Z-Transform Introduction

Introduction

About This Level

At the introductory level, we define the unilateral Z-transform, derive basic transform pairs using geometric series, and learn the most fundamental properties (linearity and time shifting). We then introduce the concepts of transfer functions, poles, and stability through a digital filter application.

The region of convergence (ROC), bilateral Z-transform, convolution theorem, and further properties are covered in the Basic Level.

Contents

Chapter 1: Definition and Applications of the Z-Transform

Unilateral Z-transform definition, basic transform pairs derived from geometric series, transfer functions, poles, stability, and digital filter applications with stem plots, block diagrams, and pole-zero plots.

Unilateral Z-transform Geometric series Transfer function Poles and stability Block diagram

Chapter 2: Basic Properties of the Z-Transform

The most fundamental properties of the Z-transform — linearity and time shifting — with proofs and worked examples.

Linearity Time shifting z⁻¹ = delay

Prerequisites

  • Sum formula for geometric series
  • Basic complex numbers (Euler's formula $e^{j\theta} = \cos\theta + j\sin\theta$)