EE-317: Information Theory
Fall 2008
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Organizational Details
Class Meeting:
7:00-9:45pm, Tuesday, Room SCIT213
Instructor: Dr. Igor Aizenberg
Office: Science and Technology Building, 115
Phone (903 334 6654)
e-mail: igor.aizenberg@tamut.edu
Office hours:
Wednesday, Thursday, Friday 12-30 – 2-00 pm
Monday: by appointment
Class Web Page: http://www.eagle.tamut.edu/faculty/igor/EE-317.htm
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Dr. Igor Aizenberg: self-introduction
• MS in Mathematics from Uzhgorod National University (Ukraine),
1982
• PhD in Computer Science from the Russian Academy of Sciences,
Moscow (Russia), 1986
• Areas of research: Artificial Neural Networks, Image Processing and
Pattern Recognition
• About 100 journal and conference proceedings publications and
one monograph book
• Job experience: Russian Academy of Sciences (1982-1990);
Uzhgorod National University (Ukraine,1990-1996 and 1998-1999);
Catholic University of Leuven (Belgium, 1996-1998); Company
“Neural Networks Technologies” (Israel, 1999-2002); University of
Dortmund (Germany, 2003-2005); National Center of Advanced
Industrial Science and Technologies (Japan, 2004); Tampere
University of Technology (Finland, 2005-2006); Texas A&M
University-Texarkana, from March, 2006
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Text Books
• [1] "An Introduction to Information
Theory" by Fazlollah M. Reza, Dover
Publications, 1994, ISBN 0-486-68210-2
(main book).
• [2]“An Introduction to Information
Theory" by John R. Pierce, Dover
Publications, 1980, ISBN 0-486—24061-4
(supportive book).
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Control
Exams (open book, open notes):
Midterm 1:
Midterm 2:
Final Exam:
October 7, 2007
November 4, 2007
December 9, 2007
Homework
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Grading
Grading Method
Homework and preparation:
Midterm Exam 1:
Midterm Exam 2:
Final Exam:
10%
25%
30%
35%
Grading Scale:
90%+  A
80%+  B
70%+  C
60%+  D
less than 60%  F
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What we will study?
• Basic concepts of information theory
• Elements of sets theory and probability
theory
• Measure of information and uncertainty.
Entropy
• Basic concepts of communication channels
organization
• Basic principles of encoding
• Error-detecting and error-correcting codes
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Information
• What does a word “information” mean?
• There is no some exact definition, however:
• Information carries new specific knowledge,
which is definitely new for its recipient;
• Information is always carried by some specific
carrier in different forms (letters, digits, different
specific symbols, sequences of digits, letters, and
symbols , etc.);
• Information is meaningful only if the recipient is
able to interpret it.
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Information
• According to the Oxford English
Dictionary, the earliest historical meaning
of the word information in English was the
act of informing, or giving form or shape to
the mind.
• The English word was apparently derived
by adding the common "noun of action"
ending "-ation”
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Information
• The information materialized is a message.
• Information is always about something (size of a
parameter, occurrence of an event, etc).
• Viewed in this manner, information does not have
to be accurate; it may be a truth or a lie.
• Even a disruptive noise used to inhibit the flow of
communication and create misunderstanding
would in this view be a form of information.
• However, generally speaking, if the amount of
information in the received message increases,
the message is more accurate.
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Information Theory
• How we can measure the amount of
information?
• How we can ensure the correctness of
information?
• What to do if information gets corrupted by
errors?
• How much memory does it require to store
information?
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Information Theory
• Basic answers to these questions that formed
a solid background of the modern information
theory were given by the great American
mathematician, electrical engineer, and
computer scientist Claude E. Shannon in his
paper “A Mathematical Theory of
Communication” published in “The Bell
System Technical Journal” in October, 1948.
