Interesting Paper available on Moog VCF

[Tom May]

Don Tillman <don at till.com> writes:

>    Date: Fri, 20 Sep 1996 13:15:51 -0400 (EDT)
>    From: Eli Brandt <eli at ux3.sp.cs.cmu.edu>
> 
>    Take the case where we've got harmonic partials and they're all in
>    phase.  So the input signal x(t) is a sum of cos(kt) terms, out to K.
>    Each term a_k cos(kt) can be expanded, using the Chebychev* T_n, to a
>    degree-k polynomial in cos(t).  Waveshaping their sum with a degree-D
>    polynomial gives one of degree-KD.  Then translate each cos^n(t) back
>    to a sum of cos(kt) terms, k<=n.  So we have a sum of cos(kt) terms,
>    k<=KD.
> 
> Wow, I have no idea what you're talking about!

Neither did I.  The interesting thing about Chebyshev polynomials is
this: using a Chebyshev polynomial as a transfer function has the
property that if you put cos(x) through the Nth order Chebyshev
polynomial, you get cos(Nx) out.  It works for sin, too; the phase
doesn't matter.  So you can "dial in" the harmonic content that your
waveshaping function gives by using a linear combination of the
C. polynomials, where the weight for each polynomial is just the
amount of that harmonic you want.  Note, however, that you only get
the desired harmonic mix when the amplitude of the input is one.

For the record, the zeroth and first order Chebyshev polynomials are
trivial to find: C_0(x) = 1 and C_1(x) = x.  And for some strange
reason, the rest of them can be computed from the relation C_n(x) =
2x*C_{n-1}(x) - C_{n-2}(x).

Tom.