Digital Signal Processing Reference
In-Depth Information
Left-half s-plane Re
s
<
0
Substituting Re
{
s
}= σ<
0 in Eq. (16.9) yields
z
<
1
.
Therefore, we observe that the left-half s-plane is mapped to the interior
of the unit circle. We now show that the mapping
z
=
e
sT
is not a unique
one-to-one mapping and that different strips of width 2
π/
T
are mapped into
the same region within the unit circle
z
<
1
.
Consider the set of points
s
= σ
0
+
j2
k
π/
T
, with
k
=
0
,
1
,
2
,...,
in the
s-plane. Substituting
s
= σ
0
+
j2
k
π/
T
in Eq. (16.7) yields
=
e
T
(
σ
0
+
j2
k
π/
T
)
=
e
σ
0
T
e
j2
k
π
=
e
σ
0
T
.
z
(16.10)
In other words, the set of points
s
+
j2
k
π/
T
are all mapped to the same
point
z
=
exp(
σ
0
T
) in the z-plane. Equation (16.8) is, therefore, not a unique,
one-to-one mapping, and different strips of width 2
π/
T
in the left-half s-plane
are mapped to the same region within the interior of the unit circle.
= σ
0
We now illustrate the procedure used to obtain an equivalent
H
(
z
) from an
impulse response
h
(
t
) through Examples 16.1 and 16.2.
Example 16.1
Use the impulse invariance method to convert the s-transfer function
H
(
s
)
=
1
s
+ α
into the z-transfer function of an equivalent LTID system.
Solution
Calculating the inverse Laplace transform of
H
(
s
) yields
−α
t
u
(
t
)
.
h
(
t
)
=
e
Using impulse train sampling with a sampling interval of
T
, the impulse
response of the LTID system is given by
−α
kT
u
(
kT
)or
h
[
k
]
=
e
−α
kT
u
[
k
]
.
h
(
kT
)
=
e
The z-transfer function of the equivalent LTID system is given by
1
−α
T
.
H
(
z
)
=
z
h
[
k
]
=
−
1
,
ROC :
z
>
e
1
−
e
−α
T
z
Figure 16.2 compares the impulse response
h
(
t
) and transfer function
H
(
s
)
of the LTIC system with the impulse response
h
[
k
] and transfer function
H
(
z
) of the equivalent LTID system obtained using the impulse invariance
method. A sampling period of
T
=
0
.
1 s and
α =
0
.
5 are used. Comparing
the CT impulse response
h
(
t
), plotted in Fig. 16.2(a), with the DT impulse
response
h
[
k
], plotted in Fig. 16.2(c), we observe that
h
[
k
] is a sampled ver-
sion of
h
(
t
), and the shapes of the impulse responses are fairly similar to each
other.
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