[sdiy] FM math question

[Magnus Danielson]

Ian Fritz wrote:
> At 11:39 AM 10/16/2008, Mattias Rickardsson wrote:
>> A saw core vco with a saw->tri->sin shaper is a vc sine oscillator. You
then modulate this sine with another sine, and get an appropriate 
result.
>> Am I missing something?
>
> Well, you don't modulate the final sine.  The modulation is of the core 
saw, before the wave multiplication.  Since the multiplication is a 
nonlinear process it would seem to me that the results would be 
different for the two cases.  Maybe not. <shrug>

The key issue is if the non-linear mapping has memory effects or not. To
be strict, real applications always have memory effects, but usually  we
can approximate things not to have memory effect. Then it becomes a  an
aspect of f(p(t)) where p(t) is the phase function over time and f(x)  is
the waveform function on the phase x. For most non-linear waveshapers  we
are memoryless mapping-functions. Step-wise non-linear mappings of  g(x)
followed by h(x) becomes h(g(x)) which we can reduce as a combined  f(x) =
h(g(x)) without changing anything, thus if we realize f(x)  directly or
indirectly through g(x) and h(x) is not relevant as long as  they are
equalent. Thus, an analysis done for f(x) will be valid for  h(g(x)) as
long as f(x) = h(g(x)). This is what I assumed all along.

If either h(x) or g(x) has memory effects, they depend on t as well and 
then it all becomes ugly in a new level.

Cheers,
Magnus