*Lecturer: Dr R.J. Gibbens*
(`rg31@cl.cam.ac.uk`)

*No. of lectures:* 4

*This course is a prerequisite for Computer Graphics and Image
Processing, Computer Vision *(*Part II and Diploma*)*,
Information Theory and Coding *(*Part II*)* and Neural
Computing *(*Part II*)*.*

**Aims**

The aims of this course are to review some key concepts and operations defined in continuous mathematics involving real and complex-valued functions. Focus is on the use and implementation of these notions as encountered in computing.

**Lectures**

**Review of analysis.**Limits, continuity and differentiability. Power series and transcendental functions. Taylor series. Complex variables.**Fourier series.**Introduction. General properties. Uses and applications.**Linear vector spaces and decompositions.**Expansions and basis functions. Orthogonality, inner products and completeness. Useful expansion bases for functions.**Signals and systems.**Fourier transforms and their inverses: introduction and general properties. Uses and applications. Brief introduction to wavelet analysis and its comparison with Fourier analysis.

**Objectives**

At the end of the course students should

- understand how data or functions can be
represented in terms of their projections onto basis
functions
- be fluent in the use and properties of
complex variables
- grasp key properties and uses of Fourier
analysis, transforms, and wavelets

**Reference books**

Kaplan, W. (1992). *Advanced Calculus*.
Addison-Wesley (4th ed.).

Oppenheim, A.V. & Willsky, A.S. (1984). *Signals
and Systems*. Prentice-Hall.