Digital Signal Processing Reference
In-Depth Information
or
m
=−∞
x
1
[
m
]
z
∞
p
=−∞
x
2
[
p
]
z
∞
←→
−
m
−
p
,
x
1
[
k
]
∗
x
2
[
k
]
which proves Eq. (13.24).
Like the DTFT convolution property discussed in Chapter 11, the time-
convolution property of the z-transform provides us with an alternative approach
to calculate the output
y
[
k
] when a DT sequence
x
[
k
] is applied at the input of
an LTID system with the impulse response
h
[
k
]. The procedure for calculating
the output
y
[
k
] of an LTID system in the complex z-domain consists of the
following four steps.
(1) Calculate the z-transform
X
(
z
) of the input sequence
x
[
k
]. If the input
sequence and the impulse response are both causal functions, then the
unilateral z-transform is used. If either of the two functions is non-causal,
the bilateral z-transform must be used.
(2) Calculate the z-transform
H
(
z
) of the impulse response
h
[
k
] of the LTID
system. The z-transform
H
(
z
) is referred to as the z-transfer function of
the LTID system and provides a meaningful insight into the behavior of the
system.
(3) Based on the convolution property, the z-transform
Y
(
z
) of the resulting
output
y
[
k
] is given by the product of the z-transforms of the input signal
and the impulse response of the LTID system. Mathematically, this implies
that
Y
(
z
)
=
X
(
z
)
H
(
z
).
(4) Calculate the output response
y
[
k
] in the time domain by taking the inverse
z-transform of
Y
(
z
) obtained in step (3).
Example 13.9
The exponential decaying sequence
x
[
k
]
=
a
k
u
[
k
]
,
0
≤
a
≤
1, is applied at the
input of an LTID system with the impulse response
h
[
k
]
=
b
k
u
[
k
], 0
≤
b
≤
1.
Using the z-transform approach, calculate the output of the system.
Solution
Based on Table 13.1, the z-transforms for the input sequence and the impulse
response are given by
1
1
−
az
−
1
1
1
−
bz
−
1
.
X
(
z
)
=
and
H
(
z
)
=
The z-transform of the output signal is, therefore, calculated as follows:
1
Y
(
z
)
=
H
(
z
)
X
(
z
)
=
−
1
)
.
(1
−
az
−
1
)(1
−
bz
Search WWH ::
Custom Search