# Information Theory

## Unit aims

To give a rigorous and modern introduction into Shannon's theory of information, with emphasis on fundamental concepts and mathematical techniques.

## Unit description

Shannon's information theory is one of the great intellectual achievements of the 20th century, which half a century after its conception continues to inspire communications engineering and to generate challenging mathematical problems. Recently it has extended dramatically into physics as quantum information theory. The course is about the fundamental ideas of this theory: data compression, communication via channels, error correcting codes, and simulations.

It is a statistical theory, so notions of probability play a great role, and in particular laws of large numbers as well as the concept of entropy are fundamental (and will be discussed in the course). The course contains a discussion of data compression, leading to entropy as the crucial quantity; an introduction to noisy channels and error correcting codes, culminating in Shannon's coding theorems.

The course aims at demonstrating information theoretical modelling, and the mathematical techniques required will be rigorously developed.

It is a natural companion to the Quantum Information course offered in Mathematics (MATH M5610), and to a certain degree to Cryptography B (COMSM 0007), offered in Computer Science, and Communications (EENG 22000), in Electrical Engineering. It may also be interesting to physicists having attended Statistical Physics (PHYS 30300).

## Relation to other units

The probabilistic nature of the problems considered and of the mathematical modellings in information theory relates this unit to the probability and statistics units at Levels 4, 5 and 6. It is very much suited as a companion to the Quantum Information unit.

Related courses in Computer Science: Cryptography B, in Electrical Engineering: Communications, and in Physics: Statistical Physics.

## Learning objectives

This unit should enable students to:

- Understand how information problems are modeled and solved
- Model and solve problems of information theory: data compression and channel coding
- Discuss basic concepts such as entropy, mutual information, relative entropy, capacity
- Use information theoretical methods to tackle problems arising in a number of settings

## Transferable skills

Mathematical - Knowledge of basic information theory; probabilistic reasoning.

General skills - Modelling, problem solving and logical analysis. Assimilation and use of complex and novel ideas.

## Syllabus

- Information sources and review of probability
- Variable length and block data compression; entropy
- Noisy channels
- Error correcting codes
- Shannon's channel coding theorem; mutual information and relative entropy
- Further topics (time permitting); e.g., Gaussian channels and signal-to-noise ratio, cryptography, ...

## Reading and References

There exist many textbooks on the elements of information theory.

- R B Ash. Information Theory, Dover Publications, 1990
- D J C Mackay. Information Theory, Inference, and Learning Algorithms, CUP, 2003 - free download from http://www.inference.phy.cam.ac.uk/itprnn/book.html
- T M Cover & J A Thomas. Elements of Information Theory, Wiley Interscience, 1991

Other useful references are:

- C E Shannon & W Weaver. The Mathematical Theory of Communication, University of Illinois Press, 1963
- I Csiszar & J Koerner. Information Theory: Coding Theorems for Discrete Memoryless Systems (2nd ed.), Akademiai Klado, Budapest, 1997

The course only requires elementary probability theory, but students who have taken further probability will find some of the course content easier. A very good reference is

- G R Grimmett & D Welsh. Probability: An Introduction, Oxford University Press, 1986

**Unit code:** MATH34600

**Level of study:** H/6

**Credit points:** 10

**Teaching block (weeks):** 1 (7-12)

**Unit director:** Dr Oliver Johnson

**Lecturer:** Dr Oliver Johnson

## Pre-requisites

MATH11300 Probability 1 OR Level 2 Physics (MATH 11400 Statistics 1 is helpful, but not necessary)

## Co-requisites

None

## Methods of teaching

Lectures. Exercises to be done by students, problem classes.

## Methods of Assessment

The pass mark for this unit is 40.

The final mark is calculated as follows:

- 100% from a 1 hour 30 minute exam in January

NOTE: Calculators of an approved type (non-programmable, no text facility) are allowed.

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.