This module develops the learners knowledge of the use of the Laplace and Fourier transforms and introduces discrete transforms and the convolution and correlation of discrete signals in engineering systems analysis. This will allow them to proceed to the remaining mathematical modules and to confidently handle the mathematical content of other modules in the degree. The module serves as part of the ongoing mathematical training required for an Engineering graduate.
| Differential equations | Review of solving Differential Equations using the Laplace transform and the definition of a transfer function and the impulse response. Modelling with differential equations and the application of the Laplace Transform in the derivation of the input/output relationship for Linear systems. 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. The discrete Fourier transform and its inverse. Analytical calculation of the DFT of simple functions, including the single pulse. The relation of the DFT to the Fourier transform. Matrix representation of the DFT. Convolutions and correlation of discrete sequences including circular convolution and cross-correlation. | ||||
| Discrete Systems | 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. | ||||
| Applications of the discrete transforms | 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. |
Differential equations
Review of solving Differential Equations using the Laplace transform and the definition of a transfer function and the impulse response. Modelling with differential equations and the application of the Laplace Transform in the derivation of the input/output relationship for Linear systems.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. The discrete Fourier transform and its inverse. Analytical calculation of the DFT of simple functions, including the single pulse. The relation of the DFT to the Fourier transform. Matrix representation of the DFT. Convolutions and correlation of discrete sequences including circular convolution and cross-correlation.
Discrete Systems
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.
Applications of the discrete transforms
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.
The calculus element of this module is best thought by introducing the methods of solution for ODEs, Definitions of Transforms, Impulse reposes methods and so on in lectures, explaining how and why they work and then practising as many examples as feasibly possible in the time allowed in tutorials. Where possible, the discrete transforms will be implemented as examples, illustrating how the transforms behave and what information they draw out.
| Module Content & Assessment | |
|---|---|
| Assessment Breakdown | % |
| Formal Examination | 70 |
| Other Assessment(s) | 30 |