[sdiy] Pole Mixing

David Moylan dave at expeditionelectronics.com
Sun Apr 11 19:06:08 CEST 2021


That's the denominator I have with g normalized to 1.   But then the 
numerator for the A term (dry mix signal) would end up with that k term 
as well when you multiply it to get everything over the common 
denominator.  That's why my 4P HP ends up as

s^^4 + k
---------------------
(s+1)^^4 + k

(In all of the modes mixing dry in the dry mix coefficient is 1).  Am I 
missing something that cancels the k term in the numerator?  I tried it 
without the k in the numerator and the HP curves remain as HP curves 
rolling off at low frequencies, and seem to match Dixon's.  It seems 
that is the major discrepancy now just trying to figure out why.

On 4/11/21 11:49 AM, Andrew Simper wrote:
> You can also write the denominators as:
>
> (g^4 (1 + k) + 4 g^3 s + 6 g^2 s^2 + 4 g s^3 + s^4) = (g + s)^4 + g^4 k
>
> and
>
> (g^3 (1 + k) + 3 g^2 s + 3 g s^2 + s^3) = (g + s)^3 + g^3 k
>
> for the 4 pole and 3 pole versions respectively, which are a bit 
> shorter, but hide the terms that need cancelling for various responses 
> to happen. I'm not sure the numerators get much better than I've 
> already posted.
>
> Cheers,
>
> Andy
>
> On Sun, 11 Apr 2021 at 22:22, Andrew Simper <andy at cytomic.com 
> <mailto:andy at cytomic.com>> wrote:
>
>     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 <mailto: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 <mailto: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 <mailto: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
>         <https://expeditionelectronics.com/Diy/Polemixing>
>         >>>
>         >>> Cheers.
>         >>>
>         >>> --
>         >>> David Moylan
>         >>> Expedition Electronics
>         >>> sonic adventures!
>         >>>
>         >>> _______________________________________________
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>         >>
>         >> _______________________________________________
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>         >
>         >
>
>         -- 
>         David Moylan
>         Expedition Electronics
>         sonic adventures!
>
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>

-- 
David Moylan
Expedition Electronics
sonic adventures!

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