Signal Synthesis and Analysis
Subject code: SIP 7015

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Lecturer
Dr John van der Hoek
University of Adelaide

Mode of delivery
On-line with possibility of weekly lectures.

Assumed knowledge
Basic knowledge of Fourier transforms and Z-transforms, Linear Algebra.

Aim/Learning Objectives
The course is a basic course for other sudies in signal processing, providing in a coherent and reasonably comprehensive way the L² theory of signals

Signals can be analysed into component signals which we will do through Fourier related ideas (Fourier Transforms, Fourier Series, Fast Fourier Transforms and so on) and through Wavelet representations. In fact because of its importance, the study of wavelet related ideas will comprise the biggest portion of the course.

Synthesis will refer to the construction of signals -- this could refer to the reconstruction of a signal from its components, from sampling, or reconstruction from a compressed transmitted signal, and so on.

In preparing these lectures, a search was made of the relevant literature and the lecture notes attempt to cover the major tools of this area. However when discussing wavelets, we do not get into a discussion of the most recent topics like ``wavelet packets'', which should be covered in a more advanced course.

Content
Hilbert space: Inner product, completeness, L₂, orthogonality and Reisz basis, Parsevaal's theorem, linear operators and resolutions of unity.
Fourier Series: Basis, L₂(Rⁿ), Plancherel Theorem, Uncertainty Theorem, Multidimensional Fourier transform, Short Time Fourier transform. Discrete Fourier Transform Properties, DFT Matrix, factorisation, Fast Fourier transform, sampling and Interpolation, Shannon sampling.
Wavelets Multiresolution Analysis: Scaling function and dilation, orthogonal wavelets, compact supported wavelets, Quadrature Mirror filters, Finite discrete wavelet transform, wavelet design. Overview of other transforms.

The Course includes example(s) like how wavelet analysis can be used with coding for data transmission.

Assessment
50% examination (mid-term and final), 50% assignments (5), essay (1), however these percentages are indicative only and may be varied at the lecturer's discretion. Details of the actual assessment used in a given year can be found in the study guide provided at the start of the semester.

Resources
All the materials necessary for the course will be availabe on-line. The lecture notes also include an extended bibliography for further reading on the subject.