This module serves to introduce the learner to the concepts and properties of the Laplace and Fourier transforms and the associated ideas of transfer functions, filters and the impulse response used in engineering systems analysis. The module then introduces discrete transforms and the application of transforms to the analysis of the behaviour of engineering systems.
The Laplace Transform
The Laplace Transform of standard functions; the constant function, powers, the exponential function and the sine and cosine functions. The properties of the Laplace transform; the transform of the derivative, the shift properties of the Laplace Transform and the transform of periodic functions. The transform of piecewise linear functions such as the square wave and sawtooth functions.
Applications of Transforms
Solution of Ordinary Differential equations and the Laplace and Fourier transforms. The concept of an engineering system and its modelling as a differential equation, with input and output functions and their transforms identified, and the role of the transform domain in this analysis. The concept of a transfer function and the impulse response. The convolutions of functions and the properties of transforms of convolutions. Ordinary Differential equations and Filters. The use and definition of poles and zeros in the transform domain. The analysis of engineering systems and their representation by the Block Diagram.
Discrete Transforms
The definition of the Z-transform and the z transform of simple discrete functions. Analytical calculation of the z-transform of simple functions, including the single pulse. The link between the Z- and Laplace transforms in the s-plane.
Discrete Systems and Applications
The mathematical definition of sampling a continuous signal. Discrete systems, difference equations and their role in engineering systems analysis. The solution of difference equations by substitution and characterising their behaviour. The discrete transforms and their use in solving difference equations. The sample of the transform of a continuous signal and the Discrete transform of the sampled signal. Modelling engineering systems as discrete systems and interpreting the behaviour of solutions. Characterisation of poles and zeros in the solution of difference/differential equations. Impulse response and transfer functions for discrete systems.
| Module Content & Assessment | |
|---|---|
| Assessment Breakdown | % |
| Other Assessment(s) | 40 |
| Formal Examination | 60 |