Module Details

Module Code: MATH8002
Title: Discrete Time Mathematics
Long Title: Discrete Time Mathematics
NFQ Level: Advanced
Valid From: Semester 1 - 2020/21 ( September 2020 )
Duration: 1 Semester
Credits: 5
Field of Study: 4610 - Mathematics
Module Delivered in: 1 programme(s)
Module Description: Introduction to discrete transform theory (Z-transforms and Discrete Fourier Transforms) and to error-control coding theory.
 
Learning Outcomes
On successful completion of this module the learner will be able to:
# Learning Outcome Description
LO1 Encode and decode using linear block codes.
LO2 Solve first- and second-order difference equations with constant coefficients using the method of Z-transforms.
LO3 Carry out a pole-zero analysis of a discrete time linear system.
LO4 Find the Discrete Fourier Transform of an N-point sequence using the DFT and FFT.
LO5 Use a computer algebra system to assist with computations in discrete transform theory and in error-control coding theory.
Dependencies
Module Recommendations

This is prior learning (or a practical skill) that is strongly recommended before enrolment in this module. You may enrol in this module if you have not acquired the recommended learning but you will have considerable difficulty in passing (i.e. achieving the learning outcomes of) the module. While the prior learning is expressed as named MTU module(s) it also allows for learning (in another module or modules) which is equivalent to the learning specified in the named module(s).
14757 MATH7031 Transform Methods for E.Eng
14763 MATH8002 Discrete Time Mathematics
Incompatible Modules
These are modules which have learning outcomes that are too similar to the learning outcomes of this module. You may not earn additional credit for the same learning and therefore you may not enrol in this module if you have successfully completed any modules in the incompatible list.
No incompatible modules listed
Co-requisite Modules
No Co-requisite modules listed
Requirements

This is prior learning (or a practical skill) that is mandatory before enrolment in this module is allowed. You may not enrol on this module if you have not acquired the learning specified in this section.
No requirements listed
 
Indicative Content
Introduction to Error-Control Coding
Binary Symmetric Channel. Introduction to error-control coding - message, redundancy, codeword, Hamming distance. Error detection and error correction. Repetition codes and parity codes, distance of code. Linear block codes - generator matrix, parity check matrix, encoding, decoding. The Hamming (7,4) code. Reliability analysis.
Z-transforms, Discrete-Time Systems
Discrete functions and sequences - direct formula, recursive formula. Examples of discrete signals, including the delta function, step function, and discrete sinusoids. LTI systems and convolution. Z-transform - definition, and radius of convergence. Discussion of properties of the Z-transform including linearity and time-shifting properties. Construction of a short table of Z-transforms. Find the inverse transform using the table and partial fractions. The solution of difference equations using Z-transforms. Applications of Z-transform to LTI systems - discrete transfer function, convolution pole-zero diagram and stability. Relationship between the Laplace transform and the Z-transform.
Discrete Fourier Analysis
Overview of Fourier transforms - definition of the Fourier transform for a non-periodic signal, frequency spectra. Definition of the Discrete Fourier Transform (DFT) and of the inverse transform. Matrix representation of N-point DFT. Amplitude spectrum, phase spectrum. Matrix representation of the Inverse Discrete Fourier Transform (IDFT). Application to 4-point and 8-point sequences. Fast Fourier Transform (FFT) - discussion of decimation-in-time and decimation-in-frequency algorithms. FFT butterfly.
Practical programming
Use of packages such as WolframAlpha and MATLAB to illustrate, consolidate and extend the learning in this module.
Module Content & Assessment
Assessment Breakdown%
Coursework30.00%
End of Module Formal Examination70.00%

Assessments

Coursework
Assessment Type Short Answer Questions % of Total Mark 10
Timing Week 4 Learning Outcomes 1
Assessment Description
Test 1 - Coding theory
Assessment Type Short Answer Questions % of Total Mark 10
Timing Week 9 Learning Outcomes 2,3
Assessment Description
Test 2 - Z-transforms and discrete-time systems
Assessment Type Practical/Skills Evaluation % of Total Mark 10
Timing Week 12 Learning Outcomes 5
Assessment Description
Test 3 - Computer algebra system exam
End of Module Formal Examination
Assessment Type Formal Exam % of Total Mark 70
Timing End-of-Semester Learning Outcomes 1,2,3,4
Assessment Description
End-of-Semester Final Examination
Reassessment Requirement
Repeat examination
Reassessment of this module will consist of a repeat examination. It is possible that there will also be a requirement to be reassessed in a coursework element.

The University reserves the right to alter the nature and timings of assessment

 

Module Workload

Workload: Full Time
Workload Type Contact Type Workload Description Frequency Average Weekly Learner Workload Hours
Lecture Contact Formal delivery of module content Every Week 3.00 3
Lab Contact Mathematical computer software laboratory Every Second Week 0.50 1
Tutorial Contact Questions on lecture material Every Second Week 0.50 1
Independent & Directed Learning (Non-contact) Non Contact Review of lecture material, completion of exercise sheets Every Week 3.00 3
Total Hours 8.00
Total Weekly Learner Workload 7.00
Total Weekly Contact Hours 4.00
Workload: Part Time
Workload Type Contact Type Workload Description Frequency Average Weekly Learner Workload Hours
Lecture Contact Formal delivery of module content Every Week 2.00 2
Lab Contact Mathematical computer software laboratory Every Second Week 0.50 1
Tutorial Contact Questions on lecture material Every Second Week 0.50 1
Independent & Directed Learning (Non-contact) Non Contact Review of lecture material, completion of exercise sheets Every Week 4.00 4
Total Hours 8.00
Total Weekly Learner Workload 7.00
Total Weekly Contact Hours 3.00
 
Module Resources
Recommended Book Resources
  • R. Hill. (1986), A First Course in Coding Theory, Clarendon Press, [ISBN: 0-19-853803-0].
  • G.James. (2018), Advanced Modern Engineering Mathematics, 5th. Pearson Education Limited, [ISBN: 9781292174341].
Supplementary Book Resources
  • S.Singh. (2000), The Code Book: The Secret History of Codes and Code-breaking, Fourth Estate, [ISBN: 1857028899].
  • A. C. Grove. (1991), An introduction to the Laplace transform and the z transform, Prentice Hall, New York, [ISBN: 0-13-488933-9].
This module does not have any article/paper resources
Other Resources
 
Module Delivered in
Programme Code Programme Semester Delivery
CR_EELES_8 Bachelor of Engineering (Honours) in Electronic Engineering 7 Mandatory