IEEE Solid-States Circuits Magazine - Summer 2020 - 14
Next, we select v 1 0 to ensure
stabilit y in CT systems. Consequently, z = e TCK v e TCK j~ has a magnitude less than unity [Figure 18(b)].
H2 (z )
2
H1 (z )
fCK
0
fCK
4
In our analysis
of DT systems,
we will often
draw intuition
from their CT
counterparts.
f
2
(a)
2
1
u [k ]
x
0
1
2
3
H2 (z )
k
y
0
1
2
3
k
This indicates that DT systems must
be designed such that their z-transform displays poles only inside the
unit circle. For example, the unitdelay stage is characterized by z -1
and hence a pole at the origin.
The third case occurs if v 2 0
and z falls outside the unit circle, Figure 18(c), causing oscillation with a
growing amplitude. In other words,
if the poles of H(z) are too far from
the origin, the system becomes unstable. Figure 18(d) depicts an example where a pole of H(z) is varied
from z = 0 along the real axis, and
its counterpart in the s domain is
shown. As the former crosses point
A, the latter goes from the left half
plane to the right half plane.
(b)
FIGURE 16: (a) The frequency responses of H1 and H2 and (b) the step response of H2.
H (z )
2
LPF
HPF
1 - z -2
0
fCK
2
fCK
4
FIGURE 17: The frequency response of
1 − z −2.
s Plane
f
those in the right half plane cause
a growing sinusoidal response.
We must now see how these phenomena manifest themselves in the
z domain.
R e c a l l t h a t z = e TCK s, w h e r e
s = v + j~. If v = 0, s appea rs on
the j~ axis while z = e TCK j~ has a
unity magnitude and is located on
the unit circle [Figure 18(a)]. Thus,
if the z-transform contains poles on
the unit circle, the system exhibits
constant-amplitude oscillation.
z Plane
Im {z }
jω
s Plane
z Plane
Im {z }
jω
1 Re{z }
σ
1 Re{z }
σ
(a)
s Plane
(b)
1
σ
s Plane
z Plane
Im {z }
z Plane
Im {z }
jω
Re{z }
A
1
Re{z }
(c)
jω
A
0
σ
(d)
FIGURE 18: The (a) poles on the j~ axis mapping to the unit circle, (b) poles in the left half plane mapping to a region inside the unit circle, (c)
poles in the right half plane mapping to a region outside the unit circle, and (d) correspondence between the real poles in the z and s planes.
14
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IEEE SOLID-STATE CIRCUITS MAGAZINE
IEEE Solid-States Circuits Magazine - Summer 2020
Table of Contents for the Digital Edition of IEEE Solid-States Circuits Magazine - Summer 2020
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IEEE Solid-States Circuits Magazine - Summer 2020 - Cover1
IEEE Solid-States Circuits Magazine - Summer 2020 - Cover2
IEEE Solid-States Circuits Magazine - Summer 2020 - Contents
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