electiveHQ

General systems essentially defy analysis. On the other hand, linear, time-invariant (LTI) systems often provide a very useful first approximation of the behaviour of systems close to an operating region and moreover are amenable to analysis. This module offers an introduction to the analysis of LTI systems both in continuous and discrete-time. It is impossible to provide an analysis procedure for systems without introducing signals and the tools used for their analysis. Within the module, mathematical ideas are described, which underpin the very important concept of the frequency content of a signal and the frequency response of a system. The module covers the mathematics required to undertake a study of dynamics, communication theory, signal processing, advanced circuit theory and control theory, with engineering examples. Above all the module also provides advanced, but ultimately systematic techniques for the solution of linear, constant coefficient, ordinary differential and difference equations, motivated by engineering applications in which the equations of motion are approximately of this form, at least close to an operating point. Module Outline: General continuous-time systems. Approximate equations of motion close to an operating point. Motivation of study of Linear, Time-Invariant systems. Laplace Transform, solution of linear, constant-coefficient ordinary differential equations, theory of LTI continuous-time systems. Spectral Theory. Fourier series and Fourier transform. Frequency content of continuous-time signals. Frequency response of continuous-time systems. Fourier analysis. Discrete-time Systems. Z-transform. Discrete-time Fourier series and Fourier transform. Fourier analysis in discrete-time. Justification of Learning Outcomes: LO1/LO2: The mathematical description of the frequency content of a signal and the related mathematical description of the frequency response of a system comprises some of the most important practical mathematics yet discovered. It is the enabling theoretical backdrop for music theory, signal processing, vibration analysis, design and understanding of electrical filters and much more. LO3/LO4: While ordinary differential equations provide non-local, perhaps even global models of systems, in practice a system will operate close to some desired operating point so that all that is required of the modeller is to provide a valid model in the locality of this operating point. Such models serendipitously, often turn out to be linear and time-invariant, i.e. linear, constant-coefficient ordinary differential equations. These it transpires are effectively the only ODEs for which there is a general theory, and this theory provides us with significant insights into the behaviour of practical systems. LO5: MATLAB and Python are ubiquitous, and knowledge of at least one of these packages will well serve the student.