| Author |
Message |
Guest
|
 Posted:
Sun Nov 20, 2005 1:16 am Post subject:
Re: question about non-uniform sampling? |
|
|
David Tweed wrote:
| Quote: | The bandwidth is defined as the bandwidth of the perfect brick-wall
low-pass or band-pass antialias filter you ran the signal through
*prior* to sampling it. The impulse response of this filter is the
interpolation function used in the system of equations you have to
solve as shown in my writeup.
-- Dave Tweed
|
The bandwidth of the signal could be less then output of the filter
could be less then the bandwidth of the filter and I think such
problems are also of interest. Do we lose generality by dealing with
the case they are the same?
Anyway the link you gave wasn't much of a proof. It was more like two
sample problems with perhaps some had waving in the solutions. I'm
trying to recall why we interpolate with sinc functions. I believe it
is a consequence of sign x of x compensation which effectively reduces
the distortion caused by a sample an hold output.
Also why are all the sinc functions the same width. Shouldn't the
width be dependent upon the time interval the sample is held for? You
mentioned something about the error in the matrix inversion but I think
only vaguely in the since that small elements in the matrix lead to a
large inverse, which cause a large error. Perhaps, with the use of
singular value decomposition you could develop a better expression for
the error in the inverse. |
|
| Back to top |
|
 |
zzbunker@netscape.net
Guest
|
| Posted:
Sun Nov 20, 2005 7:16 am Post subject:
Re: question about non-uniform sampling? |
|
|
JohnCreighton_@hotmail.com wrote:
| Quote: | David Tweed wrote:
The bandwidth is defined as the bandwidth of the perfect brick-wall
low-pass or band-pass antialias filter you ran the signal through
*prior* to sampling it. The impulse response of this filter is the
interpolation function used in the system of equations you have to
solve as shown in my writeup.
-- Dave Tweed
The bandwidth of the signal could be less then output of the filter
could be less then the bandwidth of the filter and I think such
problems are also of interest. Do we lose generality by dealing with
the case they are the same?
Anyway the link you gave wasn't much of a proof. It was more like two
sample problems with perhaps some had waving in the solutions. I'm
trying to recall why we interpolate with sinc functions. I believe it
is a consequence of sign x of x compensation which effectively reduces
the distortion caused by a sample an hold output.
|
Well, it has more to do with that the definite integral
of sinc has a value on every interval.
The compensation isn't really a compensation of anything.
It's better known as Gibb's phenomenon in physics.
Siince the sample and hold isn't actually a distortion.
Musicians call it a distortion, basically since the only
thing any of them knew about sampling or physics
is the directions to the nearest mall.
| Quote: |
Also why are all the sinc functions the same width. Shouldn't the
width be dependent upon the time interval the sample is held for? You
mentioned something about the error in the matrix inversion but I think
only vaguely in the since that small elements in the matrix lead to a
large inverse, which cause a large error. Perhaps, with the use of
singular value decomposition you could develop a better expression for
the error in the inverse. |
|
|
| Back to top |
|
|
glen herrmannsfeldt
Guest
|
| Posted:
Sun Nov 20, 2005 7:24 am Post subject:
Re: question about non-uniform sampling? |
|
|
Jerry Avins wrote:
| Quote: | glen herrmannsfeldt wrote:
|
(snip)
| Quote: | Given all the samples of his work and the first three as input, one
might compute in some weeks the most probable fourth movement.
|
| Quote: | Including the choral parts, which were unprecedented for a Beethoven
symphony?
|
I was thinking about the people who analyze written work to identify
the author. I believe such analysis is also done on musical
compositions. It seems, then, that the actual fourth movement is
not the most probable one.
