[sdiy] Walsh Organ??

[Magnus Danielson]

From: Tim Ressel <madhun2001 at yahoo.com>
Subject: [sdiy] Walsh Organ??
Date: Sun, 7 Apr 2002 23:02:54 -0700 (PDT)

> Here's a line of what I like to call "thinking":
> Should the Walsh Organ mimic the Hammond drawbar
> scheme? That is to say, should the drawbars represent
> harmonic content just like the Hammonds? This would
> take some pretty manipulation, and I'm not sure its
> even possible.
> 
> Any math weenies out there care to take a whack at
> proving or disproving the possibility?

I don't feel too comfortable with being called a "math weenie" but
other than that...

Sure it's possible!

Since both the Fourier and Walsh transforms are orthogonal linear
transforms, they both can represent the same information in a lossless
manor (hey! In theory). Just think back to the Fourier vs. LaPlace
discussions just recently (I never got an answer back... :( ), which
also fits under the same umbrella.

So, what you have to do is to do an Inverse-Walsh and Forward-Fourier,
then fiddle with draw-bars, then an Inverse-Fourier. Fortunately you
can slam the Inverse-Walsh and Forward-Fourier transforms into a
common transform. These transforms can be relalized using op-amps and
resistors forming a fixed matrix.

There is also a simpler way, by Walsh-transform the Fourier
frequencies you want, you get the mix-vector from the Walsh that forms
that sine/cosine. So you can create a full matrix of these and process
through the drawbars before summing.

However... if you have a say 16 signal Walsh (where one is an exciting
DC), that will become 8 Fourier frequencies (yeat again where one is
an exciting DC but 6 of them got phase i.e. is both cosine and sine
waveforms) and in the end, what you've got is equalent of a 16 sample
sine/cosine... for the base frequency.

This level of alias distorsion haven't been heard of since the early
PPG days! ;O)

> Since I'm short on math (and on sleep, can you tell?),
> I would prove it the hard way. Generate the 72
> possible waveforms from the Hammond drawbars, then
> turn around and wash those waveforms thru the
> not-yet-written Walsh Coef generator software. Then
> take those results and wash it thru the walsh demo
> software I have written to generate the resultant
> waveform. Then compare waveforms. 
> 
> Call me Sisyphus.
> 
> Time to see if I can get some sleep.

You can't, you've got Walsh generators on your mind...

Meanwhile, here spring happends...

Cheers,
Magnus