question about non-uniform sampling?

Jerry Avins wrote:
> Ron N. wrote:
> > cs_post...@hotmail.com wrote:
> >
> >>>Ron N. wrote:
> >>>
> >>>>For this counterexample to work, the response of the lowpass filter
> >>>>for each sample would have to be over an hours worth of samples wide
> >
> > ...
> >
> >>to which Jerry Avins replied:
> >>
> >>>What if it's a request show and the second half-hour's music isn't known
> >>>until after the sampling stops? Suppose even that the program is known
> >>>in advance. Do you suggest that it might be possible to reconstruct the
> >>>last movement of Beethoven s 9th Symphony from a gross oversampling of
> >>>the first three?
> >>
> >>Yes, provided you pass the samples through the anti-causal low pass
> >>filter that Ron postulated.
> >
> >
> > It wouldn't have to be anti-causal.  A symmetric low-pass FIR filter
> > with enough taps and bits of precision would work.  Of course there
> > would be a significant filter delay (roughly the length of the entire
> > symphony) before you would see the first output samples.
>
> I'd be happy to wait a whole week for a filter to compose the last
> movement of Beethoven's Ninth, given only the the first three. Think how
> much work (and time!) such a filter would have saved Beethoven himself!

How would the filter have saved any work?  If fed a last movement
of silence, that's what would be reproduced, given a few billion bits
of precision per sample and a lowpass filter an equivalent number
of dB down in the stop band.

What you think is the first sample of the first movement, can't be
correctly bandlimited to the precision required unless the low pass
filter is fed the last sample of the last movement.


IMHO. YMMV.
-- 
rhn A.T nicholson d.O.t C-o-M

Jerry Avins wrote:
> Ron N. wrote:
> > cs_post...@hotmail.com wrote:
> >
> >>>Ron N. wrote:
> >>>
> >>>>For this counterexample to work, the response of the lowpass filter
> >>>>for each sample would have to be over an hours worth of samples wide
> >
> > ...
> >
> >>to which Jerry Avins replied:
> >>
> >>>What if it's a request show and the second half-hour's music isn't known
> >>>until after the sampling stops? Suppose even that the program is known
> >>>in advance. Do you suggest that it might be possible to reconstruct the
> >>>last movement of Beethoven s 9th Symphony from a gross oversampling of
> >>>the first three?
> >>
> >>Yes, provided you pass the samples through the anti-causal low pass
> >>filter that Ron postulated.
> >
> >
> > It wouldn't have to be anti-causal.  A symmetric low-pass FIR filter
> > with enough taps and bits of precision would work.  Of course there
> > would be a significant filter delay (roughly the length of the entire
> > symphony) before you would see the first output samples.
>
> I'd be happy to wait a whole week for a filter to compose the last
> movement of Beethoven's Ninth, given only the the first three. Think how
> much work (and time!) such a filter would have saved Beethoven himself!

How would the filter have saved any work?  If fed a last movement
of silence, that's what would be reproduced, given a few billion bits
of precision per sample and a lowpass filter an equivalent number
of dB down in the stop band.

What you think is the first sample of the first movement, can't be
correctly bandlimited to the precision required unless the low pass
filter is fed the last sample of the last movement.


IMHO. YMMV.
-- 
rhn A.T nicholson d.O.t C-o-M

Jerry Avins wrote:
> Ron N. wrote:
> > cs_post...@hotmail.com wrote:
> >
> >>>Ron N. wrote:
> >>>
> >>>>For this counterexample to work, the response of the lowpass filter
> >>>>for each sample would have to be over an hours worth of samples wide
> >
> > ...
> >
> >>to which Jerry Avins replied:
> >>
> >>>What if it's a request show and the second half-hour's music isn't known
> >>>until after the sampling stops? Suppose even that the program is known
> >>>in advance. Do you suggest that it might be possible to reconstruct the
> >>>last movement of Beethoven s 9th Symphony from a gross oversampling of
> >>>the first three?
> >>
> >>Yes, provided you pass the samples through the anti-causal low pass
> >>filter that Ron postulated.
> >
> >
> > It wouldn't have to be anti-causal.  A symmetric low-pass FIR filter
> > with enough taps and bits of precision would work.  Of course there
> > would be a significant filter delay (roughly the length of the entire
> > symphony) before you would see the first output samples.
>
> I'd be happy to wait a whole week for a filter to compose the last
> movement of Beethoven's Ninth, given only the the first three. Think how
> much work (and time!) such a filter would have saved Beethoven himself!

How would the filter have saved any work?  If fed a last movement
of silence, that's what would be reproduced, given a few billion bits
of precision per sample and a lowpass filter an equivalent number
of dB down in the stop band.

What you think is the first sample of the first movement, can't be
correctly bandlimited to the precision required unless the low pass
filter is fed the last sample of the last movement.


IMHO. YMMV.
-- 
rhn A.T nicholson d.O.t C-o-M

Ron N. wrote:
> Jerry Avins wrote:
> > Ron N. wrote:
> > > cs_post...@hotmail.com wrote:
> > >
> > >>>Ron N. wrote:
> > >>>
> > >>>>For this counterexample to work, the response of the lowpass filter
> > >>>>for each sample would have to be over an hours worth of samples wide
> > >
> > > ...
> > >
> > >>to which Jerry Avins replied:
> > >>
> > >>>What if it's a request show and the second half-hour's music isn't known
> > >>>until after the sampling stops? Suppose even that the program is known
> > >>>in advance. Do you suggest that it might be possible to reconstruct the
> > >>>last movement of Beethoven s 9th Symphony from a gross oversampling of
> > >>>the first three?
> > >>
> > >>Yes, provided you pass the samples through the anti-causal low pass
> > >>filter that Ron postulated.
> > >
> > >
> > > It wouldn't have to be anti-causal.  A symmetric low-pass FIR filter
> > > with enough taps and bits of precision would work.  Of course there
> > > would be a significant filter delay (roughly the length of the entire
> > > symphony) before you would see the first output samples.
> >
> > I'd be happy to wait a whole week for a filter to compose the last
> > movement of Beethoven's Ninth, given only the the first three. Think how
> > much work (and time!) such a filter would have saved Beethoven himself!
>
> How would the filter have saved any work?  If fed a last movement
> of silence, that's what would be reproduced, given a few billion bits
> of precision per sample and a lowpass filter an equivalent number
> of dB down in the stop band.
>
> What you think is the first sample of the first movement, can't be
> correctly bandlimited to the precision required unless the low pass
> filter is fed the last sample of the last movement.

