question about non-uniform sampling?

Jerry Avins wrote:
> 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.

For this counterexample to work, the response of the lowpass filter
for each sample would have to be over an hours worth or samples wide,
and with enough bits of precision so that the last half hours
information are reflected in the filters output of the first half hour.
Otherwise the signal would not be bandlimited to the precision
required.

So the limits of S/N, bit precision and low pass filter width limit
the degree of sampling on-uniformity which might still allow
practical reconstruction.


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

Steve Underwood wrote:
>
> The practicality of non-uniform sample isn't a whole lot different
> whether we are talking about minor non-uniformity or some extreme. As
> soon as sampling is even a little non-uniform it is highly sensitive to
> sampling error and converter noise. As it becomes more non-uniform it
> quickly becomes totally impractical to make sense of the kind of samples
> you can get in the real world. 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.
>

I thought statistics was the science of making sense of precisly those
sorts of non-uniform real-world samples...

Ron N. wrote:
> Jerry Avins wrote:
> 
>>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.
> 
> 
> For this counterexample to work, the response of the lowpass filter
> for each sample would have to be over an hours worth or samples wide,
> and with enough bits of precision so that the last half hours
> information are reflected in the filters output of the first half hour.
> Otherwise the signal would not be bandlimited to the precision
> required.
> 
> So the limits of S/N, bit precision and low pass filter width limit
> the degree of sampling on-uniformity which might still allow
> practical reconstruction.

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? Nonuniform sampling conveys information, but there are 
limits to how much can be extracted.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
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Ron N. wrote:
> Jerry Avins wrote:
> 
>>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.
> 
> 
> For this counterexample to work, the response of the lowpass filter
> for each sample would have to be over an hours worth or samples wide,
> and with enough bits of precision so that the last half hours
> information are reflected in the filters output of the first half hour.
> Otherwise the signal would not be bandlimited to the precision
> required.
> 
> So the limits of S/N, bit precision and low pass filter width limit
> the degree of sampling on-uniformity which might still allow
> practical reconstruction.

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? Nonuniform sampling conveys information, but there are 
limits to how much can be extracted.

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

Ron N. wrote:
> Jerry Avins wrote:
> 
>>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.
> 
> 
> For this counterexample to work, the response of the lowpass filter
> for each sample would have to be over an hours worth or samples wide,
> and with enough bits of precision so that the last half hours
> information are reflected in the filters output of the first half hour.
> Otherwise the signal would not be bandlimited to the precision
> required.
> 
> So the limits of S/N, bit precision and low pass filter width limit
> the degree of sampling on-uniformity which might still allow
> practical reconstruction.

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? Nonuniform sampling conveys information, but there are 
limits to how much can be extracted.

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

> 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 or 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.

cs_posting@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 or 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.

I Ron lends me one of those for a couple of days, I'll take it to 
Hialeah Park for a few days and pay him handsomely. In the meanwhile, 
I'll have to content myself with disassembling a solid sphere into an 
infinite number of pieces -- only some of them infinitesimal -- and 
reassembling the pieces to make two solid spheres, each the size of the 
original. (If the spheres are made of gold, that can be profitable too, 
even though the hourly return is rather low.)

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

Jerry Avins wrote:

> 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

If that is the case, then the speech signal is not bandlimited -- and
if that is the case, then you're talking about an entirely different
thing than I was talking about.

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)

Carlos
--

Jerry Avins wrote:

>>  [...]
>> 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 

No it's not.  Not only is it not an *adequate* counterexample;
it's not even a counterexample.

The above reasoning has nothing to do with practical applicability
of the issue.  The above reasoning is purely mathematical -- or
I should rather say, it works purely at the mathematical level.

If the music you're talking about is truly *band-limited*, then
it spans from time -infinity to +infinity.  So, assuming that
the signal is *strictly bandlimited* between DC and 20kHz, then
yes, taking one hour worth of samples (at least 1 + 40000*3600
samples) in an interval of 1 microsecond right before the hour
of music began *is* enough to fully, completely, and perfectly
(i.e., 100% accurately) reconstruct the whole hour of music;
provided that the remaining infinity of samples before and
after the sampleless hour is there, and provided that the
1+40000*3600 samples are distinct, and at times different from
all the remaining uniform samples).

Again:  *mathematically* speaking, the signal is fully recoverable
(analytically;  or numerically, if we could count on "infinite
precision" representation of real numbers).

Carlos
--

Carlos Moreno wrote:
> Jerry Avins wrote:
> 
>> 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
> 
> 
> If that is the case, then the speech signal is not bandlimited -- and
> if that is the case, then you're talking about an entirely different
> thing than I was talking about.

I'm talking about precisely what you are. A signal that is truly 
bandlimited isn't time limited, and vice versa. In the real world, 
signals with finite duration can bandlimited well enough so that we can 
deal with them. But when one becomes pedantic about what is 
theoretically possible, on must be likewise aware of what is 
theoretically impossible.

> 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?


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