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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