Digital Signal Processing Reference
In-Depth Information
H
LP
(
e
j
w
)
H
HP
(
e
j
w
)
1
1
w
w
−
w
c
w
c
−
p
p
−
p
−
w
c
w
c
p
0
0
(
a
)
(
b
)
H
BP
(
e
j
w
)
H
BS
(
e
j
w
)
1
1
w
w
−
p
−
w
c
2
−
w
c
1
w
c
1
w
c
2
p
−
p
−
w
c
2
−
w
c
1
w
c
1
w
c
2
p
(
c
)
(
d
)
Figure 5.5
Magnitude responses of four ideal filters. (Reprinted from Ref. 9, with per-
mission from John Wiley & Sons, Inc.)
Another form for the Fourier series coefficients is
ω
c
π
sinc
(ω
c
n)
sin
(ω
c
n)
πn
c
LP
(n)
=
=
;
∞
<n<
∞
(5.32)
Note that sinc
(ω
c
n)
=
1when
n
=
0, so we find another way of listing the
coefficients as
⎨
ω
c
π
;
n
=
0
c
LP
(n)
=
(5.33)
sin
(ω
c
n)
πn
⎩
|
|
;
n
>
0
The Fourier series coefficients for the ideal HP, BP, and BS filter responses
shown in Figures 5.5b-d can be similarly derived as follows:
⎨
⎩
ω
c
π
;
1
−
n
=
0
c
HP
(n)
=
(5.34)
sin
(ω
c
n)
πn
−
;
|
n
|
>
0
⎨
⎩
ω
c
2
−
ω
c
1
;
n
=
0
π
c
BP
(n)
=
(5.35)
1
πn
[sin
(ω
c
2
n)
−
sin
(ω
c
1
n)
]
;
|
n
|
>
0
⎨
⎩
(ω
c
2
−
ω
c
1
)
−
;
n
=
1
0
π
c
BS
(n)
=
(5.36)
1
πn
[sin
(ω
c
1
n)
−
sin
(ω
c
2
n)
]
;
|
n
|
>
0
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