Revisiting the Basics - Digital Signal Processing: A Practitioner's Approach - page 29

Unit circle | z | = 1

1

1 − z −1 0.9136 e j 1.12

r

θ

1

1 − z −1 0.9136 e − j 1.12

Figure 2.5 Polar plot of the second-order system in (2.17)

2.2.5 Pole-Zero Representation
The polynomials A ð z Þ and B ð z Þ are set equal to zero and solved for z. The roots of
the numerator polynomial are called zeros and the roots of the denominator
polynomial are called poles of the system. Note that the continuous system
represented by (2.10) is defined by only two parameters ½; ! n but by converting
into discrete form we get three parameters ½ 0 12 , as shown in (2.17)

and (2.19). The addition of a third parameter is due to the sampling time T s .The
parameter

: 2596 ; 0 : 9136 ; 1 :

!

represents the natural angular frequency;
they roughly translate into the radial position of the pole and the angular position of
the pole, respectively.

represents damping and

2.2.6 Continuous to Discrete Domain
Converting from the continuous domain to the discrete domain also affects the pole
positions due to sampling rate. Figure 2.6 depicts the pole positions

ð r ;Þ for

200

1

180

0.8

100

0.05 to 0.9

0.6

0

0.4

0.05 to 0.9

−100

0.2

30° to 120°

0

−200

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

Sampling time

Sampling time

(a) (b)

Figure 2.6 Continuous to discrete sensitivities of the system in (2.11)