-- glen |
|
| Back to top |
|
|
Matt Timmermans
Guest
|
| Posted:
Sun Nov 20, 2005 9:15 am Post subject:
Re: question about non-uniform sampling? |
|
|
JERRY> It is impossible to reconstruct a signal from a set of samples that
have
| Quote: | large gaps in time, even if the total number of samples is larger than
the number of uniformly spaced samples that are enough. Although
mathematics tells us that it is possible if the signal is perfectly
bandlimited, mathematics also tells us that such perfectly bandlimited
signals don't exist. The larger the gaps in the sampling sequence, the
worse the reconstruction will be. Where the samples are matters.
|
RON> We don't get mathematically perfectly bandlimited signals in the
| Quote: | real world. However, engineers succeed quite well at resampling
signals using imperfectly bandlimited signals.
|
The trouble is that as Jerry's sample gap increases in size, the
reconstruction of the signal in that gap becomes much more sensitive to
sampling filter leakage and other out-of-band noise. The growth in
sensitivity is exponential, so improving the sampling filter and quantizer
resolution by some given number of dB will only increase your usable gap
size by some fixed time.
Jerry's examples of "impossible stuff" with large gaps really are
impossible. To get feel for how quickly these things become impossible, try
this thought experiment:
Imagine a signal, filtered and sampled non-uniformly with a big gap.
Now, take the impulse response of your sampling filter, representing a spike
in the original singal, and add it to the filtered signal, centered in the
gap.
How much does that addition change the sample values, relative to its
amplitude? How big do you think you can make the gap before those
perturbations are so small that you can't reconstruct the difference in the
signal?
--
Matt |
|
| Back to top |
|
|
Ron N.
Guest
|
| Posted:
Mon Nov 21, 2005 1:15 am Post subject:
Re: question about non-uniform sampling? |
|
|
Matt Timmermans wrote:
| Quote: | JERRY> It is impossible to reconstruct a signal from a set of samples that
have
large gaps in time, even if the total number of samples is larger than
the number of uniformly spaced samples that are enough. Although
mathematics tells us that it is possible if the signal is perfectly
bandlimited, mathematics also tells us that such perfectly bandlimited
signals don't exist. The larger the gaps in the sampling sequence, the
worse the reconstruction will be. Where the samples are matters.
RON> We don't get mathematically perfectly bandlimited signals in the
real world. However, engineers succeed quite well at resampling
signals using imperfectly bandlimited signals.
The trouble is that as Jerry's sample gap increases in size, the
reconstruction of the signal in that gap becomes much more sensitive to
sampling filter leakage and other out-of-band noise. The growth in
sensitivity is exponential, so improving the sampling filter and quantizer
resolution by some given number of dB will only increase your usable gap
size by some fixed time.
Jerry's examples of "impossible stuff" with large gaps really are
impossible. To get feel for how quickly these things become impossible, try
this thought experiment:
Imagine a signal, filtered and sampled non-uniformly with a big gap.
Now, take the impulse response of your sampling filter, representing a spike
in the original singal, and add it to the filtered signal, centered in the
gap.
How much does that addition change the sample values, relative to its
amplitude? How big do you think you can make the gap before those
perturbations are so small that you can't reconstruct the difference in the
signal?
|
I think the gap can be made quite wide before the arithmetic and
quantization precision required exceeds the number of particles
in the universe, or the filter width even comes close to the length
of time since the big-bang.
Of course, I am speaking theoretically. I don't actually have a PC
with that much memory. But I assume Jerry's example was just a
thought experiment.
IMHO. YMMV.
--
rhn A.T nicholson d.O.t C-o-M |
|
| Back to top |
|
|
Ron N.
Guest
|
| Posted:
Mon Nov 21, 2005 1:15 am Post subject:
Re: question about non-uniform sampling? |
|
|
Jerry Avins wrote:
| Quote: | Ron N. wrote:
Jerry Avins wrote:
David Tweed wrote:
Jerry Avins wrote:
How non-uniform can the sampling be and still support reconstruction?
...
It is impossible to reconstruct a signal from a set of samples that have
large gaps in time, even if the total number of samples is larger than
the number of uniformly spaced samples that are enough. Although
mathematics tells us that it is possible if the signal is perfectly
bandlimited, mathematics also tells us that such perfectly bandlimited
signals don't exist. The larger the gaps in the sampling sequence, the
worse the reconstruction will be. Where the samples are matters.
We don't get mathematically perfectly bandlimited signals in the
real world. However, engineers succeed quite well at resampling
signals using imperfectly bandlimited signals. How well one can
resample depends on how close the real world bandlimiting gets to
the ideal. The better the pre-sampling bandlimit filter, the better
one can resample that "almost" bandlimited signal, including some
ability to deal with wider gaps in the uniformity of the sampling.
