EEEN30110 Signals and Systems

Academic Year 2024/2025

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.

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Curricular information is subject to change

Learning Outcomes:

On successful completion of this subject the student will be able to:

1. Explain the mathematical basis for the frequency content of a signal with particular reference to the Fourier series and the Fourier transform.

2. Explain the mathematical basis of the frequency response of a linear, time-invariant system, continuous-time (analog) or discrete-time.

3. Derive mathematical models for and analyse the response of linear, time-invariant systems, continuous-time (analog) or discrete-time.

4. Effectively solve linear, constant coefficient ordinary differential and difference equations.

5. Effectively employ MATLAB, Python or both in the analysis of signals and systems.

Indicative Module Content:

The topics outlined in the content will be complemented with examples. Computer-based simulations and experiments will complement the learning.

Lec. 0: Background Mathematics: calculus and algebra of complex numbers.
Lec. 1 – Lec. 3: Introduction, linearisation and motivation.
Lec. 4: Introduction to frequency response and partial sample report.
Lec. 5: Linear, Time-invariant Systems.
Lec. 6: Dirac Delta function, Convolution Integral and Impulse Response.
Lec. 7: Definition of Laplace Transform.
Lec. 8 – Lec. 13: Laplace Transform and solution of linear, constant coefficient ordinary differential equations.
Lec. 14: Laplace Transform and Frequency Response.
Lec. 15 – Lec. 17: Fourier Series.
Lec. 18 – Lec. 19: Fourier Series and Numerical Fourier Series.
Lec. 20: Fourier Analysis.
Lec. 21: Fourier Transform.
Lec. 22: Numerical Fourier Transform
Lec. 23 : Discrete-time Linear, Time-invariant Systems.
Lec. 24 – Lec. 25: Z-Transform.
Lec. 26: Discrete-time Fourier Series.
Lec. 27: Discrete-time Fourier Transform.
Lec. 28: Discrete-time Fourier Analysis.

Week 1: Practical Computer Laboratory: introduction to MATLAB/Python
Week 2: Practical Computer Laboratory: specialised MATLAB/Python for Signals and Systems
Week 5: Practical Computer Laboratory: Laplace Transform
Week 8: Practical Computer Laboratory: Fourier Theory
Week 11: Practical Computer Laboratory: Discrete-time systems

Student Effort Hours: 
Student Effort Type Hours
Lectures

28
Laboratories

10
Specified Learning Activities

17
Autonomous Student Learning

58
Total

113
Approaches to Teaching and Learning:
Problem-based learning.
Application of appropriate software tools to enable efficient resolution of problems. 
Requirements, Exclusions and Recommendations
Learning Requirements:

Differential and Integral Calculus to advanced level 1 or better.
Differential Equations to advanced level 1 or better.
Algebra, vectors and complex numbers to advanced level 1 or better.


Module Requisites and Incompatibles
Not applicable to this module.
 
Assessment Strategy  
Description Timing Open Book Exam Component Scale Must Pass Component % of Final Grade
Report(s): Practical/Computer Laboratory occurs in week 11 with report due end of week 12. n/a Graded No

33
Report(s): Practical/Computer Laboratory in week 5, report due end of week 6. n/a Graded No

33
Report(s): Practical/Computer Laboratory occurs in week 8, report due end of week 9. n/a Graded No

33

Carry forward of passed components
No
 
Resit In Terminal Exam
Spring No
Please see Student Jargon Buster for more information about remediation types and timing. 
Feedback Strategy/Strategies

• Feedback individually to students, on an activity or draft prior to summative assessment
• Feedback individually to students, post-assessment
• Group/class feedback, post-assessment

How will my Feedback be Delivered?

Grading Scheme included with laboratory description. Individual feedback on progress within associated laboratory offering. Graded reports available when full set of reports graded, with some individual feedback in annotated reports. Common errors document made available to class on Brightspace after release of grades including more general feedback to class as a whole.