Mathematics

ECM2707 - Systems, Series and Transforms (2015)

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MODULE TITLESystems, Series and Transforms CREDIT VALUE15
MODULE CODEECM2707 MODULE CONVENERDr Frank Kwasniok (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 0 11 weeks 0
Number of Students Taking Module (anticipated) 110
DESCRIPTION - summary of the module content

This module will uncover the amazing and beautiful mathematics that underpins the miniaturised digital revolution of the last couple of decades. Most of today's digital equipment depends on signal processing techniques. Even the most techno-phobic make daily use of mobile phones, mp3 players, digital TVs and cameras, and so are highly dependent on signal processing techniques. 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.
 

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.

Prerequisite module: ECM1709, ECM2702 or equivalent

AIMS - intentions of the module

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. This module will uncover the amazing and beautiful mathematics which underpins the miniaturised digital revolution of the last couple of decades. By the end of this module, you will be able to 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. You will also be able to reconstruct a signal from its frequency content, and be aware of the fundamental limitations of reconstructing such a signal.


INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)

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


Discipline Specific Skills and Knowledge:
4 reveal sufficient knowledge of signal processing techniques for subsequent applications in modeling and data analysis arising in Stage 3 modules and beyond.
Personal and Key Transferable/ Employment Skills and  Knowledge:
5 reason using abstract ideas;
6 formulate and solve problems;
7 communicate reasoning and solutions effectively in writing;
8 use learming resources appropriately;


9 exhibit self management and time-management skills.

 

SYLLABUS PLAN - summary of the structure and academic content of the module

- data: examples of data, seismographs, time series, continuous vs. discrete time modelling and analysis, time-and frequency-sampled data;

- function approximations;

- Chebyshev and Legendre polynomials;

- orthogonal polynomials;

- Weierstrass theory;

- best approximations and function space norms;

- Matlab applications;

- series, Maclaurin and Taylor series, periodic functions, Fourier series, Parseval’s identity, calculus of Fourier series, basis properties in L2, Gram-Schmidt orthogonalisation;

- systems, discrete-time systems, ARMA models, matrix models, impulse response function, Markov chains, eigenvalues, Populations Projection Models, Matlab applications;

- transforms, Laplace transforms and linear inhomogeneous differential equations, poles and zeros, Fourier transforms, Plancherel's theorem, convolution theorem and applications, transfer function analysis of series and systems, serial and parallel connections of systems;

- fundamental limitations of signal processing: discrete Fourier transform and aliasing, frequency-temporal limitations, Shannon’s sampling theorem, wavelets.

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
Scheduled Learning & Teaching Activities 44.00 Guided Independent Study 106.00 Placement / Study Abroad 0.00
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
Category Hours of study time Description
Scheduled learning and teaching activities 33 Lectures including examples classes
Scheduled learning and teaching activities 11 Tutorials
Guided independent study 106 Lecture and assessment preparation; wider reading

 

ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
Form of Assessment Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Three examples sheets 15 hours, 6-10 questions per sheet (four sheets) 1-9 Comments on each script, general comments uploaded to ELE, sketch solutions uploaded to ELE
       
       
       
       

 

SUMMATIVE ASSESSMENT (% of credit)
Coursework 20 Written Exams 80 Practical Exams
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Written exam – closed book 80 2 hours 1-7 Verbal on request
Coursework – based on assessed questions set on example sheets 20 10 hours, 3-5 assessed questions per problem sheet (three sheets) 1-9 Comments on each script, general comments uploaded to ELE, solutions uploaded to ELE
         
         
         

 

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
Original Form of Assessment Form of Re-assessment ILOs Re-assessed Time Scale for Re-reassessment
All above Written exam (100%) All August Ref/Def period
       
       

 

RE-ASSESSMENT NOTES

If a module is normally assessed entirely by coursework, all referred/deferred assessments will normally be by assignment.


If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 40% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener

ELE-http://vle.exeter.ac.uk

Reading list for this module:

Type Author Title Edition Publisher Year ISBN Search
Set McMahon D Signals and Systems Demystified 1 McGraw Hill 2007 978-0071475785 [Library]
CREDIT VALUE 15 ECTS VALUE 7.5
PRE-REQUISITE MODULES ECM2702, ECM1709
CO-REQUISITE MODULES
NQF LEVEL (FHEQ) 5 AVAILABLE AS DISTANCE LEARNING No
ORIGIN DATE Friday 09 January 2015 LAST REVISION DATE Friday 09 January 2015
KEY WORDS SEARCH Fourier series; Fourier transforms; ARMA models; linear systems; ODEs; PDEs; signal processing.