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Claude Elwood Shannon (1916-2001)
The father of information theory
The father of practical digital circuit design theory
Bell Laboratories (1941-1972), MIT(1956-2001)
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Information Content
• What is the information content of any
message?
• Shannon’s answer is: The information content
of a message consists simply of the number of
1s and 0s it takes to transmit it.
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Information Content
• Hence, the elementary unit of information is a
binary unit: a bit, which can be either 1 or 0;
“true” or “false”; “yes” or “know”, “black” and
“white”, etc.
• One of the basic postulates of information
theory is that information can be treated like a
measurable physical quantity, such as density
or mass.
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Information Content
• Suppose you flip a coin one million times and
write down the sequence of results. If you want
to communicate this sequence to another
person, how many bits will it take?
• If it's a fair coin, the two possible outcomes,
heads and tails, occur with equal probability.
Therefore each flip requires 1 bit of information
to transmit. To send the entire sequence will
require one million bits.
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Information Content
• Suppose the coin is biased so that heads occur only 1/4 of the time,
and tails occur 3/4. Then the entire sequence can be sent in
811,300 bits, on average This would seem to imply that each flip of
the coin requires just 0.8113 bits to transmit.
• How can you transmit a coin flip in less than one bit, when the only
language available is that of zeros and ones?
• Obviously, you can't. But if the goal is to transmit an entire
sequence of flips, and the distribution is biased in some way, then
you can use your knowledge of the distribution to select a more
efficient code.
• Another way to look at it is: a sequence of biased coin flips contains
less "information" than a sequence of unbiased flips, so it should
take fewer bits to transmit.
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Information Content
• Information Theory regards information as only
those symbols that are uncertain to the receiver.
• For years, people have sent telegraph messages,
leaving out non-essential words such as "a" and
"the."
• In the same vein, predictable symbols can be left
out, like in the sentence, "only infrmatn esentil to
understandn mst b tranmitd”. Shannon made
clear that uncertainty is the very commodity of
communication.
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Information Content
• Suppose we transmit a long sequence of one
million bits corresponding to the first example.
What should we do if some errors occur
during this transmission?
• If the length of the sequence to be
transmitted or stored is even larger that 1
million bits, then 1 billion bits… what should
we do?
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Two main questions of
Information Theory
• What to do if information gets corrupted by
errors?
• How much memory does it require to store
data?
• Both questions were asked and to a large
degree answered by Claude Shannon in his
1948 seminal article:
use error correction and data compression
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Shannon’s basic principles of
Information Theory
• Shannon’s theory told engineers how much
information could be transmitted over the
channels of an ideal system.
• He also spelled out mathematically the principles
of data compression, which recognize what the
end of this sentence demonstrates, that “only
infrmatn esentil to understadn mst b tranmitd”.
• He also showed how we could transmit
information over noisy channels at error rates we
could control.
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Why is the Information Theory Important?
• Thanks in large measure to Shannon's insights,
digital systems have come to dominate the world
of communications and information processing.
–
–
–
–
–
Modems
satellite communications
Data storage
Deep space communications
Wireless technology
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Channels
• A channel is used to get information
across:
Source
0,1,1,0,0,1,1
Receiver
binary channel
Many systems act like channels.
Some obvious ones: phone lines, Ethernet cables.
Less obvious ones: the air when speaking, TV screen
when watching, paper when writing an article, etc.
All these are physical devices and hence prone to errors.
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Noisy Channels
A noiseless binary channel
transmits bits without error:
A noisy, symmetric binary
channel applies a bit-flip
01 with probability p:
0
0
0
1
1
1–p
p
0
p
1
1–p
1
What to do if we have a noisy channel and
you want to send information across reliably?
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Error Correction pre-Shannon
• Primitive error correction (assume p<1/2):
Instead of sending “0” and “1”, send “0…0” and “1…1”.
• The receiver takes the majority of the bit values
as the ‘intended’ value of the sender.
• Example: If we repeat the bit value three times,
the error goes down from p to p2(3–2p).
