[sdiy] XOR as 'digital' ring modulator

[cheater cheater]

Hi John,
a typical multiplier works like this:

z = x * y

Now if you expand this to show the signs, it looks like this:

z = sgn(x) * |x| * sgn(y) * |y|
= sgn(x)*sgn(y) * |xy|

the xor table is as follows:

F \/ F == F
F \/ T == T
T \/ F == T
T \/ T == F

The table of sgn(x) * sgn(y) is as follows:

-1 * -1 = 1
-1 * 1 = -1
1 * -1 = -1
1 * 1 = 1
-1 * 0 = 0
1 * 0 = 0
0 * 1 = 0
0 * -1 = 0

So, if False is represented by 1 and True is represented by -1, then
we have a match for the first 4 rows of the table. However there is a
third item here. Of course to represent the whole function of a ring
modulator you'll at least need an operator that works in ternary
logic. Those are, from what I understand, possible to wire using
binary gates.

It appears this can be achieved in the ternary logic with null. See
the table here:

http://www.comsci.us/db/sqlnotes/sql03.html

Binary Ternary Logic Operators
Operands   AND OR XOR
null	null 	null	null	null
null	false 	false	null	null
null	true 	null	true	null
false	null 	false	null	null
false	false 	false	false	false
false	true 	false	true	true
true	null 	null	true	null
true	false 	false	true	true
true	true 	true	true	false

As you can see, the value combinations are the same as sgn(x) * sgn(y).

A ternary logic XOR can be created in this way: two signals are
ternary and encoded with -5V = False, 5V = True, 0V = Null.

Because the ternary logic XOR operator is an extension of the binary
logic XOR operator, we can use the usual XOR circuit in a very
specific monad; that means a circuit like this:

1. either signal gets checked for being 0 (Null). There is a wire that
shoots off to the side and bypasses the xor circuit completely, and
then goes to a multiplier, say an AND gate or transistor or op-amp. If
that bypass wire signals 0, then the AND outputs 0. If it doesn't
signal 0, the AND outputs what it gets from the XOR circuit. This wire
is our monad sub-channel.

2. After the signals got checked for being 0 with the special wire
coming off, there is a level shift to convert the signal to binary
logic. We ignore Null and replace it with e.g. False, or True, it
doesn't matter. Then we send that converted signal to the XOR block
which is able to process it. This is our converter, and we're "in the
monad" now.

3. After the signals have been processed in the monad by the XOR
block, we up-convert them to the ternary signal again. True gets
converted to 5V, False gets converted to -5V.

4. After this up-conversion we go to the multiplier which checks the
sub-channel. When the sub-channel says 0, the multiplier outputs 0V
(which is Null).

This should produce the behavior that you are looking for.

HTH,
D.

On 10/11/2010, Speth, John <John.Speth at coherent.com> wrote:
> Of course, it's the good sound we're after when all is said and done.  The
> MOTM-120 (http://www.synthtech.com/motm120.html) features a digital
> ring-mod.  I socketed the digital chip in my M120 so I could experiment with
> XOR vs NOR easily.
>
> To my ears they both sounded the same when two waveforms were driven into
> it.  The XOR will sound the squared wave version of the sounding input if
> the other input is silent (IOW, out = in).
>
> It was my opinion that a ring-mod should be silent when one of the inputs is
> silent so I left the NOR chip installed.  I still like it that way.  I
> wouldn't say a digital ring-mod is any kind of sub for the analog version.
> It just sounds too different.
>
> John Speth
> mailto:john.speth at coherent.com
>
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