question about non-uniform sampling?

Ron N. wrote:
> Jerry Avins wrote:
> 
>>If a symphony is played in a forest and there is no one to sample the
>>last movement, how can the filter "hear" it?
> 
> 
> If it's not input to the filter (with, say, 100 hours of group delay)
> then
> it's not output and can't be reconstructed.

Earlier, you wrote:

 > True.  But what I am hypothesizing is that a low pass filter on
 > the order of a symphony duration in impulse response might
 > not leak enough of that Hz which you mention below to allow
 > pulling out the movement in "hiding".  Then Jerry won't have
 > a problem with non-causality, because a filter with this long
 > an impulse response will have to hear the end of the symphony
 > before sending the start (group delay) to the sampler.

In response, I believed, to my

"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."

What did you mean?

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

Ron N. wrote:
> Jerry Avins wrote:
> 
>>But which last movement? At least two composers have written completions
>>for Schubert's 9th, the "Unfinished".
> 
> 
> The one that's input to the low pass bandlimiting filter is the one
> that affects
> the output.

My point was that none of them get to the filter. The premise of this 
discussion is that samples may occur anywhere in the signal so long as 
the total is an adequate number.

Here is Carlos's original claim that my examples are intended to refute: 
irregular sampling can describe a signal provided that the number of 
samples meets a modified Nyquist criterion. The number of samples must 
exceed the product of the duration of the signal and twice the highest 
frequency in it. Differently put, The Nyquist criterion dictates how 
many samples must be taken, but they may be taken at any (known) time.

I provided a counterexample to dispute that claim: A long signal is 
oversampled by a factor S, placing all the samples in the first 1/S part 
of the signal. According to Carlos's original assertion, it is possible 
to reproduce the entire signal. You seemed to be responding to that, but 
I now believe you missed the point.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

Jerry Avins wrote:
> Ron N. wrote:
> > Jerry Avins wrote:
> >
> >>But which last movement? At least two composers have written completions
> >>for Schubert's 9th, the "Unfinished".
> >
> >
> > The one that's input to the low pass bandlimiting filter is the one
> > that affects
> > the output.
>
> My point was that none of them get to the filter. The premise of this
> discussion is that samples may occur anywhere in the signal so long as
> the total is an adequate number.

And that the signal is bandlimited to a much higher degree than
typical.

Which probably requires a low pass filter with a very long delay.
So long that any samples taken during the first half of a symphony
only record the patrons seating themselves several hours before
it starts, not the first half of the symphony.  If you want to actually
sample the first half, the filter has already heard the last half.

- rhn

Jerry Avins wrote:
> Ron N. wrote:
> > Jerry Avins wrote:
> >
> >>But which last movement? At least two composers have written completions
> >>for Schubert's 9th, the "Unfinished".
> >
> >
> > The one that's input to the low pass bandlimiting filter is the one
> > that affects
> > the output.
>
> My point was that none of them get to the filter. The premise of this
> discussion is that samples may occur anywhere in the signal so long as
> the total is an adequate number.
>
> Here is Carlos's original claim that my examples are intended to refute:
> irregular sampling can describe a signal provided that the number of
> samples meets a modified Nyquist criterion. The number of samples must
> exceed the product of the duration of the signal and twice the highest
> frequency in it. Differently put, The Nyquist criterion dictates how
> many samples must be taken, but they may be taken at any (known) time.
>
> I provided a counterexample to dispute that claim: A long signal is
> oversampled by a factor S, placing all the samples in the first 1/S part
> of the signal. According to Carlos's original assertion, it is possible
> to reproduce the entire signal. You seemed to be responding to that, but
> I now believe you missed the point.
>


You continue to miss the point that the signal must be bandlimited.
You appear to be confusing a signal that can be adequately approximated
by a band limited signal with a band limited signal.  A typical
symphony
can be adequately approximated by a band limited signal, but it is
not a band limited signal.  (And given a symphony, you cannot
construct the band limited signal that approximates the entire
symphony, without first hearing the entire symphony).

                             - William Hughes

William Hughes wrote:

> And given a symphony, you cannot
> construct the band limited signal that approximates the entire
> symphony, without first hearing the entire symphony.

It's too late to try anyway though, you already missed the big bang at
t=0.

Ron N. wrote:


> Which probably requires a low pass filter with a very long delay.
> So long that any samples taken during the first half of a symphony
> only record the patrons seating themselves several hours before
> it starts, not the first half of the symphony.  If you want to actually
> sample the first half, the filter has already heard the last half.

Not is the last half wasn't sampled. This is becoming tedious.

The question is not how long the process takes, but how much of the 
signal can be sampled well below the rate corresponding to half the 
bandwidth and nevertheless be properly reconstructed.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

William Hughes wrote:


> You continue to miss the point that the signal must be bandlimited.
> You appear to be confusing a signal that can be adequately approximated
> by a band limited signal with a band limited signal.  A typical
> symphony
> can be adequately approximated by a band limited signal, but it is
> not a band limited signal.  (And given a symphony, you cannot
> construct the band limited signal that approximates the entire
> symphony, without first hearing the entire symphony).

I didn't miss that. Carlos and I already agreed on that and moved on.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

cs_posting@hotmail.com wrote:
> William Hughes wrote:
>
> > And given a symphony, you cannot
> > construct the band limited signal that approximates the entire
> > symphony, without first hearing the entire symphony.
>
> It's too late to try anyway though, you already missed the big bang at
> t=0.

I conjecture that the bandlimit filter to reconstruct the last
half of a symphony would not need to have infinite extent
back to the big-bang; but perhaps only a few hours or
days in impulse response width might be sufficient, and
the sampler would of course have to wait until this input
filter settles.  You'd also have to keep a lot of bits of
precision to make the tails of that filter count.  Can only
be done either in a universe not quantized by Planck, or
perhaps digitally using a very long (days) FIR and infinite
precision arithmetic for the bandlimit filter.

Then the samples of the first half would include sufficient
information from the second half already input to the low
pass filter.

- rhn

But, if the speech were repeated many times, for ever, then the signal
would be band-limited and the speech (with all its repetitions) could
be recnstructed from a finite segment.

I think this is what is done in politics, and in TV soap operas, which
is why in these special cases you can in fact know what is going to be
said before it actually is.

Chris
==========================
Chris Bore
BORES Signal Processing
www.bores.com

Not having read the whole thread diligently, but...

If all the samples are taken in some (short) time at the beginning of
the signal, then does that not limit the frequency resolution?
So even granted the possibility that one might be able to recontruct
the last part of the signal, it would in fact be smeared frequentially.
I have tended to assume that one must sample for a total time that is
longer than the cycle time of the slowest frequency component of the
signal -
though I am not confident that is valid.

Chris
===============
Chris Bore
BORES Signal Processing
www.bores.com