### Introduction

In this we apply z-transforms to the solution of certain types of difference equation. We shall see that this is done by turning the difference equation into an ordinary algebraic equation. We investigate both first and second order difference equations.

A key aspect in this process in the inversion of the z-transform. As well as demonstrating the use of partial fractions for this purpose we show an alternative, often easier, method using what are known as residues.

#### Prerequisites

- have studied carefully Section 21.2
- be familiar with simple partial fractions

#### Learning Outcomes

- invert z-transforms using partial fractions or residues where appropriate
- solve constant coefficient linear difference equations using z-transforms

#### Contents

1 Solution of difference equations using z-transforms1.1 Solution of first order linear constant coefficient difference equations

1.2 Use of the right shift theorem in solving difference equations

2 Second order difference equations

3 Inversion of z-transforms using residues

3.1 Pole of a function of a complex variable

3.2 Residue at a pole

3.3 Inverse z-transform formula

4 An application of difference equations – currents in a ladder network