Digital Signal Processing Reference
In-Depth Information
0.3
0.2
l
g
r
,
y
i
d
.
,
©
,
L
s
0.1
−
30
−
20
−
10
0
10
20
30
−
0.1
Figure 5.11
Impulse response (portion) of the ideal lowpass filter,
ω
=
π/
3.
c
absolutely summable; this means that any FIR approximation of the ideal
lowpass obtained by truncating
h
needs a lot of samples to achieve some
accuracy and that, in any case, convergence to the ideal frequency response
is only be in the mean square sense. An immediate consequence of these
facts is that, when designing realizable filters, we will take an entirely differ-
ent approach.
Despite these practical difficulties, the ideal lowpass and its associated
DTFT pair are so important as a theoretical paradigm, that two special func-
tion names are used to denote the above expressions. These are defined as
follows:
[
n
]
1
|
x
|≤
1
/
2
rect
(
x
)=
(5.33)
0
|
x
|
>
1
/
2
⎧
⎨
⎩
sin
(
π
x
)
x
=
0
sinc
(
x
)=
π
x
(5.34)
1
x
=
0
Note that the sinc function is zero for all integer values of the argument ex-
cept zero. With this notation, and with respect to the bandwidth of the filter,
the ideal lowpass filter's frequency response between
−
π
and
π
becomes
rect
ω
ω
b
e
j
ω
)=
H
lp
(
(5.35)
(obviously 2
π
-periodized over all
). Its impulse response in terms of band-
width becomes
sinc
ω
b
2
n
]=
ω
b
2
h
lp
[
n
(5.36)
π
π
