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

[Aaron Lanterman]

On Jun 3, 2009, at 9:12 AM, Jerry Gray-Eskue wrote:

> As I understand it, A Fourier series define the theoretical  
> combination of sine waves that produce any repeating waveform.

Yup. This combination of sine waves has a special property that all  
the wave frequencies are integer multiples of the fundamental. (Any of  
the harmonics might be missing).

> 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.

Uhm, sorry I lost you there. It is true that isn't any _finite_  
combination of sine waves that will result in this "sigmoid" waveform,  
but if you let yourself use an infinite number you can (more or less -  
the math details of what it means for a Fourier series to converge are  
subtle and tricky, but for horseshoes and handgrades using an equal  
sign is close enough).

That big discontinuity in our sigmoid waves means that the harmonics  
will drop off relatively slowly - you need lots of strong harmonics to  
generate that discontinuity.

> 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?

Yup.

- Aaron