# Fourier Series and Transforms (E1-10)

Mike Brookes
8 lectures in the Autumn Term

## Aims

The aim of this module is to introduce the basic mathematical objects and tools to support signal processing modules covering Fourier series and Fourier transforms. This short module should emphasize a rigorous theoretical treatment leaving the applications to courses such as EE1-06 Introduction to Signals and Communications and EE2-05 Signals and Linear Systems.

## Syllabus

Sums of geometric series, single and double summations, periodic waveforms, averages of sin, cos and complex exponentials. Real and complex forms of the Fourier series, Dirichlet conditions, Fourier analysis, Fourier series or even, odd and anti-periodic waveforms, Parseval's theorem, Fourier series of a product of two waveforms, Gibbs' Phenomenon, Fourier series of the derivative and integral of a waveform, periodic extension of a waveform with finite support. Fourier transform, Dirac delta function and its properties, Fourier transform of a periodic signal, Convolution theorem, Parseval's theorem, energy and power signals, cross-correlation and autocorrelation, Cauchy-Schwartz inequality, Wiener Khinchin theorem. Alternative Fourier transform definitions.

## Textbooks

•  Textbook: "Mathematical Methods for Physics and Engineering" by K. F. Riley, M. P. Hobson and S. J. Spence, 3rd Ed. CUP 2006. ISBN 978-0-521-67971-8 [Amazon]

## Lecture List

1. Sums and Averages [slides, handouts]
2. Fourier Series [slides, handouts]
3. Complex Fourier Series [slides, handouts]
4. Parseval's Theorem and Convolution [slides, handouts]
5. Gibbs Phenomenon [slides, handouts]
6. Fourier Transforms [slides, handouts]
7. Convolution, Energy Conservation, Power Spectra [slides, handouts]
8. Correlation, Autocorrelation [slides, handouts]

Complete set of handouts (1.8 MB)

## Problem Sheets

1. Sums and Averages [probs, solutions]
2. Fourier Series  [probs, solutions]
3. Parseval, Convolution and Gibbs  [probs, solutions]
4. Fourier Transforms  [probs, solutions]

Complete Set of Problems + Solutions

## Revision Lecture

Revision Lecture [slides, handouts]

## Exam

The Mathematics IA exam will consist of four compulsory questions. Question 1 comprises ten short parts and carries 40% of the total marks. Questions 2, 3 and 4 are longer and each carry 20% of the marks. A formula sheet will be issued. Calculators are not permitted in the exam.