Digital Signal Processing Reference
In-Depth Information
Inverse DTFT of the ideal lowpass filter
0.25
0.2
0.15
0.1
0.05
0
0.05
−
−
0.1
−
21
−
16
−
11
−
6
0
4
9
14
19
Value of the index n
Figure 3.22
The inverse DTFT of an ideal lowpass filter.
This is a line spectrum which is shown in Figure 3.22. It is interesting to com-
pare the general shape of the rectangular pulse function
x
r
(n)
and its frequency
response
X
r
(e
jω
)
with the frequency response
H(e
jω
)
of the lowpass filter and
its inverse DTFT
h(n)
which are derived above. However, it should be noted that
they are not duals of each other, because
X
r
(e
jω
)
is not exactly a sinc function
of
ω
.
3.4.2 Multiplication Property
When two discrete -time sequences are multiplied, for example,
x(n)h(n)
y(n)
,
the DTFT of
y(n)
is the convolution of
X(e
jω
)
and
H(e
jω
)
that is carried
out in the frequency domain as an integral over one full period. Choosing the
period [
−
=
ππ
] in the convolution integral, symbolically denoted by
X(e
jω
)
*
H(e
jω
)
Y(e
jω
)
, we have the property
=
π
1
2
π
X(e
jς
)H (e
j(ω
−
ς)
)dς
x(n)h(n)
⇔
(3.56)
−
π
Remember that we will use (3.55) and (3.56) in the design of FIR filters discussed
in Chapter 5.
Example 3.12
The properties and the DTFT-IDTFT pairs discussed here are often used in
frequency-domain analysis of discrete-time systems,
including the design of
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