Higher quality bandlimiting filters which allow gap replacement
also have a longer delay (before sampling) so that causality is
not violated.
It's not a binary decision.
Your thought experiment fails because you failed to hypothesize
a "good enough" pre-sampling filter.
OK, do it then. Sample a signal uniformly at four times the prescribed
Nyquist rate, then discard a contiguous group consisting of half of
them. There remains twice the minimum number needed for reconstruction.
Reconstruct. Display the result and the original. When I see (or hear)
it, I'll believe that the locations of the samples in time can be arbitrary.
|
Take 8 samples of 1 second of a 1 Hz sine signal plus DC. Throw away
the last 4 samples. Find any reconstruction using just those 4 samples
which has absolutly no frequency content above 1 Hz when repeating
those 4 samples every 1 second over an infinite time extent. Use
closet fit if needed to overcome quantization error.
Take any 8 contiguous samples for 1 second at an 8 Hz sampling rate
of any signal. Perfectly filter those 8 samples so that they contain
no content above 1 Hz (using either an FFT/IFFT or a symmetric FIR
of width much greater than 16 samples). Call that signal X. Continue
as above to reconstruct X from any 4 samples, including only
the first 4 samples
Repeat for any N*8 samples plus perfect prefiltering to below N Hz
before discarding N*4 samples. etc. Use an arbitrary precision math
package so that the tails of a FIR prefilter or FFT are never discarded
in quantization noise before reconstruction of the 0 to N Hz signal.
As N doubles, increase the precision of the of the coefficients and
filtered samples as required.
Increase N to 4 * 44100 and time from 1 second to the length of
a symphony. Filter. Throw away the last half of the samples after
perfect filtering. Reconstruct. The number of bits of precision
required is left as an exercise for the student.
IMHO. YMMV.
--
rhn A.T nicholson d.O.t C-o-M |
|
| Back to top |
|
|
Jerry Avins
Guest
|
| Posted:
Mon Nov 21, 2005 1:15 am Post subject:
Re: question about non-uniform sampling? |
|
|
Ron N. wrote:
...
| Quote: | I think the gap can be made quite wide before the arithmetic and
quantization precision required exceeds the number of particles
in the universe, or the filter width even comes close to the length
of time since the big-bang.
Of course, I am speaking theoretically. I don't actually have a PC
with that much memory. But I assume Jerry's example was just a
thought experiment.
|
Not exactly. I was trying to show that from a set of real-world
non-uniform samples in which the nonuniformity (i.e., the ratio of the
longest interval to the shortest) exceeds a fairly small number, you
can't satisfactorily deduce the underlying continuous function.
Jerry
--
Engineering is the art of making what you want from things you can get.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ |
|
| Back to top |
|
|
Ron N.
Guest
|
| Posted:
Mon Nov 21, 2005 1:15 am Post subject:
Re: question about non-uniform sampling? |
|
|
Jerry Avins wrote:
| Quote: | Ummm... If the second half isn't sampled, how is it fed to the filter?
|
The actual sampler has to go after the low pass filter unless
the data is already bandlimited. Any "samples" before the filter
aren't bandlimited samples, just intermediate voltages or data
in the pre-sampler low pass filter.
| Quote: | I note that you said samples "from" the first half, not "during" the
first half. They're different due to the low pass filter delay.
The output of the filter may be delayed, but there's no delay at the
input. Sampled _from_ the first half are put in _during_ the first half
with at most trivial delay.
|
But for band-limited samples, you have to sample after the filter at
the filter output, and after waiting for the filter lag if you want
synchronized time-stamping.
-- rhn |
|
| Back to top |
|
|
Jerry Avins
Guest
|
| Posted:
Mon Nov 21, 2005 1:15 am Post subject:
Re: question about non-uniform sampling? |
|
|
Ron N. wrote:
| Quote: | Jerry Avins wrote:
Ummm... If the second half isn't sampled, how is it fed to the filter?
