Digital Signal Processing Reference
In-Depth Information
We know from the modulation theorem in (5.22) that the Fourier trans-
formof (7.2) is the convolution (in the space of 2
π
-periodic functions) of the
Fourier transforms of
h
[
n
]
and
w
[
n
]
:
l
g
r
,
y
i
d
.
,
©
,
L
s
π
1
2
H
e
j
ω
)=
e
j
ω
)
e
j
(
ω
−
θ
)
)
(
H
(
W
(
d
θ
π
−
π
e
j
ω
)
It is easy to compute
W
(
as
sin
N
1
2
ω
+
N
e
j
ω
)=
e
−j
ω
n
sin
2
W
(
=
(7.4)
n
=
−
N
An example of
W
(
e
j
ω
)
for
N
=
6 is shown in Figure 7.6. By analyzing the
e
j
ω
)
form of
W
(
for arbitrary
N
, we can determine that:
e
j
ω
)
•
the first zero crossing of
W
(
occurs at
ω
=
2
π/
(
2
N
+
1
)
;
•
thewidth of themain lobe of themagnitude response is
Δ=
4
π
(
2
N
+
1
)
;
•
there aremultiple
sidelobes
, an oscillatory effect around themain lobe
and there are up to 2
N
sidelobes for a 2
N
+
1-tap window.
12
9
6
3
0
-3
-
π
-3
π/
4
-2
π/
4
-
π/
4
0
π/
4
2
π/
4
3
π/
4
π
Figure 7.6
Fourier transform of the rectangular window for
N
=
6.
In order to understand the shape of the approximated filter, let us go
back to the lowpass filter example and try to visualize the effect of the con-
volution in the Fourier transform domain. First of all, since all functions
are 2
-periodic, everything happens circularly, i.e. what “goes out” on the
right of the
π
[
−
π
,
π
]
interval “pops” immediately up on the left. The value at
0
of
H
e
j
ω
)
e
j
ω
)
ω
(
is the integral of the product between
H
(
and a version of
