Unit 5: Pole-Zero Analysis

In Unit 4.4 The Inverse Laplace Transform we introduced the idea of poles and zeros and in Unit 4.6: Transfer Functions we introduced the idea of a transfer function which represents the impulse response of a system \(H(s)\). The transfer function \(H(s)\) has a numerator \(N(s)\) and a denominator \(D(s)\):

(44)\[H(s) = \frac{N(s)}{D(s)}\]

Both the numerator and the denominator can be factorised

(45)\[H(s) = \frac{(s - z_1)\ldots(s - z_m)}{(s - p_1)\ldots(s - p_n)}\]

The \(z_i\), which are the values for which \(N(s)=0\) are called the zeros of \(H(s)\). The values \(p_i\) are the zeros of the denominator \(D(s)\) at which \(H(s) = \infty\). The zeros of \(D(s)\) are called the poles of the system \(H(s)\).

In Unit 4.4 The Inverse Laplace Transform we categorised the poles of the system into four cases:

• The case of the distinct real poles

• The case of the complex poles

• The case of the repeated poles

• The case of the improper rational polynomial

However, we didn’t really discuss the implications of these cases. In this unit we aim to correct this omission. At the end of this unit, we hope that you will be able to discuss the qualitative behaviour of a system with any number of poles and zeros. This will be very useful going forward.

Table of Contents

• Unit 5.1: Qualitative Properties of Signals and Transfer functions

• Unit 5.2: More on the Qualitative and Quantitative Response of First- and Second-Order Poles