### Introduction

z-transform of a sequence. We then obtain the z-transform of some important sequences and discuss useful properties of the transform.

Most of the results obtained are tabulated at the end of the Section.

The z-transform is the major mathematical tool for analysis in such areas as digital control and digital signal processing.

#### Prerequisites

- understand sigma ( $\Sigma $ ) notation for summations
- be familiar with geometric series and the binomial theorem
- have studied basic complex number theory including complex exponentials

#### Learning Outcomes

- define the z-transform of a sequence
- obtain the z-transform of simple sequences from the definition or from basic properties of the z-transform

#### Contents

1 The z-transform2 Commonly used z-transforms

2.1 Unit impulse sequence (delta sequence)

2.2 Unit step sequence

2.3 The geometric sequence $\left\{{a}^{n}\right\}$

3 Linearity property and applications

3.1 Linearity property

3.2 Trigonometric sequences

4 Further $z$ -transform properties

4.1 Multiplication of a sequence by ${a}^{n}$

4.2 Multiplication of a sequence by $n$

5 Shifting properties of the z-transform

5.1 Right shift theorems

5.2 Left shift theorems