The actual sampler has to go after the low pass filter unless
the data is already bandlimited. Any "samples" before the filter
aren't bandlimited samples, just intermediate voltages or data
in the pre-sampler low pass filter.
|
Naturally. I don't see the connection except, of course, that you don't
need to low pass an interval that won't be sampled.
| Quote: | I note that you said samples "from" the first half, not "during" the
first half. They're different due to the low pass filter delay.
The output of the filter may be delayed, but there's no delay at the
input. Sampled _from_ the first half are put in _during_ the first half
with at most trivial delay.
But for band-limited samples, you have to sample after the filter at
the filter output, and after waiting for the filter lag if you want
synchronized time-stamping.
|
Sure, but once you get to the part that you don't intend to sample,
however long that takes, you can stop filtering. Keep fixed on the
structure of the counterexample I proposed: one oversamples the first N
minutes of a 2N-minute signal by a factor of more than two, then
reconstructs all N minutes worth from those samples. Reconstruction can
begin as soon as the requisite number of samples have been collected.
There are still knowledgeable people here who think I am recalcitrant
for claiming that this won't work. I can only conclude that I haven't
adequately communicated the proposed conditions. Shall I try again?
Jerry
--
Engineering is the art of making what you want from things you can get.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ |
|
| Back to top |
|
|
William Hughes
Guest
|
| Posted:
Mon Nov 21, 2005 9:15 am Post subject:
Re: question about non-uniform sampling? |
|
|
Jerry Avins wrote:
| Quote: | Ron N. wrote:
Jerry Avins wrote:
Ummm... If the second half isn't sampled, how is it fed to the filter?
The actual sampler has to go after the low pass filter unless
the data is already bandlimited. Any "samples" before the filter
aren't bandlimited samples, just intermediate voltages or data
in the pre-sampler low pass filter.
Naturally. I don't see the connection except, of course, that you don't
need to low pass an interval that won't be sampled.
I note that you said samples "from" the first half, not "during" the
first half. They're different due to the low pass filter delay.
The output of the filter may be delayed, but there's no delay at the
input. Sampled _from_ the first half are put in _during_ the first half
with at most trivial delay.
But for band-limited samples, you have to sample after the filter at
the filter output, and after waiting for the filter lag if you want
synchronized time-stamping.
Sure, but once you get to the part that you don't intend to sample,
however long that takes, you can stop filtering.
|
And since the filter lag is the length of the entire signal you
cannot stop filtering until after the entire signal has been
fed into the filter.
| Quote: | Keep fixed on the
structure of the counterexample I proposed: one oversamples the first N
minutes of a 2N-minute signal by a factor of more than two, then
reconstructs all N minutes worth from those samples. Reconstruction can
begin as soon as the requisite number of samples have been collected.
|
But the vaiues of the requisite number of samples cannot be determined
until after the entire signal has been processed. So reconstruction
cannot begin until after the entire signal. There is no causality
paradox.
| Quote: | There are still knowledgeable people here who think I am recalcitrant
for claiming that this won't work. I can only conclude that I haven't
adequately communicated the proposed conditions. Shall I try again?
|
Perhaps you should to figure out what the
conditions that others are talking about are (the fact
that you cannot reconstruct a real world signal with
samples taken in the first half is accepted by everyone).
The simple point is that you can only predict the entire
signal from samples taken in the first half if this
signal is band limited. And you cannot find a band
limited signal that adequately approximates your real
world signal without processing the entire signal. So
there is no causality paradox.
-William Hughes |
|
| Back to top |
|
|
robert bristow-johnson
Guest
|
| Posted:
Mon Nov 21, 2005 9:15 am Post subject:
Re: question about non-uniform sampling? |
|
|
in article 437EAA05.352D9355@acm.org, David Tweed at dtweed@acm.org wrote on
11/18/2005 23:41:
| Quote: | robert bristow-johnson wrote:
David Tweed at dtweed@acm.org wrote on 11/12/2005 09:32:
lucy wrote:
Can non-uniform sampled signal be used to perfectly reconstruct the
original continuous time signal?
Yes, but it isn't easy.
What is the Nyquist sampling rate in the non-uniform case?
