ee-317: information theory fall 2008 1. organizational details class meeting: 7 :00-9:45pm, tuesday,...

EE-317: Information Theory

Fall 2008

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Organizational DetailsClass Meeting: 7:00-9:45pm, Tuesday, Room SCIT213

Instructor: Dr. Igor Aizenberg

Office: Science and Technology Building, 115Phone (903 334 6654)e-mail: [email protected]

Office hours:Wednesday, Thursday, Friday 12-30 – 2-00 pmMonday: by appointment

Class Web Page: http://www.eagle.tamut.edu/faculty/igor/EE-317.htm

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http://www.eagle.tamut.edu/faculty/igor/EE-305.htm

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: October 7, 2007

Midterm 2: November 4, 2007

Final Exam: December 9, 2007

Homework

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Grading

Grading MethodHomework and preparation: 10%Midterm Exam 1: 25%Midterm Exam 2: 30%Final Exam: 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 theoryThe father of practical digital circuit design theory

Bell Laboratories (1941-1972), MIT(1956-2001)13

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 ofInformation 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

Channels• A channel is used to get information

across:Source

binary channel

0,1,1,0,0,1,1 Receiver

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:

0 0

1 1

A noisy, symmetric binary channel applies a bit-flip 01 with probability p:

0 0

1 1

1–p

1–p

p

p

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 onebit of information across and this will get worse ifwe want to reduce the error rate further…

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Channel Rate

• When correcting errors, we have to be mindful ofthe 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 00r and 11r 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 arbitrarilysmall 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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• 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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• 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 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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• 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:

• The desired amount of information associated with the selection of a particular model must be a function of the probability of choosing :

1 2, ,..., nx x x

( )kI x

kx

kx

( )k kI x f P x

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• 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:

1 1/kI x f n

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• 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 is :

2 1/j jI c f P c f m

jc

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• The selection may be done in two ways: • Select the equipment and then the color

independently of each other

• Select the equipment and its color at the same time as one selection from mn possible choices: & 1/k jI x c f mn

1 1& ( ) ( ) 1/ 1/k j k jI x c I x I c f n f m

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• Since these amounts of information are identical, we obtain:

• Among several solutions of this functional equation, the most important for us is:

• Thus, when a statistical experiment has n eqiuprobable outcomes, the average amount of information associated with an outcome is log n

logf x x

1/ 1/ 1/f n f m f mn

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• 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 and provides a unit of information known as a bit.

2 2log 1/ 2 log 2 1

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ee-317: information theory fall 2008 1. organizational details class meeting: 7 :00-9:45pm, tuesday,...

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