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Wed Jun 10 15:35:37 CEST 2009

## SD and edge-triggered representation

```What is nice about the SD form is the ease at which it can be
integrated into mixed analog/digital electronics.  In fact, it can be
attached directly to a set of speakers, or any analog circuit with
low-pass characteristic.

However, I wonder if moving to an edge-only representation will make
things simpler.  This will give a pure FM representation of a signal:
value directly proportional to square wave frequency.  It eliminates
an element from the data representation: the width of a pulse.  This
is essentially an arbitrary component and can be easily reconstructed
later.

The important question is: does it make things really simpler?

Let's re-interpret the elementary operations explored in [1] in terms
of square wave signal.

FM signals are trivially generated from an accumulator as the
accumulator's MSB.  Converting an FM signal to analog can be done in
many ways [2], but the simplest ways seem to convert the edge signal
back to a pulse signal either unclocked using a one-shot, or clocked
using a bit delay.

Since a signal's value is directly proportional to the density of
transitions, addition is represented by the XOR operation.  The
advantage here is that addition is "dumb".  There is no gating
required as in the pulsed case.

This seems simpler, but just as the pulse representation, this might
lead to loss of information due to aliasing.  However, the form of
aliasing is different and consists of glitches.  Proper "de-glitching"
needs some clocked logic, and can only be done exactly if there is
room in the output signal (if signals don't change during de-glitching
pulse width).  This brings us back to the complexity of dealing with
bit slots as in the SD case.

Attenuation could be implemented statistically as "selectively
forgetting" to toggle the switch.  There isn't much difference here.

So, it looks like most operations can again be implemented using
simple logic gates, conditionally on the uncorrelated nature of the
input signal.  I'm not really convinced about this since it seems to
be harder to qualify: the extra work that needs to be done in a
clocked/pulsed representation seems to bring at least some order that
is harder to reconstruct in the FM case.  The FM case however is
probably simpler to interface to special puropose (non-clocked) analog
circuits.

Conclusion: FM representation is in some respects simpler, but harder
to control using statistical time-multiplexing tricks as described in
[1].  Since conversion between the two formats is rather
straightforward in clocked logic, it seems that sticking with SD is
best.  Then FM can be used when it is convenient, i.e. when
interfacing to an unclocked or asynchronously clocked circuit.

[1] entry://20090609-111303
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