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

[Jerry Gray-Eskue]

<<... there is no combination of sine waves that will result in sine wave
short of 4 quadrants.>>

I phrased that poorly, what I meant to say was

there is no combination of sine waves ** below the fundamental ** that will
result in a sine wave short of 4 quadrants ** over the period of the
fundamental **.

To say this in another way, If you have a sine wave and shorten the period
without changing the wave form so that you have clipped off part of the
original wave form ( it is no longer a full 4 quadrants of sine wave) you
have changed the fundamental frequency even though you can see most of the
original sine wave unchanged.

- Jerry

-----Original Message-----
From: synth-diy-bounces at dropmix.xs4all.nl
[mailto:synth-diy-bounces at dropmix.xs4all.nl]On Behalf Of Aaron Lanterman
Sent: Wednesday, June 03, 2009 2:04 PM
To: sdiy DIY
Subject: Re: [sdiy] Harmonic content of the "sigmoid" half-sine wave


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

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