Digital Signal Processing Reference
In-Depth Information
The numerator and denominator factors can then be converted.
( )
()
[
a
+
a
cos
Ω
+
a
cos(
2
Ω
)]
+
j
⋅
[
a
sin(
Ω
)
+
a
sin(
2
Ω
)]
j
Ω
0
1
2
1
2
H
(
e
)
=
(6.28)
[
b
+
b
cos
Ω
+
b
cos(
2
Ω
)]
+
j
⋅
[
b
sin(
Ω
)
+
b
sin(
2
Ω
)]
0
1
2
1
2
We see in Figure 6.4 that the specifications have been met (at 1,000 Hz
the −1 dB gain = 0.89125, and at 5,000 Hz the −20 dB gain = 0.1).
Example 6.7 Chebyshev Bilinear Transform Filter Design
Problem:
Use WFilter to completely design a Chebyshev digital IIR filter
and display the magnitude response. The specifications for this filter are
a
pass
= −1 dB,
a
stop
= −60 dB,
f
pass
= 10 kHz,
f
stop
= 20 kHz, and
f
samp
= 50 kHz
Solution:
We can supply these values to WFilter and WFilter will calculate
the magnitude response of the filter, as shown in Figure 6.5. The digital IIR
coefficients and the pole and zero locations are shown in Figure 6.6.
In Figure 6.5 we see the effect of sampling on the frequency response. In
particular, we see the lowpass response replicated in mirror image form at the
sampling frequency of 50 kHz. In addition, we see the lower half of the reflection
at twice the sampling frequency of 100 kHz. If we had chosen a larger-frequency
scale, we would see these replications reproduced at all multiples of the sampling
frequency. Of course, in the typical discrete-time system, the antialiasing filter
will be set to one-half of the sampling frequency (25 kHz in this case), and the
user would not see the effects of the higher-frequency components.
Figure 6.5
Magnitude response from WFilter.
