# 4

# z-Transform and Discrete Systems

## 4.1 Introduction

The *z*-transform is a frequency domain representation that makes solution, design, and analysis of discrete linear systems simpler. It also gives some insights about the frequency contents of signals where these insights are hard to see in real-time systems. There are other important uses for the *z*-transform but we will concentrate only on the issues described in this introduction.

## 4.2 Bilateral *z*-Transform

The *z*-transform of the signal *x*(*n*) is given by

$X\left(z\right)={\displaystyle \sum _{n=-\infty}^{+\infty}x\left(n\right){z}^{-n}}$ |
(4.1) |

where *z* is the complex variable. If we try to expand Equation 4.1, we get

$X\left(z\right)=\cdots +x\left(-2\right){z}^{2}+x\left(-1\right){z}^{1}+x\left(0\right){z}^{0}+x\left(1\right){z}^{-1}+x\left(2\right){z}^{-2}+\cdots $ |
(4.2) |

You can see that in Equation 4.1 the power of *z* indicates the position of ...

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