1. EEET 5101 Information Theory Chapter 1
Introduction
Probability Theory
BY Wai (W2-4)

2. Basic Course Information
Lecturers:
Dr Siu Wai Ho, W2-4, Mawson Lakes
Dr Badri Vellambi Ravisankar, W1-22, Mawson Lakes
Dr Roy Timo, W1-7, Mawson Lakes
Office Hour: Tue 2:00-5:00pm (starting from 27/7/2010)
Class workload:
Homework Assignment 25%
Mid-term 25%
Final 50%
Textbook: T. M. Cover and J. M. Thomas, Elements of Information Theory, 2nd, Wiley-Interscience, 2006.

3. Basic Course Information
References:
OTHER RELEVANT TEXTS (Library):
1. Information Theory and Network Coding by Raymond Yeung
2. Information Theory: Coding Theorems for Discrete Memoryless Systems by Imre Csiszar and Janos Korner.
OTHER RELEVANT TEXTS (Online):
3. Probability, Random Processes, and Ergodic Properties by Robert Gray
4. Introduction to Statistical Signal Processing Robert Gray and L. Davisson
5. Entropy and Information Theory by Robert Gray

4. The Beginning of Information Theory
In 1948, Claude E. Shannon published his paper “A Mathematical Theory of Communication” in the Bell Systems Technical Journal.
He introduced two fundamental concepts about “information”:
Information can be measured by entropy
Information to be transmitted is digital
Information
Source
Transmitter
Message
Receiver
Destination
Signal
Received
Noise Source

5. The Beginning of Information Theory
In the same paper, he has answered two fundamental questions in communication theory:
What is the ultimate data compression ?
How to minimize the compression rate m/n with Pr{u  v} = 0.
What is the ultimate transmission rate of communication?
How to maximize the transmission rate n/m with Pr{k  k’}  0.
Source
Receiver
Encoder
Decoder
u = u1 … un
v = v1 … vn
x1 … xm
k  {1,…,2n}
x1 … xm
y1 … ym k’
Source
Receiver
Encoder
Channel
Decoder

6. The Idea of Channel Capacity
Example [MacKay 2003]: Suppose we are now provided a noisy channel
We test it times and find the following statistics
Pr{y=0|x=0} = Pr{y=1|x=1} = 0.9; Pr{y=0|x=1} = Pr{y=1|x=0} = 0.1
The occurrence of difference is independent of the previous use
Suppose we want to send a message: s =
The error probability = 1 – Pr{no error} = 1 – 0.97 
How can we get a smaller error probability?
Channel x y
0.9
0.1 1

7. The Idea of Channel Capacity
Method 1: Repetition codes
[R3] To replace the source message by 0  000; 1  111
The original bit error probability Pb : The new Pb : = 3  0.9  = 0.028
bit error probability  0  rate  0 ?? s 1 t
000
111 n
001
101
r=tn
010 s’
t: transmitted symbols
n: noise
r: received symbols
Majority vote at the receiver

8. The Idea of Channel Capacity
Method 1: Repetition codes
pb  0  rate  0

9. The Idea of Channel Capacity
Method 2: Hamming codes
[(7,4) Hamming Code] group 4 bits into s. E.g., s =
Here t = GTs = , where

10. The Idea of Channel Capacity
Method 2: Hamming codes
Is the search of a good code an everlasting job? Where is the destination?

11. The Idea of Channel Capacity
Information theory tells us the fundamental limits.
It is impossible to design a code with coding rate and error probability on the right side of the line.
Shannon’ s
Channel Coding
Theorem

12. Intersections with other Fields
Information theory shows the fundamental limits in different communication systems
It also provides insights on how to achieve these limits
It also intersects other fields [Cover and Thomas 2006]

13. Content in this course 2) Information Measures and Divergence:
2a) Entropy, Mutual Information and Kullback-Leibler Divergence
-Definitions, chain rules, relations
2b) Basic Lemmas & Inequalities:
-Data Processing Inequality, Fano’s Inequality.
3) Asymptotic Equipartition Property (AEP) for iid Random Processes:
3a) Weak Law of Large Numbers
3b) AEP as a consequence of the Weak Law of Large Numbers
3c) Tail event bounding:
-Markov, Chebychev and Chernoff bounds
3d) Types and Typicality
-Strong and weak typicality
3e) The lossless source coding theorem

14. Content in this course 4) The AEP for Non-iid Random Processes:
4a) Random Processes with memory
-Markov processes, stationarity and ergodicity
4b) Entropy Rate
4c) The lossless source coding theorem
5) Lossy Compression:
5a) Motivation
5b) Rate-distortion (RD) theory for DMSs (Coding and Converse theorems).
5c) Computation of the RD function (numerical and analytical)
How to minimize the compression rate m/n with u and v satisfying certain distortion criteria.
Source
Receiver
Encoder
Decoder
u = u1 … un
v = v1 … vn
x1 … xm

15. Content in this course 6) Reliable Communication over Noisy Channels:
6a) Discrete memoryless channels
-Codes, rates, redundancy and reliable communication
6b) Shannon’s channel coding theorem and its converse
6c) Computation of channel capacity (numerical and analytical)
6d) Joint source-channel coding and the principle of separation
6e) Dualities between channel capacity and rate-distortion theory
6f) Extensions of Shannon’s capacity to channels with memory (if time permits)

16. Content in this course
7) Lossy Source Coding and Channel Coding with Side-Information:
7a) Rate Distortion with Side Information
-Joint and conditional rate-distortion theory, Wyner-Ziv coding, extended Shannon lower bound, numerical computation
7b) Channel Capacity with Side Information
7c) Dualities
8) Introduction to Multi-User Information Theory (If time permits):
Possible topics: lossless and lossy distributed source coding, multiple access channels, broadcast channels, interference channels, multiple descriptions, successive refinement of information, and the failure of source-channel separation.

