottom up construction):
1. Add two least probable symbols
2. Repeat step 1 until no further combination is possible

27. Example Huffman encoding
Find the Huffman code words for the following source
P(S1) = 0.3 ; P(S2) = 0.2 ; P(S3) = 0.15 ; P(S4) = 0.1 ; P(S5) = 0.1 ; P(S6) = 0.05 ; P(S7) = 0.05 ; P(S8) = 0.05
Solution
Because each code word is assigned a leaf of the tree, no code word is the prefix of any other. The code is instantaneous. Calculation of the average length and the efficiency of this code
H(source) = 2.70 ; L = 2.75 ;  = 98%

28. Example Huffman encoding
1.00 1
0.30
0.20
0.15
0.10
0.05 S1 00 S2 10 S3
010 S4
110 S5
111 S6
0110 S7
01110 S8
01111
0.60
0.40
0.3
0.15
0.20
0.10