As far as I could remember, with sampled signals we always try to forget
values *between* samples, for they are always uniquely determined by values
*at* the samples, so if we get these right, those must be right as well.
The problem with mimicking an analogue filter with digital is that the
analogue one is never band-limited below Nyquist frequency, so it doesn't
have a digital equivalence at all. Some textbooks include impulse invariance
filter design as an alternative to bilinear transform, which would've given
the perfect digital equivalence had the analogue prototype been
band-limited. But with rational transfer functions this never happens.
Xue
-----Original Message-----
From: Theo Verelst
Sent: Monday, November 19, 2012 9:43 PM
To: music-dsp@music.columbia.edu
Subject: Re: [music-dsp] stuck with filter design
Remember the main rules:
Sampled signals can be powerfully processed are nicely fixed (no
"analog" noise" and the bits and words specify exact signals), but
sampling theory must be understood to enforce some main limitations: the
signals of course must have no higher frequency components than half the
sampling frequency, and to get back the "original" signal (as it was
before sampling the analog signal into a digital set of samples), the
reconstruction filter is quite complicated when high accuracy is needed.
So no matter which digital filter equivalent is used to mimic analog
filters, getting proper signal values *between* samples requires quite a
complicated computation (adding a lot of "sinc" functions). All filters
being designed without taking this int consideration (and some do) are
going to sound similar in the sense that they use the fixed delay
between subsequent samples to form delay elements, and that is audible
(and measurable in the properly formed self-correllation signal of the
output).
So no matter whether you use equivalency of digital filters with analog
filter networks (which in the linear case is a well defined part of
"Network Theory"), and the mathematical design tools for electronic
amplifiers (the bode diagrams and such like integral function theory),
the digital filters are going to have properties not easily put
completely in line with their analog "equivalent".
Moreover, it is hard to *at any rate* make some sort of perfect filter,
be it analog, digital, with fourier transforms, etc, only the
complicated (well known, continuous, so NOT the FFT) Fourier Theory can
predict the workings of analog and well made digital filters, and say
theoretically (and 100%) accurate how certain filters will behave with
given signals and of course there are since the beginnings of radio all
kinds of books on how to design certain branches of the filter design tree.
So, are there perfect orthogonal filters, for instance? Yes, but
unfortunately most of them are very complicated to be theoretically all
correct (like in theoretical physics), and all of the digital filters
are highly causal in the sense of costing time to compute and to
reconstruct the correct (emphasis on correct) analog signal. In theory
most perfect filters take close to forever to compute, so to engineer
some great filters, usually the theoretical limitations are quickly in
sight, and even a lot of hard work isn't going to create a
communications receiver or a great audio synthesizer, ever. Even though
of course those jobs *can* be done!
T.Verelst
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