Believe it or not, it's the same as the uniform case ... the number
of samples over the time interval must exceed twice the bandwidth
of the signal.
See here (questions 2 and 3) for a little more detail:
http://www.circuitcellar.com/library/eq/136/index.asp
hi Dave,
could you take a look at the paper that Bob Adams did in 1992 that i
reference here:
http://groups.google.com/group/comp.dsp/msg/ae7fe00eb3c8622b
i haven't cracked your brief analysis, but does that accomplish what
i was hoping would be shown that if your average sample rate is more
than twice the bandwidth, then random sampling will also be sufficient
for reconstruction?
Sorry for not replying sooner -- things got kind of busy around here
this past week.
I can't easily get at the paper itself, but your description in that
message leads me to believe that it does. Obviously, the samples have
to be unique.
|
| Quote: | I just browsed down through the rest of the thread -- it's amazing
what it turned into, thanks mainly to Jerry's stubbornness about
causality and how it relates to strict band-limiting.
|
well, i generally identify with Jerry both technically and in terms of
attitude. i took a brief look at your circuitcellar.com answer and i think
i understand it. it seems to me that you need to solve a system of an
infinite number of equations that have an infinite number of unknowns,
unless you consider a windowed sinc(), and then the theory is not perfect.
i think what you did is a generalization of what Bob Adams did (his paper
involved the periodic removal of a sample and reconstructing) and, if i can
find the paper, i'll post a synopsis of it here.
--
r b-j rbj@audioimagination.com
"Imagination is more important than knowledge." |
|
| Back to top |
|
|
Guest
|
| Posted:
Mon Nov 21, 2005 3:26 pm Post subject:
Re: question about non-uniform sampling? |
|
|
Jerry Avins wrote:
| Quote: | Ron N. wrote:
Jerry Avins wrote:
Ummm... If the second half isn't sampled, how is it fed to the filter?
The actual sampler has to go after the low pass filter unless
the data is already bandlimited. Any "samples" before the filter
aren't bandlimited samples, just intermediate voltages or data
in the pre-sampler low pass filter.
Naturally. I don't see the connection except, of course, that you don't
need to low pass an interval that won't be sampled.
I note that you said samples "from" the first half, not "during" the
first half. They're different due to the low pass filter delay.
The output of the filter may be delayed, but there's no delay at the
input. Sampled _from_ the first half are put in _during_ the first half
with at most trivial delay.
But for band-limited samples, you have to sample after the filter at
the filter output, and after waiting for the filter lag if you want
synchronized time-stamping.
Sure, but once you get to the part that you don't intend to sample,
however long that takes, you can stop filtering. Keep fixed on the
structure of the counterexample I proposed: one oversamples the first N
minutes of a 2N-minute signal by a factor of more than two, then
reconstructs all N minutes worth from those samples. Reconstruction can
begin as soon as the requisite number of samples have been collected.
There are still knowledgeable people here who think I am recalcitrant
for claiming that this won't work. I can only conclude that I haven't
adequately communicated the proposed conditions. Shall I try again?
|
You don't need to, since what you're doing isn't signal samplimg.
it's
infornmation sampling.
And you conveniently picked the onle application that is does work
well,
with telecommunications equipment.
i.e. you're essentially not sampling anything physical,
but rather than Goedel's Theroem.
And what you're doing is usually called reverese-engineering,
rather than reconstruction.
| Quote: |
Jerry
--
Engineering is the art of making what you want from things you can get.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ |
|
|
| Back to top |
|
|
jim
Guest
|
| Posted:
Mon Nov 21, 2005 5:15 pm Post subject:
Re: question about non-uniform sampling? |
|
|
William Hughes wrote:
| Quote: |
Nit. You can only produce a bandlimited approximation (unless
the concert was bandlimited (unlikely))
|
Whether you think it resembles "the concert" or not is not relevant.
The important thing is we would have a bandlimited function which we can
hypothetically sample for the purpose of analyzing the process of
non-uniform sampling.
| Quote: | You can now re-sample that continuous function with a
non-uniform scheme where you end up with a new set of samples of the
same number as before clustered in the first half of the concert. Can
you now reconstruct the entire concert? No, you can't unless you know
the exact time interval of the whole concert. It's not enough to simply
say the samples represent a bandlimited signal, you need to know what
the actual frequency limits are.
