On 19/11/2012 9:24 AM, Bjorn Roche wrote:
(Shashank wrote:)
I have one more question:
Why so many people use analog prototypes to get a digital filter
? Why not just put a few constraints on location of poles/zeros
on Z plane and get done with it ?
This is a really great question.
Indeed. I hope someone tries to answer the mathematical aspect of the
question...
Perhaps student engineers (and their instructors) prefer to do a little
algebra than calculus on complex-domain rational functions?
One answer is that analog filter design is a highly developed art,
and therefore serves as an excellent starting point.
I'm not sure how excellent it is given the problem of trying to map an
infinite frequency space on to a periodic space.
It may well be an excellent starting point if you've undertaken a
traditional education where you get one or two years of calculus and
differential equations before you even look at a discrete system or a
difference equation. But if you bypassed that and started programming
computers before you got told the analog world was more important than
the digital one...
All that said, some folks have started to think along the same lines
as you. After all 1. there may be unique digital solutions (and I'm
not just talking about FIR filters), and 2. you should be able to
learn digital filter design without also having to learn analog
filter design. To that end, here is one interesting
paper:
>http://www.elec.qmul.ac.uk/people/josh/documents/Reiss-2011-TASLP-ParametricEqualisers.pdf
Haven't seen that one... but a good paper in this direction is "A
Generalization of the Biquadratic Parameteric Equaliser" by Knud Bank
Christensen
http://www.aes.org/e-lib/browse.cfm?elib=12429
As a side note, there are, indeed, problems associated with filters
designed with an analog prototype. For example, let's say you design
a bell filter in the analog domain, and map it into the digital
domain with a sample rate of, say 44100 Hz. Let's further assume that
in the analog domain, the gain at the niquist frequency of this
filter is 1 dB. That means the filer will boost 20 kHz by 1 dB. When
you use the bilinear transform, though, the resulting filter will
have a gain a the niquist frequency of zero. This is usually not a
serious problem, and often not a problem at all. However, if you are
designing a parametric EQ, the difference will be noticeable at
extreme settings.
This problem is addressed directly by the Christensen paper I just linked.
You could say this is one reason to do audio
production at sample rates > 44100 Hz but there are arguments both
ways there as well.
Another thing to keep in mind is that even if you get the magnitude
response correct without oversampling, the phase response will not be
the same as the analog prototype. And high frequency phase coherence can
be important in audio applications.
Ross.
P.S. Shashank: my blog is at rossbencina.com , not much DSP there
though, mainly programming and random soapboxing.
--
dupswapdrop -- the music-dsp mailing list and website:
subscription info, FAQ, source code archive, list archive, book reviews, dsp
links
http://music.columbia.edu/cmc/music-dsp
http://music.columbia.edu/mailman/listinfo/music-dsp
