of Microprocessors

DIGITAL-TO-ANALOG AND ANALOG-TO-DIGITAL CONVERSION 391
-ol========---~==~""""=""""'~
-5
-10
-15
odB = BUTTERWORTH
-20
-25
iii
~ -30
:2
~
'" -35
-40
-45
-50 ---------------------------------------------------- ------
-55
- 60 l--,.----...,...,..,c---=--c-----::-::-::--=-=----=--;---;:--;:---;;-::-----;c-;:--;-:;--:-;;-'-"--':-':-;-::-;-';~~
0.1 0.12 0.15 0.2 0.25 0.3 0.4 0.5 0.6 0.8 1.0
FREQUENCY (kHz)
Fig. 12-19. Five-section Chebyshev filter response
Figure 12-19 shows two five-section Chebyshev filters with differing
amounts of ripple allowed plus a Butterworth, which can be considered to be
a Chebyshev with zero ripple, for comparison. Note that by merely allowing
0.25 dB of ripple, a reasonable amount for high-fidelity audio, that a 50-dB
attenuation is achieved at 1. 30 times the cutoff frequency rather than 1. 62.
This in turn would allow a sample rate as low as 2.3 times the cutoff
frequency.
Elliptical Response
Believe it or not, there is still something that can be done to improve
cutoff slope. The idea is to follow the basic low-pass filter with a band-reject
(notch) filter tuned just a little beyond the cutoff frequency in an attempt to
make the cutoff sharper. Essentially, the notch depresses the response just
after cutoff in much the same way that peaks improve the flatness just before
cutoff.
Unfortunately, to be effective the notch must be narrow. Thus, on the
other side of the notch, the response curve starts back up toward what it
would have been without the notch filter. At this point, another notch can be
inserted to press the curve back down again. This is continued until the
original filter curve has dropped low enough to be satisfactory without
further help. Figure 12-20 shows the response curve of a 7th order (it is no
longer meaningful to give section or pole counts) elliptical filter. With such a
filter, the sample rate could be a mere 2.19 times the cutoff frequency, the
best yet.