Digital Signal Processing Reference
In-Depth Information
functions is obtained from the convolution of
(e
jω
)
with the frequency response
H
LP
(e
jω
)
of the ideal, desired, frequency response.
π
1
2
π
H
M
(e
jω
)
H
LP
(e
jϕ
)(e
j(ω
−
ϕ)
)dϕ
=
(5.40)
−
π
The mainlobe of
(e
jω
)
, centered at
ω
=
0, has a width defined by the first
1
)
2
]=
zero crossings on either sides of
ω
=
0, which occur when [(2
M
+
±
π
,
that is, when
ω
1
)
.
As
M
increases, the width of the mainlobe and the sidelobes decreases, giving
rise to more sidelobes or ripples in the same frequency band. At the same time,
the peak amplitudes of the mainlobe and the sidelobes increase such that the
area under each lobe remains constant. These features of
(e
jω
)
directly reflect
on the behavior of
H
M
(e
jω
)
when it is convolved with
H
LP
(e
jϕ
)
. The effect
of convolution between
H
LP
(e
jω
)
and
(e
jω
)
is illustrated by looking at the
overlapping interval over which the product
H
LP
(e
jϕ
)(e
j(ω
−
ϕ)
)
is integrated,
for four different values of
ω
, in Figure 5.8. It is obvious that if the width of the
mainlobe is extremely narrow, the resulting
H
M
(e
jω
)
will have a sharp drop at
ω
=
2
π(
2
M
+
1
)
so that the width of the mainlobe is 4
π/(
2
M
+
ω
c
. If the number of sidelobes or their peak values in
(e
jω
)
increases, so
also will the number of ripples and the maximum error in
H
M
(e
jω
)
.
=
H
id
(
q
)
y
(
w − q
)
w = p
y
(
w − q
)
w
c
<
w
<
p
q
q
−
w
c
w
c
−
w
c
w
c
p
−p
p
y
(
w
q
)
w = w
c
−
y
(
w
q
)
0 <
w
<
w
c
−
q
q
−
w
c
w
c
−
w
c
w
c
p
−p
p
H(
w
)
p
w
−w
c
w
c
Figure 5.8
Convolution of the frequency response of a rectangular window with an ideal
filter. (Reprinted from Ref. 9, with permission from John Wiley & Sons, Inc.)
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