Indeed. (There is of course a similar problem in the uniform case,
but it is not as extreme as the fact that you know the samples
are uniform gives you a lot of information, so it is possible
to calculate a frequency limit below which there is no ambiguity.
In the non uniform case we cannot do this without knowing
the length of the signal to be reconstructed.)
|
Yes, but that's far from a trivial problem. In the uniform case you may
not know what the original sample rate was but its easy to imagine that
the concert samples could be analyzed and enough clues found to
reconstruct and play back where the listener would not even notice. With
the non-uniform case if you don't pretty much know the frequency content
of the signal before hand, its game over. In the example I gave above of
non-uniform sampling, you couldn't even do a good job of reconstructing
the half of the concert that the samples were taken from. Knowing that
the samples came from a bandlimited signal is pretty much useless info.
-jim
----== Posted via Newsfeeds.Com - Unlimited-Unrestricted-Secure Usenet News==----
http://www.newsfeeds.com The #1 Newsgroup Service in the World! 120,000+ Newsgroups
----= East and West-Coast Server Farms - Total Privacy via Encryption =---- |
|
| Back to top |
|
|
Jerry Avins
Guest
|
| Posted:
Mon Nov 21, 2005 5:16 pm Post subject:
Re: question about non-uniform sampling? |
|
|
William Hughes wrote:
...
| Quote: | Indeed. (There is of course a similar problem in the uniform case,
but it is not as extreme as the fact that you know the samples
are uniform gives you a lot of information, so it is possible
to calculate a frequency limit below which there is no ambiguity.
In the non uniform case we cannot do this without knowing
the length of the signal to be reconstructed.)
|
From this I conclude you mean that spacing the samples nonuniformly
reduces the information they can yield. I've been saying that if the
nonuniformity is great enough, they become essentially useless. We're on
the same track. "No problem" becomes "no way".
Jerry
--
Engineering is the art of making what you want from things you can get.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ |
|
| Back to top |
|
|
Jerry Avins
Guest
|
| Posted:
Mon Nov 21, 2005 5:16 pm Post subject:
Re: question about non-uniform sampling? |
|
|
William Hughes wrote:
...
| Quote: | But the vaiues of the requisite number of samples cannot be determined
until after the entire signal has been processed. So reconstruction
cannot begin until after the entire signal. There is no causality
paradox.
|
I can't imagine how causality entered into this discussion. As I
understood the original assertion about nonuniform sampling,
reconstruction is possible so long as the average sampling interval (or
was it rate?) meets the Nyquist criterion. For uniform sampling of a
signal of bandwidth B Hz and duration D seconds, at least 2BD samples
are needed. The assertion stated that 2BD samples also suffices for
nonuniform sampling, and that the location of the samples within the
signal is irrelevant.
| Quote: | There are still knowledgeable people here who think I am recalcitrant
for claiming that this won't work. I can only conclude that I haven't
adequately communicated the proposed conditions. Shall I try again?
|
Consider a two-second telephone monolog. B = 3.5 kHz; D = 2. By sampling
at 8 KHz, there is some processing margin. With uniform sampling, there
will be 16,000 samples, and according to the assertion, 16,000 samples
taken anywhere in the monolog suffice to characterize it.
Before sampling, the signal is passed through a very sharp lowpass
filter that cuts off at 3.5 KHz. The cutoff of the analog filter is so
sharp and deep -- its impulse response is so long -- that the delay
through it is several days, but we're patient. At the appropriate time,
sampling begins; not at 8 KHz, but 16. After one second of sampling, all
of the necessary 16,000 samples have been recorded. Although the filter
continues to produce output, the sampler can stop.
I claim that the 16,000 samples so obtained meet the conditions of the
assertion but do not support the reconstruction of the entire monolog.
Do think my claim is false?
Jerry
--
Engineering is the art of making what you want from things you can get.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ |
|
| Back to top |
|
|
|
|
|
|
FAQ
Memberlist
Register
Profile
Log in to check your private messages
Log in