17. Prerequisites – Probability Theory
Let X be a discrete random variable taking values from the alphabet 
The probability distribution of X is denoted by pX = {pX(x), x  X}, where
pX(x) means the probability that X = x.
pX(x)  0
x pX(x) = 1
Let SX be the support of X, i.e. SX = {x  X: p(x) > 0}.
Example :
Let X be the outcome of a dice
Let  = {1, 2, 3, 4, 5, 6, 7, 8, 9, …} equal to all positive integers. In this case,  is a countably infinite alphabet
SX = {1, 2, 3, 4, 5, 6} which is a finite alphabet
If the dice is fair, then pX(1) = pX(2) =  = pX(6) = 1/6.
If  is a subset of real numbers, e.g.,  = [0, 1],  is a continuous alphabet and X is a continuous random variable

18. Prerequisites – Probability Theory
Let X and Y be random variable taking values from the alphabet X and Y, respectively
The joint probability distribution of X and Y is denoted by pXY and
pXY(xy) means the probability that X = x and Y = y
pX(x), pY(y), pXY(xy)  p(x), p(y), p(xy) when there is no ambiguity.
pXY(x)  0
xy pXY(x) = 1
Marginal distributions: pX(x) = y pXY(xy) and pY(y) = x pXY(xy)
Conditional probability: for pX(x) > 0, pY|X(y|x) = pXY(xy)/ pX(x) which denotes the probability that Y = y given the conditional that X = x
Consider a function f: X  Y
If X is a random variable, f(X) is also random. Let Y = f(X).
E.g., X is the outcome of a fair dice and f(X) = (X – 3.5)2
What is pXY?
PY|X X Y

19. Expectation and Variance
The expectation of X is given by E[X] = x pX(x)  x
The variance of X is given by E[(X – E[X])2] = E[X2] – (E[X])2
The expected value of f(X) is E[f(X)] = x pX(x)  f(x)
The expected value of k(X, Y) is E[k(X, Y)] = xy pXY(xy)  k(x,y)
We can take the expectation on only Y, i.e., EY[k(X, Y)] = y pY(y)  k(X,y) which is still a random variable
E.g., Suppose some real-valued functions f, g, k and l are given.
What is E[f(X, g(Y), k(X,Y))l(Y)]?
xy pXY(xy) f(x, g(y), k(x,y))l(y) which gives a real value
What is EY[f(X, g(Y), k(X,Y)]l(Y)?
y pY(y) f(X, g(y), k(X,y))l(y) which is still a random variable.
Usually, this can be done only if X and Y are independent.

20. Conditional Independent
Two r.v. X and Y are independent if p(xy) = p(x)p(y) x, y
For r.v. X, Y and Z, X and Z are independent conditioning on Y, denoted by X  Z | Y if
p(xyz)p(y) = p(xy)p(yz) x, y, z (1)
Assume p(y) > 0,
p(x, z|y) = p(x|y)p(z|y) x, y, z (2)
If (1) is true, then (2) is also true given p(y) > 0
If p(y) = 0, p(x, z|y) may be undefined for a given p(x, y, z).
Regardless whether p(y) = 0 for some y, (1) is a sufficient condition to test X  Z | Y
p(xy) = p(x)p(y) is also called pairwise independent

21. Mutual and Pairwise Independent
Mutual Indep. : p(x1, x2, …, xn) = p(x1)p(x2)    p(xn)
Mutual Independent  Pairwise Independent
Suppose we have i, j s.t. i, j [1, n] and i  j
Let a = [1, n] \ {i, j}
Pairwise Independent  Mutual Independent

22. Mutual and Pairwise Independent
Example : Z = X  Y and Pr{X=0} = Pr{X=1} = Pr{Y=0} = Pr{Y=1} = 0.5
Pr{Z=0} = Pr{X=0}Pr{Y=0} + Pr{X=1}Pr{Y=1} = 0.5
Pr{Z=1} = 0.5
Pr{X=0, Y=0} = 0.25 = Pr{X=0}Pr{Y=0}
Pr{X=0, Z=1} = 0.25 = Pr{X=0}Pr{Z=1}
Pr{Y=1, Z=1} = 0.25 = Pr{Y=1}Pr{Z=1} ……..
So X, Y and Z are pairwise Independent
However, Pr{X=0, Y=0, Z=0} = Pr{X=0}Pr{Y=0} = 0.25
Pr{X=0}Pr{Y=0}Pr{Z=0} = 0.125
X, Y and Z are not mutually Independent but pairwise Independent