Digital Signal Processing Reference
In-Depth Information
H
lp
(e
j
w
)
⎮
⎮
l
w
c
−w
c
−
2
p
−p
0
p
2
p
2
w
(
a
)
H
lp
(e
j
w
)
⎮
⎮
−
w
c
w
c
−
p
0
p
(
b
)
H
lp
(e
j
w
)
⎮
⎮
w
c
2
p
−w
c
0
2
p
w
−
−
p
p
(
c
)
H
lp
(e
j
w
)
⎮
⎮
w
c
w
0
p
(
d)
Figure 3.7
Magnitude responses of ideal lowpass and highpass filters.
frequency 0
.
3
π
, the actual frequency can be easily computed as 30% of the
Nyquist frequency, and when the sampling period
T
or the sampling frequency
ω
s
(or
f
s
=
1
/T
) is given, we know that 0
.
3
π
represents
(
0
.
3
)(ω
s
/
2
)
rad/s or
(
0
.
3
)(f
s
/
2
)
Hz. By looking at the plot, one should therefore be able to determine
what frequency scaling has been chosen for the plot. And when the actual sam-
pling period is known, we know how to restore the scaling and find the value
of the actual frequency in radians per second or in hertz. So we will choose the
normalized frequency in the following sections, without ambiguity.
The magnitude response of the ideal filters shown in Figures 3.7 and 3.8 cannot
be realized by any transfer function of a digital filter. The term “designing a digital
filter” has different meanings depending on the context. One meaning is to find a
transfer function
H(z)
such that its magnitude
H(e
jω
)
approximates the ideal
magnitude response as closely as possible. Different approximation criteria have
been proposed to define how closely the magnitude
H(e
jω
)
approximates the
ideal magnitude. In Figure 3.9a, we show the approximation of the ideal lowpass
filter meeting the elliptic function criteria. It shows an error in the passband as
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