[sdiy] Harmonic content of the "sigmoid" half-sine wave

[Tom Wiltshire]

On 3 Jun 2009, at 14:12, Jerry Gray-Eskue wrote:

>
> Thanks Aaron
>
> I think I get the feel for what is going on, correct me if I am  
> wrong on any
> of these points.
>
> As I understand it, A Fourier series define the theoretical  
> combination of
> sine waves that produce any repeating waveform.
>
> In any case that it appears that a sine wave is being produced but  
> at less
> than all 4 quadrants over the period of the waveform, the apparent
> fractional sine wave is merely the result of the content of various  
> higher
> frequencies as there is no combination of sine waves that will  
> result in
> sine wave short of 4 quadrants.
>
> This leads me to an assumption: for any repeating waveform the  
> period of
> repeat ion is the period of the fundamental frequency - I do not  
> have a
> "warm fuzzy" on this assumption, it correct?

Yes; or at least that's my understanding too. Note that the  
fundamental frequency by this definition might not be the same as the  
heard fundamental pitch, but that's a question of psychoacoustics  
rather than physics.

As an example, imagine you add a sub-octave sine wave to a repeating  
waveform. The sub-octave waveform will cause alternate wavecycles to  
differ (since one will be added to the positive part of the wave and  
one to the negative) so the period of repetition halves along with  
the fundamental frequency.

One interesting effect is where you add together many high frequency  
harmonics, but none of the lower order ones. The resulting waveforms  
consist of a burst of energy with not much going on inbetween. The  
period of repetition is the whole thing from one burst to another,  
and this suggests a fundamental frequency which isn't present. Examples:

http://www.electricdruid.com/Lumps2.png
http://www.electricdruid.com/Lumps2_spectrum.png
http://www.electricdruid.com/Lumps4.png
http://www.electricdruid.com/Lumps4_spectrum.png

Going back to the sigmoid waveform, I once tried generating this  
waveform from harmonics for use as a distortion waveshaper (it's a  
soft-clip curve) but unfortunately I wasn't happy with the results,  
so I never kept them...otherwise I'd have a similar graph for that  
curve - Sorry!

Regards,
Tom