oversampling

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> [A philosophical aside: The nervous system communicates with "spikes" --
> all-or-nothings objects that have an associated time value, but nothing
> else. I.e., it samples the time. The only unsolved problem in
> neuroscience is which exactly byproduct of the signal at the receptors is
> being thresholded.]

To my knowledge, we thought that neurons use both pulse-coded information
as well as amplitude information.

> What follows is a very good example and I'll try to take it one step
> further. 

I personally feel that wasn't a good example... :) I took three courses
of Calculus and I still can't understand a single word of what you
wrote... :) Can you e-mail me a another description?

> No, doubling the number of samples gives you about a bit more per sample.

By this logic, I can achieve 16-bit resolution with a 2-bit A/D converter
by simply using an 8x oversampling.

I find that devastatingly hard to believe. :-)

> We know the "spectral part" -- w, we need to estimate A. Suppose we start

We know the spectral part because we know how many times the signal
crosses the 1-2 barrier, yes? f=1/t... :)

> changing A, say increasing it, and ask what happens to the observed
> sequence. Well, for a while it stays the same, then changes -- the 2 -> 3
> transition happen earlier, because now f(t) increases faster. An we have
> the sequence:
> 
> S2: 1 2 3 3 3 3 3 3 2 1 0 0 0 ...
> 
> There is a minimum A at which this happens, say A_min(S2), which is also
> A_max(S1). As we decrease A, the transition happens later:
> 
> S0: 1 2 2 2 3 3 2 2 2 1 ...
> 
> This is A_min(S1) = A_max(S0).

Ok, I can see that...

> So, by observing S1, we have bounded the value of A :
> 
> A_min(S1) <= A_max(S1) .

Well, OK -- I can accept that. HOWEVER, for 1-bit resolution, that pretty
much leaves a massively gaping hole in our ability to estimate an
amplitude... :)

> A, which effectively increases the _amplitude_ resolution, if we are lucky
> -- by a whole bit.

AAAAAA.... now it clicks. 

BUT, using the above scenario, but with a 1-bit A/D converter, we get
this:

0000000111111110000000011111111

That is, it's quantized into a square wave. Or, even using a 2-bit A/D
converter, how do you handle waveforms like this:

22332233221100110011

(I'm trying to represent f = (A1*sin(w1*t))+(A2*sin(w2*t)) where w3 =
3*w1 :-) )

> [A philosophical aside: A neuron fires a spike with a reproducible
> temporal resolution of about a millisecond about 100 times a second. People
> put this at about 3--10 bits/spike. For about 10G neurons in the brain,
> this means about 1 Tbyte/s _processed_. 

So humans aren't slow -- we just run Microsoft operating systems?? :)

(Sam ducks, and runs for cover)

Is there a web page or two that I can cross-reference on the theory behind
1-bit A/D converters? I'd like to learn more about them (since the units
you're referring to are way different than the ones I'm most familiar
with) before continuing in the original thread...

(how many people on the Internet that you know, would admit to THAT?! :D )

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