  Well that's automically incorrect. Given that they the reason even
  we invented sampling and electric guitars with feedback pedals
  and MTV to begin with was to prove to the idiots in The Filmore
  East and West and to the moronic Greatful Dead and Beatles
  bandwagon shitheads that there is no such as a first sample
  or a first movement.

  So we always like to tell the Washingtoon, San Fag ciso and
  New York Beatle idiots that when the Band played The Last Waltz,
  they didn't actually play The Last Waltz.

        They simply played the opening laser riffs of
  
              The Mick Jagger Bagdad Weight.

Jerry Avins wrote:

> Carlos Moreno wrote:

(snip)

>> The reasoning can be extended to any number N, no matter how
>> large.

>> I know this is not rigurous -- in particular, this shows that
>> the trick works for N samples taken at positions other than
>> the corresponding positions, no matter how large;  but this
>> proves nothing about an "infinity" of samples taken non-
>> uniformly...  Still, the result does suggest that you still
>> need the amount of samples that totals the same amount of
>> samples required in uniform sampling  (suggesting that your
>> Nyquist condition is given by the average sampling rate).


> It doesn't. There's a limit to how non-uniform the sampling can be 
> allowed to be. The example given above, of an hour's worth of music 
> sampled for half an hour at twice the minimum rate for the bandwidth, is 
> an adequate counterexample to what you claim is a general case.

Well, he started with an infinite number of samples.  If you subtract 
any finite number you still have an infinite number of samples, so it 
still works.   That, and that the samples can't have any quantization error.

-- glen

Jerry Avins wrote:

> Steve Underwood wrote:

(snip)
>>  There is nothing wrong with any 
>> extreme of non-uniformity in a purely mathematical sense. That is in a 
>> world with infinite sampling precision and no noise due to the 
>> converter itself.

> It's also a world where signals exist for all time. I doesn't matter how 
> precisely one can sample and how often, nothing can be known about a 
> speech yet to be given, even if the mathematics of nonuniform and highly 
> clumped sampling shows that it can.

I am not so sure what quantum mechanics says about this.

Mathematically, I believe that it works just about any time you have an 
infinite number of samples.  Consider that an infinite number of 
derivatives at a single point are also good enough to reconstruct a 
function.

All the problems appear when you don't have an infinite number of points 
with no noise or other errors in them.

-- glen

Jerry Avins wrote:

(snip)
(someone wrote)
>> But BTW, your argument does not really contradict my reasoning -- I did
>> not say (nor the conclusion I presented implies) that you can determine
>> the hour of speech before it was given.  In fact, not even right after
>> it is given.  To *fully* determine/reconstruct the continuous-time
>> speech signal you have to wait until "t = infinity" to be able to
>> reconstruct it...  (t = infinity is obviously a figure of speech, to
>> simplify the issue that the signal is given by an infinite sum)

> According to the argument you gave, samples from near the end of a 
> program can be moved to near the beginning, one at a time, without 
> compromising the reconstructed signal. How long you must have to wait 
> before the decoding is finished doesn't bear on the claim that you can 
> reconstruct what was not sampled. Do I not understand what you claimed?

This reminds me of my first thoughts hearing about FTIR, that is, 
Fourier Transform Infra Red spectroscopy.  Instead of scanning through 
frequency space, as usual for spectroscopy, you measure the output from 
a scanning interferometer which gives the FT of the spectrum.  What 
happens, then, at the points in the transform where the light source 
doesn't have any intensity?

Also, at some point the discussion should get to the Kramers-Kronig 
relations which for physical systems connect the real and imaginary 
parts, as well as time and frequency.

-- glen

Carlos Moreno wrote:

(snip)

> Well, it's not really "pedantic" -- I see it more like a matter
> of keeping in mind when some theoretical result has a direct
> applicability in the practical world.  After all, going back to
> the roots, Calculus itself could be accused of "pedantic", in
> that the real world is not really how Calculus models it (even
> though in the case of Calculus, the *results* are nicely
> correlated with the physical world, unlike this issue about
> the non-uniform sampling taken to the extreme)

If I remember right, it isn't yet determined if quantum mechanics 
quantizes time.  That may restrict the ability to do limits as delta t 
approaches zero, or it may not.

These discussions get much more interesting if you include quantum 
mechanics in them.  That is, what is actually allowed to be measured.

-- glen

Jerry Avins wrote:

(snip)

> I'd be happy to wait a whole week for a filter to compose the last 
> movement of Beethoven's Ninth, given only the the first three. Think how 
> much work (and time!) such a filter would have saved Beethoven himself!

Given all the samples of his work and the first three as input, one 
might compute in some weeks the most probable fourth movement.

-- glen

Ron N. wrote:

> What you think is the first sample of the first movement, can't be
> correctly bandlimited to the precision required unless the low pass
> filter is fed the last sample of the last movement.

There is no such thing as a bandlimited sample.  A sample is a scaled
impulse, and an impulse is by definition for all frequencies.