[sdiy] Screwing with Square Waves

[Donald Tillman]

On Oct 31, 2013, at 3:12 PM, Mattias Rickardsson <mr at analogue.org> wrote:

> So with the different phase relationships between the harmonics in the
> examples you got either discontinuous waveforms (the square wave et
> al.) or infinite spikes (the Hilbert wave et al.). Both of these are
> practically impossible to realize electrically, so I'm wondering:
> Is there any way of shifting the phases of the harmonics in a square
> or sawtooth wave so that the resulting waveform has neither
> discontinuities nor infinite spikes?

I'm not completely sure.  And that's a *fantastic* question!

We're summing an infinite number of sines, and each is at a strength of (1/i), inversely proportional to its frequency.  If this was a (1/(i^2)) series, like a Triangle Wave, that converges.  But a (1/i) series does not mathematically converge.  So that infinite energy has to go somewhere.

And as you noted, we see that energy in infinite slopes or infinite spikes.

I thought that shifting the phase of each harmonic by 2pi divided by the Golden Ratio would have a chance, but no, that's just as spikey.

It appears that an infinite number of sines of completely random phases is the closest you're going to get; no spikes, no discontinuities, but a lot of what looks like high frequency noise.  Or another way to describe it... it looks like any normal oscilloscope waveform of an acoustic source.

There should also be a more exotic way to do this; imagine putting together some metric for how spikey or discontinuous a waveform is, and then as you add in each new harmonic, adjust its phase to minimize that metric.

  -- Don

--
Don Tillman
Palo Alto, California
don at till.com
http://www.till.com