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Tue Jan 4 15:31:08 EST 2011

Nonlinear digital Moog VCF + bandlimited sawtooth.

Also mentioned in Stilson PhD[2] is Antti Huovilainen: Nonlinear
digital implementation of the Moog ladder filter[1][3] ([5][4]).
Interesting derivation, but not such a surprising result.

However, the piecewize parabola is a nice trick!  So I wonder if this
can be taken further one order by representing the remaining "bounce"
discontinuity by a 2nd order "brake" discontinuity and differentiating
twice, i.e. by using piecewize 3rd order polynomials:

   (x-1) x (x+1)

Deriving once gives the piecewize parabola, deriving again gives the
sawtooth.  The 3rd order polynomial's discontinuity is smoother so
should roll off faster (18dB/octave).

It shouldn't be too hard to implement in Pd either..
Indeed:  differentiated parabolic   (DPW)  [6]
         twice differentiated cubic (2DCW) [7]

Looks like the obvious extensions are already published[8].

  V. Välimäki, J. Nam, J. O. Smith, and J. S. Abel, “Aliassuppressed
  oscillators based on differentiated polynomial waveforms,” IEEE
  Trans. Audio, Speech, Language Processing, vol. 18, no. 4,
  pp. 786–798, May 2010.

[1] http://dafx04.na.infn.it/WebProc/Proc/P_061.pdf
[2] https://ccrma.stanford.edu/~stilti/papers/Welcome.html
[3] http://www.mitpressjournals.org/doi/abs/10.1162/comj.2006.30.2.19
[4] md5://81ef26b98b858bfc7ea351850b7f8872
[5] md5://ec26bdee832237793c6de78875942a60
[6] http://zwizwa.be/darcs/pd/abstractions/saw2~.pd
[7] http://zwizwa.be/darcs/pd/abstractions/saw3~.pd
[8] http://ieeexplore.ieee.org/Xplore/login.jsp?url=http%3A%2F%2Fieeexplore.ieee.org%2Fiel5%2F10376%2F5446581%2F05153306.pdf%3Farnumber%3D5153306&authDecision=-203


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