[sdiy] Re: Walsh Generator Release!!!

[Neil Johnson]

All,

> Can I sit in front of a Walsh console and make what I need?  ...  Or
> is it so weird that it's too weird?

Ok, my *personal* interpretation of the Walsh<->Fourier relationship works
along the lines that, for classical Fourier sine-based synthesis we build
up the result from a set of basic elements (the sines).  In Walsh we start
off with a harmonically-rich square wave and remove the higher sine-wave
harmonics by adding harmonically-related rectagular waveforms.  As a nice
contrast, to achieve a pure sinewave you need a large* number of Walsh
functions, in contrast with synthesizing a square wave from sines.

In practice it goes something like:
- to make a sine wave we start with a square wave of the same frew.  But
this also has lots of odd harmonics, going up 1/n.
- add in just enough 3rd harmonic square to knock out the 3rd, 9rd,
15th... harmonics
- now add in just enough 6th harmonic to remove the 6th, 18th, 30th...
harmonics...

And so on (although MUCH simplified!)  Oh, and don't mention phase .. it
gets horribly complicated :-(  Makes waveform analysis tricky, as the
transform is phase-sensitive, and it is difficult to separate phase from
amplitude, unlike in Fourier.

Result: Fourier synthesis can lack high-end punch, but lots of bass (see
Kurzweil K150FS), whereas Walsh has lots of higher harmonics and sounds
much brighter, if a little "artificial".  YMMV.

Maybe that's why Fourier is so popular, because its easier to do the
complex math with sin/cos, than have to handle bizarre maths that looks
horrendous!!  Shame really, as synthesis/analysis itself is dead easy,
needing nothing more than add/subtract.  Ho hum...

Hope this helps,
Neil
--

* - I don't like using the term "infinite" as the mathematicians around
here (the lab) get edgy when I use that term!  "Unbounded" seems to be
more acceptable.

--
Neil Johnson :: Computer Laboratory :: University of Cambridge ::
http://www.njohnson.co.uk          http://www.cl.cam.ac.uk/~nej22
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