[sdiy] Filter Question

[Magnus Danielson]

From: Scott Gravenhorst <music.maker at gte.net>
Subject: [sdiy] Filter Question
Date: Wed, 07 Nov 2001 15:44:08

> Can a lowpass filter be built by using a hipass filter
> output fed back to a summing amp that sums the original
> signal and the hipass output to cancel high frequencies?  
> My guess is "no" because of phase issues.  Lunatic minds
> want to know.

If you have a highpass filter which, for the sake of this discussion,
has a gain (in the passband) of 1, then you can take the input signal
and the output signal of this filter, sum them together and get a
lowpass responce.

Let's play some with algebra. Say that your highpass filter matches
this equation:

               2
              s
H  (s) = -----------
 HP       2
         s  + as + b

which also has the gain of 1 as s -> oo

Then, if we take an input signal X(s) and forms the output signal Y(s)
by the difference of the input signal and the input signal through the
highpass filter we get:

Y(s) = X(s) - H  (s) * X(s)
               HP

=>
         Y(s)
H  (s) = ---- = 1 - H  (s)
 LP      X(s)        HP

=>

                   2        2             2
                  s        s  + as + b - s      as + b
H  (s) = 1 - ----------- = ---------------- = -----------
 LP           2               2                2
             s  + as + b     s  + as + b      s  + as + b

This is a lowpass function, but only of -6 dB/Oct instead of the -12
dB/Oct.

What you *really* want is to have another tap in there, so you can
eliminate the as term too.

It is by doing exercises like these that you can cook up different
responces from a filter. The state-variable is such a nice filter in
this sence since you can tap the s^0, s^1 and s^2 zero terms as you
like and then simply mix them according to your need with chockingly
simple dimensioning formulas. This explains why they are so popular
for multimode filters, they are an easy pray. Check out the Oberheim
Xpander VCF schematic for an example of a not so easy case, but still
using the same principles as above.

An issue which migth put people up to the test is also the effect of
the Q-value, which will modify the pole equation. If one relies on
summing from the input, then the feedback term must be handled.

Naturally, cooking up all the possible multimode outputs from an Moog
ladder is the natural homelesson for the student in the basic
synthesizer VCF class.

Cheers,
Magnus