Image Processing Reference
In-Depth Information
x
a
(
t
)
x
*
(
t
)
∞
s
(
t
) =
Σ
δ(
t-nT
)
n
=-∞
FIGURE 3.7
Sampling process.
x
(
n
) ¼
x
a
(
nT
)
(
3
:
80
)
1
x*
(
t
) ¼
x
a
(
nT
)d(
t
nT
)
(
3
:
81
)
n
¼1
The Laplace transform of x*
(
t
)
is
1
x
a
(
nT
)
e
nTs
X*
(
s
) ¼
(
3
:
82
)
n
¼1
The z-transform of the discrete signal x
(
n
)
is
1
x
(
n
)
z
n
X
(
z
) ¼
(
3
:
83
)
n
¼1
Comparing Equations 3.81 and 3.82, we have
X*
(
s
) ¼
X
(
z
)j
z
¼
e
Ts
(
3
:
84
)
Thus, the z-transform of a discrete signal x
(
n
)
is the Laplace transform of the sampled
signal x*
(
t
)
with the change of variable
z
¼
e
Ts
(
3
:
85
)
The above equation de
nes a mapping from complex s-plane to complex z-plane, as
shown in Figure 3.8.
Im(
z
)
Im(
s
)
z
-plane
s
-plane
z
=
e
sT
1
Unit circle
Re(
z
)
Re(
s
)
FIGURE 3.8
Mapping from the complex s-plane to the complex z-plane.
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