oversampling

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> This signal has _very well_ determined spectral characteristics. If we
> forget about the local line noise for a while, a second of this (analog)
> signal lives in a 2x(3600-300) = 6600 dimensional subspace of the space of
> all analog functions one second long, whereby 3300 sines and 3300 cosines
> serve as the basis. So, the signal is determined by 6600 numbers, which
> may be the projections to the sines and cosines, but can be determined by
> _any_ 6600 linearly independent projections to _that subspace_. 

This makes sense... :)

> This is usually achieved with regular interval sampling, but in principle
> can be achieved by _any_ sampling -- both in the spatial, in the
> frequency, in the time domain or a combination thereof. 

hmmm...what's the difference between spatial and time domains? Perhaps
this is where I'm getting confused...

> The trick is to know how one is sampling and to calculate the
> transformation matrix correctly. Of course, one has to do the error

OK, please give a more complete example of this. I'm sorta getting what
you're talking about, and I'm sorta not.

What I'm getting confused about is how you are able to sample the
instantaneous amplitude of a signal using nothing but zero crossings... :)

> One can improve the quality of the reconstruction if one has more samples
> -- in that case the problem is called "least square fitting," or "what is
> the signal in the target subspace, closest to the measured one, which is
> in a bigger space?"

Yes, but you still need more resolution for amplitude. Let's say you have
a 2-bit A/D converter, and you sample a sine wave:

1 2 3 2 1 0 1 ...

Looks like a triangle wave to me. But that's due to the lack of time
points. Now let's double the sampling frequency:

1 2 2 3 3 3 3 2 2 1 1 0 0 0 0 1 1 2 2 ...

OK, now it's looking more like a sine wave. BUT, we reach a point of
diminishing returns here. If I were to double the sampling frequency
again, I'd get no more information out of it... :(

> In all cases oversampling measurements gives additional information so that
> the original signal can be recovered with better accuracy. 

Now my question to you is: how? :)

> With a 1-bit A/D one is measuring the zero-crossings of some function. 

OK...I'm thinking of a different KIND of 1-bit D/A converter... I'm
thinking of the converter where the output is either all-on or all-off,
and you use an integrator, followed by a LPF, to achieve the output
signal.

Upon a zero crossing, what happens? Does the state of the output
latch toggle, or does it pulse for a clock period? In other words, does
it:

0000000011111111
  ^
  \__ zero-crossing event

or does it do this:

0000000010000000
  ^___ zero-crossing event

> accurate -- i.e., when we read out that f(t)>=0, then we actually quite
> sure, that f(t)>-2^(-20), for example. And when we read out, that f(t)<0,
> we are quite sure that f(t)<+2^(-20). Then, the only uncertainty is in
> the actual time of zero-crossings, which can be made arbitrary small with
> arbitrary large oversampling.

Right. This means a simple window comparator is a sufficient 1-bit A/D
converter.

> So, by noting the exact time of the zero crossing, we have actually
> constrained the point through with the waveform passes extremely well.

Right, but you're now completely clueless of the instantaneous amplitudes
between two zero-crossing events. :)

> In other words, there aren't _that many_ possible waveforms in the
> _band-passed subspace_ that conform to those requirements. 
 ^^^^^^^^^^^^^^^^^^^^^^

That may well be true, but you're still loosing amplitude information,
even for those waveforms that ARE still in the bandpass region.

> So, if all one is after is 6600 coefficients in the Fourier expansion of
> the signal with 13 bit accuracy, one is really searching for 13 x 6600
> bits/second, or 86 Kbits/s. With a 1-bit A/D @ 5 MHz one has 5 Mbits/s,
> and with 6-bit A/D @ 5 MHz -- 30 Mbits/s. Granted they are not
> decorrelated -- long strings of zeroes and ones are expected more
> frequently than in the coin-tossing experiment, but still, there is a lot
> of information about the signal there. 

Maybe I just gotta think about it and let it sink in a bit... But I still
have never seen a 1-bit A/D or D/A converter in commercial grade telco
equipment... :)

> As I said, one need a helluva DSP to sort things out :-)

Hmmm...perhaps not. Could wavelet analysis have application in this
picture? (I'm interested in wavelets, though I've only played with Haar
wavelets.)

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