# [sdiy] Pole Mixing

Andrew Simper andy at cytomic.com
Sun Apr 11 16:22:41 CEST 2021

```The reason they grounded the first cap is the chip they were used in the
x-pander did the resonance gain in - k*lp4 internally, so the only way to
get that back out again was through losing that first cap and grabbing the
signal from that point.

The response you get when doing this is (3 pole cascade with global
feedback):

(g^3 (m0 + m1 + m2 + m3) + g^2 (3 m0 + 2 m1 + m2) s + g (3 m0 + m1) s^2 +
m0 s^3) /
(g^3 (1 + k) + 3 g^2 s + 3 g s^2 + s^3)

which requires a feedback gain of k = 8 to self oscillate, but you also
have to shift the frequency down by scaling the g by 1/sqrt(3), which you
can get by solving the frequency the denominator becomes zero with k = 8.

I seem to remember the x-pander calibrates both the 4 pole and 3 pole
filter separately and uses a chip to control it, so this scaling is
probably done in digital land, so it won't appear on the schematic.

Cheers,

Andy

On Sat, 10 Apr 2021 at 00:50, David Moylan via Synth-diy <
synth-diy at synth-diy.org> wrote:

> Working out the algebra to handle the feedback signal made me realize
> something about the Xpander implementation.  Rather than including a mix
> of the true dry input signal needed for certain modes, the Xpander
> switches the first integrator essentially into a buffer.   This makes
> the mix from the first integrator section a "dry" signal and the rest of
> the transfer function is then 3 pole based.   That means the feedback
> signal in that structure, for those modes, is also a 3 pole low pass
> signal instead of the 4 pole low pass when the integrator is engaged.
>
> Makes me curious about how that will effect the transfer functions with
> feedback applied.   Will maybe have to set up a new page dedicated to
> the exact Xpander structure after I get this feedback signal worked out
> for the original app. Down a rabbit hole...
>
> On 4/9/21 10:00 AM, Richie Burnett wrote:
> > Taking the feedback from after the 4th pole makes all of the responses
> > become 4th order when feedback is applied.  So even something
> > otherwise simple like a 1-pole low-pass filter can display resonance
> > when feedback is applied in this way.
> >
> > -Richie,
> >
> >
> > -----Original Message----- From: David Moylan via Synth-diy
> > Sent: Friday, April 9, 2021 2:19 PM
> > To: synth-diy at synth-diy.org
> > Subject: Re: [sdiy] Pole Mixing
> >
> > Thanks, Tom.   In practice, I do find quite a few interesting. LP1 +
> > Notch sounds great.  And just being able to switch between LP4, LP2, LP1
> > is also useful.  I don't get a lot of mileage out of the HP only modes,
> > usually want some LP too, but that makes HP3+LP1 mode interesting.
> > Though I agree, in the universe of all possible transfer functions the
> > list is relatively small.  It also begs the question "how many of these
> > could be approximated with 2 state variables in series?".   But that
> > ignores the feedback paths which I haven't worked into the equations
> > yet.  In the Xpander the feedback is always from the LP4 tap like it
> > would be for standard low pass filter and I think I remember someone,
> > maybe David Dixon, pointing out that it has interesting effects on the
> > curves as resonance is increased.
> >
> > If you click on the slider in question you should be able to use arrow
> > keys to step in .1 increments.  I thought that was fine enough to get
> > close to some desired curve, after that get out the pencil and paper or
> > a soldering iron!  All of the presets except the 20db LP shelf use
> > integers anyway, seems almost necessary to get the cancellations
> > required to make interesting curves.
> >
> > On 4/9/21 5:34 AM, Tom Wiltshire wrote:
> >> Absolutely agree, that is a fantastic piece of work.
> >>
> >> It makes all sorts of things about pole-mixing more obvious,
> >> including how few of the combinations are actually interesting, and
> >> as Richie said, how sensitive some of the combinations are.
> >>
> >> If you’re still working on it, would it be possible to add boxes to
> >> type in the coefficients as an alternative to the sliders? Getting
> >> specific values with the little sliders is quite fiddly.
> >>
> >> Thanks very much for this though - really great.
> >>
> >> Tom
> >>
> >> ==================
> >>         Electric Druid
> >> Synth & Stompbox DIY
> >> ==================
> >>
> >>
> >>
> >>> On 8 Apr 2021, at 23:24, David Moylan via Synth-diy
> >>> <synth-diy at synth-diy.org> wrote:
> >>>
> >>> Hi All.  I banged together a little web app to play around with
> >>> filter pole mixing, of the Oberheim Xpander type.  You can mix poles
> >>> in varying amounts and see the output magnitude shape as well as the
> >>> transfer function.  Y axis is Db and X axis is log scale based on
> >>> normalized frequency (so basically 1 equals the cutoff frequency).
> >>> Haven't done phase plot yet.
> >>>
> >>> If you have an interest in this sort of thing check it out:
> >>>
> >>> https://expeditionelectronics.com/Diy/Polemixing
> >>>
> >>> Cheers.
> >>>
> >>> --
> >>> David Moylan
> >>> Expedition Electronics
> >>>
> >>> _______________________________________________
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> >>> Synth-diy at synth-diy.org
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> >>
> >> _______________________________________________
> >> Synth-diy mailing list
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> >
> >
>
> --
> David Moylan
> Expedition Electronics
>
> _______________________________________________
> Synth-diy mailing list
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>
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