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Analog-to-Digital Filter Transformations 423
1
0.8913
Magnitude Response
0
1
Magnitude in dB
|H|
decibels
16
0.1585
0
0 0.2 0.3 0.5
Analog frequency in π units
30
0 0.2 0.3 0.5
Analog frequency in π units
1
Phase Response
Impulse Response
0.5
0.2
radians
0
ha(t)
0.1
−0.5
0
−1
0 0.2 0.3 0.5
Analog frequency in π units
0 10 20 30 40
time in seconds
FIGURE 8.18 Elliptic analog lowpass filter in Example 8.8
about phase response in our designs, phase is still an important issue in the
overall system. At the other end of the performance scale are the Butterworth
filters, which have maximally flat magnitude response and require a
higher-order N (more poles) to achieve the same stopband specification.
However, they exhibit a fairly linear phase response in their passband.
The Chebyshev filters have phase characteristics that lie somewhere in
between. Therefore in practical applications we do consider Butterworth
as well as Chebyshev filters, in addition to elliptic filters. The choice depends
on both the filter order (which influences processing speed and
implementation complexity) and the phase characteristics (which control
the distortion).
8.4 ANALOG-TO-DIGITAL FILTER TRANSFORMATIONS
After discussing different approaches to the design of analog filters, we
are now ready to transform them into digital filters. These transformations
are complex-valued mappings that are extensively studied in
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