Image Processing Reference
In-Depth Information
S
OLUTION
The solution is easily found using the delay property of
the
z-transform,
note that
6)
n
u(n
6)
4
(0
6)
n4
u(n
x(n)
¼
(0
:
4)
¼
(0
:
:
4)
Therefore,
6)
4
z
3
(z
z
z
(0
:
6)
4
z
4
X(z)
¼
(0
:
6
¼
0
:
0
:
6)
(c)
Convolution Property
: Convolution property of z-transform states that
convolution in time domain is multiplication in z-domain. This means
that the z-transform of convolution of two functions is the product of their
corresponding z-transforms. If
y
(
n
) ¼
x
(
n
)
*
h
(
n
)
(
:
)
3
54
then
Y
(
z
) ¼
X
(
z
)
H
(
z
)
(
3
:
55
)
Proof:
1
1
1
y
(
n
)
z
n
x
(
k
)
h
(
n
k
)
z
n
Y
(
z
) ¼
¼
n
¼1
n
¼1
k
¼1
1
1
h
(
n
k
)
z
n
¼
x
(
k
)
(
3
:
56
)
k
¼1
n
¼1
Using shifty property, we have
1
1
x
(
k
)
H
(
z
)
z
k
x
(
k
)
z
k
Y
(
z
) ¼
¼
H
(
z
)
¼
H
(
z
)
X
(
z
)
(
3
:
57
)
k
¼1
k
¼1
As a result of this important theorem, the z-transform of the output of a
linear shift-invariant system is the product of the z-transform of the input
signal and the z-transform of the impulse response of the system. The
z-transform of the system impulse response h
(
n
)
is the system transfer
function H
(
z
)
.
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