Revisiting the Basics - Digital Signal Processing: A Practitioner's Approach - page 30

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various values of 9 and for different sampling times. We

can use Figure 2.6 to make a number of inferences. In this figure we have shown
sampling time as 0 to 1; 0 corresponds to an infinite sampling rate whereas 1
corresponds to the Nyquist sampling rate (twice the maximum frequency). The
damping factor

in the range 0 05 0

has been varied in the same range for Figure 2.6(a) and (b). A
quick look at Figure 2.6(b) shows that the pole angle is insensitive to variations of .

. Taking advantage of
this relation, we have positioned two markers in Figure 2.6(a) at 30 and 120 ,
which correspond to 12 samples/cycle and 3 samples/cycle, respectively. The pole
plot in Figure 2.6(a) shows that the difference equation numerically behaves well in
this window.

Time has an approximately linear relation with pole angle

s- plane z- plane

jw

Im

θ

r

σ

Re

Figure 2.6A Comparing the s-plane with the z-plane

In discrete systems, roots are presented in polar form rather than rectangular form.
This is because the s-plane, which represents the continuous domain, is partitioned
into a left half and a right half by the j ! axis running from 1 to 1 (Figure 2.6A).

Such an infinite axis is folded into the sum of an infinite number of circles of unit
radius in the z-plane, which is the representation in the discrete domain. The top
half of the circle (0 to

) represents DC to Nyquist frequency and the bottom half

(0 to ) represents the negative frequencies.

In fact, periodicity and conjugate symmetry properties can be best understood by
this geometric background. If all the poles and zeros lie within the unit circle, the
system is called a minimum phase system. For good stability, all poles must be
inside the unit circle.

2.2.7 State Space Representation
Let us consider (2.14) again. We can write it as y k ¼½ B ð z Þ= A ð z Þ u k . Using the
linearity property, we can write y k ¼ B ð z Þð w k Þ where w k ¼½ 1

= A ð z Þð u k Þ . Then

w k ¼ 0 : 7961w k 1 0 : 8347w k 2 þ u k ;

w k 1 ¼ w k 1 þ 0w k 2 þ 0u k ; ð 2 : 20 Þ

¼ : þ 0 :

5193w k 1 þ 0

:

2596w k 2 : y k 0 2596w k