Prerequisite courses: Probability, Discrete Mathematics, Mathematical Methods for Computer Science

Aims

The aims of this course are to introduce the principles and applications
of information theory. The course will study how information is measured
in terms of probability and entropy, and the relationships among conditional
and joint entropies; how these are used to calculate the capacity of a
communication channel, with and without noise; coding schemes, including
error correcting codes; how discrete channels and measures of information
generalise to their continuous forms; the Fourier perspective; and extensions
to wavelets, complexity and compression.

Lectures

Foundations: probability, uncertainty, information.
How concepts of randomness, redundancy,
compressibility, noise, bandwidth, and uncertainty are
related to information.
Ensembles, random variables, marginal and conditional probabilities.
How the metrics of information are grounded in the rules
of probability.

Entropies defined, and why they are measures of information.
Marginal entropy, joint entropy, conditional entropy,
and the Chain Rule for entropy. Mutual information between ensembles
of random variables. Why entropy is the fundamental measure of
information content.

Source coding theorem; prefix, variable-, and fixed-length codes.
Symbol codes. The binary symmetric channel. Capacity of a noiseless
discrete channel. Error correcting codes.

Channel types, properties, noise, and channel capacity.
Perfect communication through a noisy channel. Capacity of a
discrete channel as the maximum of its mutual information over
all possible input distributions.

Continuous information; density; noisy channel coding theorem.
Extensions of the discrete entropies and measures to the continuous
case. Signal-to-noise ratio; power spectral density. Gaussian channels.
Relative significance of bandwidth and noise limitations.
The Shannon rate limit and efficiency for noisy continuous channels.

Fourier series, convergence, orthogonal representation.
Generalised signal expansions in vector spaces. Independence.
Representation of continuous or discrete data by complex exponentials.
The Fourier basis. Fourier series for periodic functions. Examples.

Useful Fourier theorems; transform pairs. Sampling; aliasing.
The Fourier transform for non-periodic functions. Properties of the
transform, and examples. Nyquist's Sampling Theorem derived, and the
cause (and removal) of aliasing.

Discrete Fourier transform. Fast Fourier Transform Algorithms.
Efficient algorithms for computing Fourier transforms of discrete data.
Computational complexity. Filters, correlation, modulation, demodulation,
coherence.

The quantised degrees-of-freedom in a continuous signal.
Why a continuous signal of finite bandwidth and duration has a fixed
number of degrees-of-freedom. Diverse illustrations of the principle
that information, even in such a signal, comes in quantised, countable,
packets.

Gabor-Heisenberg-Weyl uncertainty relation. Optimal ``Logons''.
Unification of the time-domain and the frequency-domain as endpoints
of a continuous deformation. The Uncertainty Principle and its optimal
solution by Gabor's expansion basis of ``logons''. Multi-resolution
wavelet codes. Extension to images, for analysis and compression.

Kolmogorov complexity. Minimal description length.
Definition of the algorithmic complexity of a data sequence, and
its relation to the entropy of the distribution from which the data
was drawn. Fractals. Minimal description length, and why this measure
of complexity is not computable.

Objectives

At the end of the course students should be able to

calculate the information content of a random variable
from its probability distribution

relate the joint, conditional, and marginal entropies
of variables in terms of their coupled probabilities

define channel capacities and properties using Shannon's Theorems

construct efficient codes for data on imperfect communication channels

generalise the discrete concepts to continuous signals on continuous
channels

understand Fourier Transforms and the main ideas of efficient
algorithms for them

describe the information resolution and compression properties of
wavelets

Recommended reading

* Cover, T.M. & Thomas, J.A. (1991). Elements of information theory. New York: Wiley.