oversampling

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On Mon, 5 Jan 1998, KC5TJA wrote:

> > [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.

The neurons that have "axons" -- traversing relatively large distances
compared to their bodies -- communicate via "spikes" -- propagating
voltage waves, driven by nonlinear equations and by external pumping of
energy along the way, i.e., no signal degradation even for the axons that
start at the head and end at the toes of Michael Jordan. 

This has been known for a long time (say for 80 years, although Luigi
Galvani first observed that fact some centuries ago), but has been
studied in detail some 30 years ago by Hodgkin and Huxley -- its hardly a
novel fact. 

The total volume of the axons is about half the total volume of the brain,
i.e., the brain is quite a messy telecom wiring closet. This has been
known since the turn of the century, especially by the Spanish Anatomist
Ramon y Cahal. 

In order to fight signal degradation, spikes have constant amplitude, i.e,
carry no amplitude information. This is an everyday fact in neuroscience
-- something like gravity in physics.

> > 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... :) 

That's because you don't need a word of Calculus to understand it -- it's
plain algebra.

> > 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.

Say, one bit per doubling. That is, with 2^(14)x or 16000x oversampling.

> I find that devastatingly hard to believe. :-)

Me too, with a 2-bit A/D and 8x oversampling you get (roughly) 5 bit
resulting resolution (in the way I described it below, that is -- there
might be more clever ways of doing it). 

> > 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... :)

No. We know the spectral part because you told us explicitly in your
previous post that this was a sine and all you needed was the amplitude.

In the case of the POST modem we know the spectral part because we know
what the telco does to the signal before it sends it to us -- namely
bandpass it between 300 and 3600 Hz. 

> > 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. 

That is exactly the reason I was cautious by using "level-crossing" 
instead of "zero-crossing."

And exactly the reason neurons don't do simple "level-crossing." 

> 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 :-) )

By finding which regions in the (A1,A2) space of RxR are consistent with
the set of inequalities:

2<=f(1)<3
2<=f(2)<3
3<=f(3)<4
...
1<=f(20)<2


> > [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? 

Not that I know of. Anybody have info?

> 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...

Units? Of what? I was using only bits as units (apart for the volume of
the head, which is in cu. inches and the head dissipation of the head,
which is in Watts). 

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

A) would admit WHAT?

B) The Internet does not contain all interesting people and/or discussion
topics :-) 


--
Penio Penev <Penev@pisa.Rockefeller.edu> 1-212-327-7423

.