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Guest
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 Posted:
Thu Nov 17, 2005 1:16 am Post subject:
Re: question about non-uniform sampling? |
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Ron N. wrote:
| Quote: | The thought experiment I was visualizing was using a very
wide FIR filter on an infinite width of sufficiently dense and
precise samples to do the proper bandlimiting. ( It's been too
long since I've soldered any analog filter components :)
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No matter how you filter, a finite duration selection of the filter
output still can't be bandlimited.
The kind of band limiting we tend to worry about as engineers is the
high frequency aliases... these are easy to deal with. But the leakage
between some small fraction of a HZ, and true zero frequency, that we
can't handle. And that is where the last movement of the symphony is
hiding. |
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Ron N.
Guest
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| Posted:
Thu Nov 17, 2005 1:16 am Post subject:
Re: question about non-uniform sampling? |
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cs_posting@hotmail.com wrote:
| Quote: | 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.
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Err... correct. I should have said a sample of a sufficiently
bandlimited signal, where the pre-sampling low-pass filter
would presumably need support from at least the duration
of the entire symphony in order to produce it from a local
group of subsamples, in theory of course.
The thought experiment I was visualizing was using a very
wide FIR filter on an infinite width of sufficiently dense and
precise samples to do the proper bandlimiting. ( It's been too
long since I've soldered any analog filter components :)
IMHO. YMMV.
--
rhn A.T nicholson d.O.t C-o-M |
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Carlos Moreno
Guest
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| Posted:
Thu Nov 17, 2005 1:16 am Post subject:
Re: question about non-uniform sampling? |
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Jerry Avins wrote:
| Quote: | The hypothesis is that the first three movements were oversampled, using
enough samples to suffice for all four. According to Carlos, it doesn't
matter where the samples are just so long as there are enough of them.
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Nit pick -- that's not exactly what I said; there are conditions, in
particular, the samples have to be at distinct positions among them
and with respect to the other samples that are at uniform intervals.
| Quote: | We can deduce from Carlos's claim that the entire symphony was sampled.
If the claim is valid, then the entire symphony can be reconstructed. I
don't suppose that the entire symphony including the choral movement can
be reconstructed from the samples. My intent is /reductio ad absurdam/.
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I hope you're not still missing the part that you still have to wait an
infinite amount of time before you can even begin to reconstruct.
I don't see that your reasoning is leading to a contradiction, even
when we use it in the practical case -- there would be a contradiction
if the reasoning led to the conclusion that you can reconstruct a part
of the signal before it has happened. But that's not what follows from
my claim. You can only reconstruct signals that have already passed,
so there's no contradiction there.
Carlos
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Ron N.
Guest
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| Posted:
Thu Nov 17, 2005 1:16 am Post subject:
Re: question about non-uniform sampling? |
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cs_posting@hotmail.com wrote:
| Quote: | Ron N. wrote:
The thought experiment I was visualizing was using a very
wide FIR filter on an infinite width of sufficiently dense and
precise samples to do the proper bandlimiting. ( It's been too
long since I've soldered any analog filter components :)
No matter how you filter, a finite duration selection of the filter
output still can't be bandlimited.
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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.
| Quote: | The kind of band limiting we tend to worry about as engineers is the
high frequency aliases... these are easy to deal with. But the leakage
between some small fraction of a HZ, and true zero frequency, that we
can't handle. And that is where the last movement of the symphony is
hiding.
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IMHO. YMMV.
--
ron A.T nicholson d.O.t C-o-M |
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Jerry Avins
Guest
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| Posted:
Thu Nov 17, 2005 8:34 am Post subject:
Re: question about non-uniform sampling? |
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cs_posting@hotmail.com wrote:
| Quote: | Ron N. wrote:
The thought experiment I was visualizing was using a very
wide FIR filter on an infinite width of sufficiently dense and
precise samples to do the proper bandlimiting. ( It's been too
long since I've soldered any analog filter components :)
No matter how you filter, a finite duration selection of the filter
output still can't be bandlimited.
