Digital Signal Processing Reference
In-Depth Information
Similary, using Euler's formula for cos
n
w
T
as a sum of two complex exponen-
tials, one can find the
ZT
of
x
(
n
)
=
cos
n
w
T
=
(
e
jn
w
T
+
e
-
jn
w
T
)/2, as
2
zz T
zz T
-
cos
cos
w
w
()
=
Xz
z
>
1
(4.11)
2
-
2
+
1
4.1.1 Mapping from
s
-Plane to
z
-Plane
The Laplace transform can be used to determine the stability of a system. If the
poles of a system are on the left side of the
j
axis on the
s
-plane, a time-decaying
system response will result, yielding a stable system. If the poles are on the right
side of the
j
w
axis, the response will grow in time, making such a system unstable.
Poles located on the
j
w
axis, or purely imaginary poles, will yield a sinusoidal
response. The sinusoidal frequency is represented by the
j
w
w
axis, and
w=
0 repre-
sents dc (direct current).
In a similar fashion, we can determine the stability of a system based on the
location of its poles on the
z
-plane associated with the
z
-transform, since we
can find corresponding regions between the
s
-plane and the
z
-plane. Since
z
=
e
sT
and
s
=s+
j
w
,
zee
=
s
T
jT
w
(4.12)
Hence, the magnitude of
z
is |
z
|
f
/
F
s
, where
F
s
is
the sampling frequency. To illustrate the mapping from the
s
-plane to the
z
-plane,
consider the following regions from Figure 4.1.
=
e
s
T
with a phase of
q=w
T
=
2
p
s<
0
Poles on the left side of the
j
axis (region 2) in the
s
-plane represent a stable
system, and (4.12) yields a magnitude of |
z
|
w
<
1, because
e
s
T
<
1. As
s
varies from
FIGURE 4.1.
Mapping from the
s
-plane to the
z
-plane.
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