AW: additive

[David MOYLAN]

On Thu, 13 Aug 1998, Eli Brandt wrote:

> Paul Perry wrote:
> > I'm not a mathematician.... but,can anyone say whether it
> > is possible to additively synthesise an arbitrary periodic
> > waveform from a series of triangles?
> 
> Excepting DC, I think so.  Take the sine spectrum {x_i} that you want;
> start off with tri1=x1 to match the fundamental and tri2=x2 to match
> the second partial.  Then you have to match the third plus cancel the
> junk you put in with tri1, so tri3=x3-x1/9.  And so forth.  Ought to
> converge.
> 
> Deconvolution of {x_i} by {1, 0, 1/9, 0, 1/25, ...} I guess.
> 
> -- 
>      Eli Brandt  |  eli+ at cs.cmu.edu  |  http://www.cs.cmu.edu/~eli/
> 
I can definitely see the use here.  Using triangle vco's would greatly
simplify an analogue design by avoiding the need for sin conversion.
Also, if we're not going for true realism (that is ignoring non-harmonic
components) all the triangles could be derived from a fundamental using
non-linear techniques (as JH suggested in a related post).  That means
one VCO, some diodes, and a bunch of VCA/EG units.  Things are getting
simpler. Now for the tough part: figure out a formula analogous to the
Fourier Transform but for triangular components.  Then program a computer
to do the analysis.

As the previous post shows, it is possible.  Actually, I think
any periodic can be synthesized by any other periodic + harmonics
(since any periodic can be decomposed to  a sum of sin waves).  Just a gut
feeling, but a strong gut feeling.  Triangles do provide a couple
great advantages though.  dave moylan