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

[Aaron Lanterman]

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