# CHAPTER *14*

# Filtering and Z-Transforms

## 14.1 INTRODUCTION

This chapter introduces the concepts of *poles* and *zeros* as the components of linear filters. Every linear, time-invariant differential equation may be characterized in terms of the poles and zeros of its frequency response function. The same notion carries over into discrete-time systems. Discrete-time versions of the continuous-time poles and zeros are used to build arbitrary discrete-time filters (assuming linearity and time invariance).

This chapter begins by analyzing the continuous-time case, starting from the general differential equation and continues by developing the discrete-time solutions, computing the gains of these discrete-time systems, and comparing them to the continuous-time systems.

## 14.2 POLES AND ZEROS FROM DIFFERENTIAL EQUATIONS

### 14.2.1 The All-Pole Equations

Every linear differential equation can be translated into the language of linear filters and vice versa ; they are equi valent concepts. Usually, physical systems are initially modeled with linear differential equations , from which the filter can be calculated. In this text we are concerned only with time -invariant systems. The general form of an *n*th-order. time-independent, linear differential equation is

In terms of the differential operator *D*, the derivative with respect to time, this can be written as

where *q* is the polynomial

The method of ...