 # Module IEA-2003:Mathematical Methods

### Module Facts

Run by School of Computer Science and Electronic Engineering

10 Credits or 5 ECTS Credits

Semester 1

Organiser: Dr Daniel Roberts

### Overall aims and purpose

To teach the principles of the Fourier Series, Fourier Transform and z-Transform, with emphasis on their application in communications engineering and signal processing.

### Course content

• Revision of trigonometric functions. periodic functions and sinusoids.

• Revision of integration by parts: orthogonality of sin(nx), cos(nx). Odd and even functions; half-range series. Definition of Fourier series and calculation of Fourier coefficients. Infinite sums.

• Revision of linear difference equations, and methods of solution of homogeneous and non-homogeneous equations.

• Laurent series and the definition of the z-transform.

• Revision of the method of partial fractions, and the solution of linear difference equations using z-transform methods. Shift theorems.

• Convolution of sequences and the z-transform of (f * g).

• Revision of complex numbers, esp.

• exp(jx) = cos(x) + j.sin(x).

• In finite integrals. The delta function and the sinc function.

• Definition of the Fourier transforms. Transforms of simple functions, properties of the Fourier transform, convolution.

### Learning outcomes mapped to assessment criteria

threshold

40%

good

60%

excellent

70%

Understand the principles of Fourier Series. Calculate Fourier coefficients.

Can sketch simple even, odd and periodic functions. Calculate fundamental angular frequencies and express harmonics as single sinusoids. Knows expressions for Fourier coefficients and can perform simple calculations. Can also make use of even/odd properties of functions to facilitate calculation. Evaluate functions like x^2.cos(nx) using integration by parts. Can obtain sums of infinite series by evaluating Fourier series at particular values. Can derive the expressions for the Fourier coefficients.

Understand the principles of z-transforms. Solve linear difference equations using traditional and z-transform methods.

Can solve quadratic, homogeneous linear difference equations and find a particular solution to a non-homogeneous equation. Know the formula for the z-transform. Apply the z-transform to simple functions. Can solve quadratic, non-homogeneous, linear difference equations using partial fractions and z-transforms. Comprehensive knowledge of the z-transform. Can make calculations involving convolution of sequences.

Understand the principles of Fourier transforms. Calculate simple transforms.

Know the formula for the Fourier transform. Be familiar with the delta function and the sinc function. Can correctly apply the Fourier transform to simple functions. Knowledge of the two shift theorems. Can correctly apply the Fourier transform to complicated functions and take advantage of odd/even properties.

### Assessment Methods

Type Name Description Weight
EXAM Examination 70
SUMMATIVE THEORETICAL ASSMT Assignment 1 10
SUMMATIVE THEORETICAL ASSMT Assignment 2 10
SUMMATIVE THEORETICAL ASSMT Assignment 3 10

### Teaching and Learning Strategy

Hours
Lecture

2 x 1 Hour lectures over 12 weeks

24
Private study 76

### Transferable skills

• Numeracy - Proficiency in using numbers at appropriate levels of accuracy
• Self-Management - Able to work unsupervised in an efficient, punctual and structured manner. To examine the outcomes of tasks and events, and judge levels of quality and importance
• Critical analysis & Problem Solving - Able to deconstruct and analyse problems or complex situations. To find solutions to problems through analyses and exploration of all possibilities using appropriate methods, rescources and creativity.

### Subject specific skills

• Solve problems logically and systematically;

### Courses including this module

#### Compulsory in courses: 