[sdiy] Screwing with Square Waves

Donald Tillman don at till.com
Sat Nov 2 21:48:35 CET 2013

On Nov 1, 2013, at 9:28 AM, Donald Tillman <don at till.com> wrote:

> 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.

Here is an example waveform, with the spectrum of a Square Wave, but with random phases:


This is summing 1000 odd harmonics, so it's up to the 2000th harmonic.  Apple's Grapher doesn't have a random function so I faked it with an arbitrary cubic function that wraps a lot.

The highest peak appears to be less than 1.6.

See?  It looks like sound.  (That's an interesting phrase.)


I believe the overall outcome here is that, while the slopes of Square Waves can get pretty steep when the harmonics are aligned as sines, the Harmonic Series rises in value so, so, slowly that 
the spikiest spikes never really get up very high. 

With 1000 harmonics, the peak value of a Hilbert Wave is 4.1.
With 1,000,000 harmonics, the peak value of a Hilbert Wave is 7.54.
With 10,000,000 harmonics, 8.6.
With 100,000,000 harmonics, 9.85.

  -- Don

Don Tillman
Palo Alto, California
don at till.com

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