Digital Signal Processing Reference
In-Depth Information
where
d
(
t
-
kT
) is the impulse (delta) function delayed by
kT
and
T
=
1/
F
s
is the
sampling period. The function
x
s
(
t
) is zero everywhere except at
t
=
kT
. The Laplace
transform of
x
s
(
t
) is
•
()
=
Ú
Ú
()
Xs
xte dt
-
st
s
s
0
•
{
}
() ()
+
()
(
)
+◊◊◊
=
xt
d
t
xt
d
t T
-
e
-
st
dt
(4.2)
0
From the property of the impulse function
•
Ú
()
(
)
()
ft
d
t kTdt
-
=
fkT
0
X
s
(
s
) in (4.2) becomes
•
Â
()
=
()
+
()
()
()
-
sT
-
2
sT
-
nsT
(4.3)
X
s
x
0
x T e
+
x
2
T e
+◊◊◊=
x nT e
s
n
=
0
Let
z
=
e
sT
in (4.3), which becomes
•
Â
0
()
=
()
-
Xz
xnTz
n
(4.4)
n
=
Let the sampling period
T
be implied; then
x
(
nT
) can be written as
x
(
n
), and (4.4)
becomes
•
Â
0
()
=
()
(
{}
-
n
(4.5)
Xz
xnz
=
ZT xn
n
=
which represents the
z
-transform (
ZT
) of
x
(
n
). There is a one-to-one correspon-
dence between
x
(
n
) and
X
(
z
), making the
z
-transform a unique transformation.
Exercise 4.1: ZT of Exponential Function x(n)
=
e
nk
The
ZT
of
x
(
n
)
=
e
nk
,
n
0 and
k
a constant, is
• •
ÂÂ
0
n
()
=
(
)
Xz
e z
nk
-
n
=
ez
k
-
1
(4.6)
n
=
n
=
0
Using the geometric series, obtained from a Taylor series approximation
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