Hence for p=0.1 we reduce the error to 0.028.
• However, now we have to send 3 bits to get one
bit of information across and this will get worse if
we want to reduce the error rate further…
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Channel Rate
• When correcting errors, we have to be mindful of
the rate of the bits that you use to encode one bit
(in the previous example we had rate 1/3).
• For the primitive encoding in the previous example
with 00r and 11r with rate 1/r, the error goes
down approximately as p rpr–1.
• If we want to send data with arbitrarily small errors,
then this requires arbitrarily low rates r, which is
costly.
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Error Correction by Shannon
• Shannon’s basic observations:
• Correcting single bits is very wasteful and inefficient;
• Instead we should correct blocks of bits.
• We will see later that by doing so we can get arbitrarily
small errors for the constant channel rate 1–H(p)
where H(p) is the Shannon entropy,
defined by H(p) = –p log2(p) – (1–p) log2(1–p).
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A model for a Communication System
• The communications systems are of a statistical
nature.
• That is, the performance of the system can never
be described in a deterministic sense; it is always
given in statistical terms.
• A source is a device that selects and transmits
sequences of symbols from a given alphabet.
• Each selection is made at random, although this
selection may be based on some statistical rule.
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A model for a Communication System
• The channel transmits the incoming symbols
to the receiver. The performance of the
channel is also based on laws of chance.
• If the source transmits a symbol A, with a
probability of P{A} and the channel lets
through the letter A with a probability
denoted by P{A|A}, then the probability of
transmitting A and receiving A is P{A}∙P{A|A}
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A model for a Communication System
• The channel is generally lossy: a part of the
transmitted content does not reach its
destination or it reaches the destination in a
distorted form.
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A model for a Communication System
• A very important task is the minimization of
the loss and the optimum recovery of the
original content when it is corrupted by the
effect of noise.
• A method that is used to improve the
efficiency of the channel is called encoding.
• An encoded message is less sensitive to noise.
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A model for a Communication System
• Decoding is employed to transform the
encoded messages into the original form,
which is acceptable to the receiver.
Encoding: F: I  F(I)
Decoding: F-1: F(I)  I
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A Quantitative Measure of Information
• Suppose we have to select some equipment
from a catalog which indicates n distinct
models: x1 , x2 ,..., xn 
• The desired amount of information I ( xk )
associated with the selection of a particular
model xk must be a function of the
probability of choosing xk :
I ( xk )  f  P xk 
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A Quantitative Measure of Information
• If, for simplicity, we assume that each one of
these models is selected with an equal
probability, then the desired amount of
information is only a function of n:
I1  xk   f 1/ n 
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A Quantitative Measure of Information
• If each piece of equipment can be ordered in
one of m different colors and the selection of
colors is also equiprobable, then the amount
of information associated with the selection of
a color c j is :


I 2  c j   f P c j   f 1/ m 
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A Quantitative Measure of Information
• The selection may be done in two ways:
• Select the equipment and then the color
independently of each other
I  xk & c j   I1 ( xk )  I1 (c j )  f 1/ n   f 1/ m 
• Select the equipment and its color at the same
time as one selection from mn possible
choices:
I  xk & c j   f 1/ mn 
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A Quantitative Measure of Information
• Since these amounts of information are identical,
we obtain:
f 1/ n   f 1/ m  f 1/ mn 
• Among several solutions of this functional
equation, the most important for us is:
f  x    log  x 
• Thus, when a statistical experiment has n
eqiuprobable outcomes, the average amount of
information associated with an outcome is log n
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A Quantitative Measure of Information
• The logarithmic information measure has the
desirable property of additivity for
independent statistical experiments.
• The simplest case to consider is a selection
between two eqiuprobable events. The
amount of information associated with the
selection of one out of two equiprobable
events is  log 2 1/ 2  log 2 2  1 and provides a
unit of information known as a bit.
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