Digital Signal Processing Reference
In-Depth Information
then the time-convolution property states that
DTFT
←−−→
x
1
[
k
]
∗
x
2
[
k
]
X
1
(
Ω
)
X
2
(
Ω
)
.
(11.47)
In other words, the convolution between two DT sequences in the time domain is
equivalent to multiplication of the DTFTs of the two functions in the frequency
domain. Note that the CTFT also has a similar property, as stated in Section
5.4.8.
Equation (11.47) provides us with an alternative technique for calculating the
convolution sum using the DTFT. Expressed in terms of the following DTFT
pairs:
DTFT
←−−→
DTFT
←−−→
DTFT
←−−→
Y
(
Ω
)
,
x
[
k
]
X
(
Ω
)
,
h
[
k
]
H
(
Ω
)
,
and
y
[
k
]
the output sequence
y
[
k
] can be expressed in terms of the impulse response
h
[
k
] and the input sequence
x
[
k
] as follows:
DTFT
←−−→
y
[
k
]
=
x
[
k
]
∗
h
[
k
]
Y
(
Ω
)
=
X
(
Ω
)
H
(
Ω
)
.
(11.48)
In other words, the DTFT of the output sequence is obtained by multiplying
the DTFTs of the input sequence and the impulse response. The procedure for
evaluating the output
y
[
k
] of an LTID system in the frequency domain therefore
consists of the following four steps.
(1) Calculate the DTFT
X
(
Ω
) of the input signal
x
[
k
].
(2) Calculate the DTFT
H
(
Ω
) of the impulse response
h
[
k
] of the LTID system.
The DTFT
H
(
Ω
) is referred to as the transfer function of the LTID system
and provides a meaningful insight into the behavior of the system.
(3) Based on the convolution property, the DTFT of the output
y
[
k
]isgiven
by
Y
(
Ω
)
=
H
(
Ω
)
X
(
Ω
).
(4) The output
y
[
k
] in the time domain is obtained by calculating the inverse
DTFT of
Y
(
Ω
) obtained in step (3).
Since the DTFTs are periodic with period
Ω
=
2
π
, steps (1)-(4) can be applied
only to the frequency range [
−π
≤
≤ π
].
Ω
Example 11.15
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 DTFT approach, calculate the output of the system.
Solution
Based on Table 11.2, the DTFTs for the input sequence and the impulse response
are given by
1
1
−
a
e
1
1
−
b
e
DTFT
←−−→
DTFT
←−−→
x
[
k
]
and
h
[
k
]
−
j
Ω
.
−
j
Ω
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