Solar flares occur on eleven-year cycles, surfers wait for “every seventh wave”, porpoise calls have a distinctive signature that can be detected against the background noise of their marine environment, bats locate prey by elaborate use of sonar. In each case, complex time-series data is decomposed into frequency-determined characteristics. This decomposition is then used for explanative and predictive purposes. The analysis of these complex frequency characteristics is at the core of systems and transforms. You will study classical mathematical techniques of Fourier and Laplace transforms, applied in a modern context of a data-rich world. You will use real data from Cornwall-based applications in marine ecology, renewable wave energy and environment and human health.

Pre-requisite modules: “Calculus and Geometry” (ECM1901), “Vector and Matrices” (ECM1902), or equivalent, and familiarity with MATLAB.

The broad aims of the module are to develop the mathematics of modern signal processing, that is the interplay between signals and series, and the systems that operate on them, and then to apply these techniques to real data arising from scientific and engineering applications. Signal processing is ubiquitous across all aspects of science and technology – smart phones, on-line music and video streaming, digital TVs and cameras could not function without signal processing, whilst many areas of applied science would struggle to function. The mathematics of signal processing finds its roots in the 19th century work of Fourier, but leads up to modern tools of discrete Z-transforms and wavelets. This module will uncover the amazing and beautiful mathematics that underpins the digital revolution of the last few decades. The module also illustrates a fundamental issue: the lead-time for technology transfer from theoretical mathematics to commercial/technological products can be decades, if not centuries.

On successful completion of this module, you should be able to:

Module Specific Skills and Knowledge:

1 Demonstrate a sound understanding of essential mathematical aspects of signal processing, including an appreciation of the temporal and frequency content in data and the representation of time series data in terms of basis functions;

2 Reconstruct signals from their frequency content;

3 Appreciate the fundamental limitations of reconstructing a signal from its frequency- or time-sampled form;

Discipline Specific Skills and Knowledge:

4 Demonstrate sufficient knowledge of signal processing techniques for applications in engineering and science;

5 Show awareness and skills in applying a range of mathematical modelling techniques;

Personal and Key Transferable / Employment Skills and Knowledge:

6 Formulate and solve problems;

7 Communicate reasoning and solutions effectively in writing;

8 Make appropriate use of learning resources;

9 Demonstrate self-management and time management skills.

Theme 1. Modelling of systems: examples of systems arising in engineering and scientific applications including electrical circuits, wave energy systems and biological systems; state space models, nonlinear systems, linearization, numerical solutions [8 hours]:

Theme 2. Laplace and inverse Laplace transforms, partial fractions and use of Laplace transform tables; using Laplace Transforms to solve and manipulate linear ordinary differential equations [8 hours];

Theme 3. Generic interpretation of system data; odd and even functions; periodic funcions; signal processing using Fourier series and Fourier transforms; the celebrated Nyquist-Shannon sampling theorem [8 hours];

Theme 4. Transfer functions and input-output systems; poles and zeros; stability; gain and phase; frequency response and Bode; block diagram algebra; Simple feedback control, Z-transforms [8 hours];

Applied group project on one of the four themes in supported matlab classes [12 hours]

Basic reading:

ELE - http://vle.exeter.ac.uk

Reading list for this module: