posted 16 April 2001 18:02            

The mysterious Feedback parameter is a multiplicative gain, taken to the 1/N power (for order 2N), used on a 1-sample delayed filter output which is fed back to and mixed with the input signal.

The equations for these filters are:

--- LPF ---
fc = !Frequency / SignalProcessor halfSampleRate
sf = fc normSin.
cf = fc normCos.

damping[i] = 2*Sin[(2*i-1) Pi/4N] i=1,2,..N
beta[i] = 0.5*(1-(0.5*damping[i]*sf))/(1+(0.5*damping[i]*sf))
gamma[i] = (0.5 + beta[i])*cf
alpha[i] = 0.25*(0.5 + beta[i] - gamma[i])
mu = 2
sigma = 1

The Biquad IIR coefficients for the direct form IIR of cascade section i:

y[n] = 2*(alpha[i]*(x[n]+mu*x[n-1]+sigma*x[n-2])+ gamma[i]*y[n-1] - beta[i]*y[n-2])

--- HPF ---

everything the same, except

mu = -2.
alpha[i] = 0.25*(0.5 + beta[i] + gamma[i])

--------------------------
The filter sections need to be computed in descending order of index i, in order to avoid internal overflows as the damping factor decreases.

I am still mystified as to why one would do the feedback thingy, given that changing its value shifts the location of the resonance peak to and fro in frequency. I can only imagine that it would be difficult to compute the required coefficients for filters with resonance peaks beyond order 2.

I also find that my filters constructed with an explicit feedback loop using a memory writer do not suffer the instabilities demonstrated by the Kyma filters, when the feedback value is increased all the way up to 1 (for HPF's -- LPF's need negative feedback else they do go ballistic). The Kyma version seems to go berserk above Feedback = 0.707.

- DM

[This message has been edited by David McClain (edited 16 April 2001).]

posted 17 April 2001 00:07            

Hence a setting of 0.5 implies a feedback gain of 0.707.

The sense is negative for LPF's (presumably to prevent buildup of large DC offsets), and positive for HPF's.

posted 18 April 2001 16:56            

But this paper explicitly shows the need for both tuning adjustment tables, and "separation" tables that modify the value of the feedback as a function of corner frequency in order to maintain nearly constant Q. And in the case of bilinear transformed designs from the continuous domain, the separation tables modifying feedback with frequency are REQUIRED in order to preserve filter stability.

So, the answer to my original query has finally been found -- Moog set the precedence for using feedback on filters to get resonance peaks at the corner frequencies. But his filters didn't shift the peak location with increased feedback. Furthermore, his filters were 4-stage single pole RC filters and not 4-pole Butterworth filters.

I guess I can now live in peace...

- DM