[sdiy] Taylor series and sums and difference tones

[Dave Manley]

Aaron Lanterman wrote:
> On Jun 4, 2009, at 3:31 PM, Ingo Debus wrote:
>
>> I think this boils down to the question: are the systems involved 
>> linear or not? If they are, then adding two sine waves gets you just 
>> the sum of two sine waves. If the sum is sent through something which 
>> is nonlinear, then components of other frequencies emerge.
>> It doesn't matter if the system is mechanical or electronic. It only 
>> matters if it's linear or not.
>
> Excellent!!!
>
> For nonlinear systems, you can think of expanding the nonlinearity in 
> terms of a Taylor series, with constant, linear, squared, cubic, 
> quartic, etc. terms.
>
> If you think of putting a sum of two sinusoids into the nonlinearity, 
> the squared term in the taylor series will give you these sum and 
> difference tones.
>
> But you'll also get all sorts of junk from the other terms.
On a related note, Grant Richter has commented that for the best 
sounding multiplication (as in a balanced modulator), you want the sine 
to be very pure, because any components that are not the fundamental 
will result in, to use Aaron's terms, "all sorts of junk".  Of course, 
it depends on how many dB down the stray components are.  A sine from a 
self-oscillating filter is better than any sine from a shaper for use in 
multiplication for this reason.

-Dave