The kind of band limiting we tend to worry about as engineers is the
high frequency aliases... these are easy to deal with. But the leakage
between some small fraction of a HZ, and true zero frequency, that we
can't handle. And that is where the last movement of the symphony is
hiding.
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But which last movement? At least two composers have written completions
for Schubert's 9th, the "Unfinished".
Jerry
--
Engineering is the art of making what you want from things you can get.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ |
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Jerry Avins
Guest
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| Posted:
Thu Nov 17, 2005 8:36 am Post subject:
Re: question about non-uniform sampling? |
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Ron N. wrote:
| Quote: | cs_posting@hotmail.com wrote:
Ron N. wrote:
The thought experiment I was visualizing was using a very
wide FIR filter on an infinite width of sufficiently dense and
precise samples to do the proper bandlimiting. ( It's been too
long since I've soldered any analog filter components :)
No matter how you filter, a finite duration selection of the filter
output still can't be bandlimited.
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.
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If a symphony is played in a forest and there is no one to sample the
last movement, how can the filter "hear" it?
...
Jerry
--
Engineering is the art of making what you want from things you can get.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ |
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Ron N.
Guest
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| Posted:
Thu Nov 17, 2005 9:15 am Post subject:
Re: question about non-uniform sampling? |
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Jerry Avins wrote:
| Quote: | But which last movement? At least two composers have written completions
for Schubert's 9th, the "Unfinished".
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The one that's input to the low pass bandlimiting filter is the one
that affects
the output.
- rhn |
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Ron N.
Guest
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| Posted:
Thu Nov 17, 2005 9:15 am Post subject:
Re: question about non-uniform sampling? |
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Jerry Avins wrote:
| Quote: | If a symphony is played in a forest and there is no one to sample the
last movement, how can the filter "hear" it?
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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.
- rhn |
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Jerry Avins
Guest
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| Posted:
Thu Nov 17, 2005 5:16 pm Post subject:
Re: question about non-uniform sampling? |
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Ron N. wrote:
| Quote: | 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:
| Quote: | 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.
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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.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ |
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Jerry Avins
Guest
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| Posted:
Thu Nov 17, 2005 5:16 pm Post subject:
Re: question about non-uniform sampling? |
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Ron N. wrote:
| Quote: | 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.
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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.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ |
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Ron N.
Guest
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| Posted:
Fri Nov 18, 2005 12:35 am Post subject:
Re: question about non-uniform sampling? |
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Jerry Avins wrote:
| Quote: | 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.
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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 |
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William Hughes
Guest
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| Posted:
Fri Nov 18, 2005 12:51 am Post subject:
Re: question about non-uniform sampling? |
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Jerry Avins wrote:
| Quote: | 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.
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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 |
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Jerry Avins
Guest
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| Posted:
Fri Nov 18, 2005 1:16 am Post subject:
Re: question about non-uniform sampling? |
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Ron N. wrote:
| Quote: | 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.
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Again?! The second half is not sampled at all. Instead, the first half
is oversampled enough so the the total number of samples -- all taken
from the first half -- suffices for the entire composition. If you don't
believe that can work, we agree.
Jerry
--
Engineering is the art of making what you want from things you can get.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ |
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Guest
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| Posted:
Fri Nov 18, 2005 1:16 am Post subject:
Re: question about non-uniform sampling? |
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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 |
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Guest
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| Posted:
Fri Nov 18, 2005 1:16 am Post subject:
Re: question about non-uniform sampling? |
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Ron N. wrote:
| Quote: | 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.
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What you are talking about isn't bandlimiting, but rather lossless data
compression or even just plain format conversion.
A band limited signal, by definition must extend for all time, so the
minute you go chopping off the duration you are not band limited.
Though if you agree to put the CD player on repeate, that's a different
story.
But data compression or format conversion is another matter. I can
easily design you a "filter" that when exposed to one hour of input
will produce a stream of output data for six minutes, containing
sufficient information for a paired "filter" to reproduce the entire
hour long composition. I might even be able to do it with an FIR
engine and carefull chosen coefficients, though I'm not sure about that. |
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