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

[Jerry Gray-Eskue]

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?



-----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 12:11 AM
To: Synth DIY
Subject: [sdiy] Harmonic content of the "sigmoid" half-sine wave



On Jun 2, 2009, at 10:55 AM, Jerry Gray-Eskue wrote:

> I have had some time to think through what the sine function would
> do in
> relation to the original ramp or saw wave and have a postulate:
>
> With a triangle wave centered on 0 volts we get all 4 quadrants of
> the sine
> wave generated over the same period so the Frequency out = Frequency
> in.

Yup.

> With a ramp or saw centered on 0 volts we get 2 quadrants of the
> sine wave
> generated over the same period so the Frequency out = Frequency in/2.

Uh, not really - the fundamental frequency will still be the same.
(Don't worry, this often confuses the daylights out of my students
too. And sometimes me.)

I gave the computation of the Fourier series of this as a homework
problem in my ECE2025: Introduction to Signal Processing class last
summer.

The complex Fourier series coefficients are

a_k = j*4*k / [pi*(4*k^2 - 1)] * (-1)^k

so the amplitude of the related cosine waves of the harmonic series are

8*k / [pi*(4*k^2 -1)]

I put the homework & solution up here:

users.ece.gatech.edu/~lanterma/hw03su08.pdf
users.ece.gatech.edu/~lanterma/hw03su08_soln.pdf

Problem 2 is what you want.

I will take them down sometime this weekend - we sometimes recycle
problems so I don't want these floating around.

- Aaron

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