CAUTION! was "What is the advantage on high-sampling rate ?"

Tim Wescott wrote:

   ...


> That reflection and repetition of the signal around the sampling 
> frequency and it's harmonics _is_ aliasing -- that's how a signal at 5/8 
> of the sample frequency ends up at 3/8 of the sample frequency (and 5/8, 
> and 1-3/8, and 1-5/8, etc.).  Since it's usually undesirable it's what 
> drives the need for an anti-aliasing filter.

I think you missed the import of the plots. It showed how something at
1/4 of the sampling frequency also shows up 3/4, at 1 3/4, at 2 1/4,
etc. even when a proper anti-alias filter is used. None of that is
aliasing, and it's all removed by the reconstruction filter.

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

In article pBVic.33755$IW1.1506386@attbi_s52, glen herrmannsfeldt at
gah@ugcs.caltech.edu wrote on 04/25/2004 16:55:

> robert bristow-johnson wrote:
> 
> (snip)
> 
>> this is very reminiscent of that periodic argument we've have about whether
>> or not the DFT inherently periodically extends the finite set of input data.
>> from the purest, simplest mathematical definition, it's pretty clear (to me,
>> at least, perhaps to Tim, and also to the O&S discussion of the topic) that
>> the DFT *does* inherently periodically extend the finite length input.  i.e.
>> the DFT views your input data as one period of a discrete periodic function
>> of "time" and returns one period of a discrete periodic function in the
>> "frequency" domain.
> 
> I agree.  Just like the Fourier series did a long time ago.

i have also maintained that the DFT and DFS are one and the same.  they be
the same damn thing, both conceptually and practically.

r b-j

Tim Wescott wrote:

(snip)

>>> Nyquist's theory basically stated that if the signal going _into_ 
>>> sampling is perfectly band-limited then you can build a perfect 
>>> reconstruction filter and get the whole thing back unmolested, which 
>>> is why 1/(2T_s) is called the Nyquist Frequency.

Nyquist was working on getting telegraph pulses through a
band limited channel and wanted to know how closely spaced
they could be and still be separated at the other end.
It is, conveniently, exactly the opposite problem and so
has the same solution.

> That reflection and repetition of the signal around the sampling 
> frequency and it's harmonics _is_ aliasing -- that's how a signal at 5/8 
> of the sample frequency ends up at 3/8 of the sample frequency (and 5/8, 
> and 1-3/8, and 1-5/8, etc.).  Since it's usually undesirable it's what 
> drives the need for an anti-aliasing filter.

I suppose so.  The important aliasing is that which ends up in
the important part of the spectrum, and can't be removed later.

While I do agree that the DFT is periodic (discussed in another thread),
I don't necessarily believe that sampling previously band limited
signals generates periodic signals in frequency space.
The sampled signal is a representation of the continuous signal.
It could, theoretically, be converted back to a continuous signal as a 
sum of sinc's.   Other reconstruction methods generate extraneous
frequency components which need to be filtered out.

-- glen

Jerry Avins wrote:

(snip)

> I think you missed the import of the plots. It showed how something at
> 1/4 of the sampling frequency also shows up 3/4, at 1 3/4, at 2 1/4,
> etc. even when a proper anti-alias filter is used. None of that is
> aliasing, and it's all removed by the reconstruction filter.

I think I agree with you, but note that band limited doesn't
imply baseband.  It may be that the 3/4 is the real signal and
the 1/4 is the alias.

Otherwise, it seems to me that the use of the term aliasing
is similar to the tree in the forest problem.  The aliasing
that you hear are the frequencies that alias into the signal
frequencies, which come from not having the proper
anti-aliasing filter.   If one uses the wrong reconstruction
filter, and so ends up with extraneous frequency components
(that may or may not be aliases), can the process be called
aliasing?   It sounds better than reconstructioning.

-- glen

In article <408922bf.1261440968@news.sf.sbcglobal.net>,
Rick Lyons <r.lyons@_BOGUS_ieee.org> wrote:
>The phrase "tones exist beyond 110KHz" is troubling.
>If the sampling rate is 44.1 kHz, no frequency above half 
>that (+22.05 kHz) has meaning.  In the world of sampled 
>signals, there is no signal energy above +22.05 kHz.

Of course frequencies above the sample rate have meaning.  It's just
that after sampling you can't distinguish them from identical frequencies
mirrored below (or around multiples of) the sample rate.

Normally one low pass filters stuff below Fs/2-e before sampling to
disambiguate these aliased frequencies, but one could just as easily
bandpass filter stuff between N*Fs+e and (N+1/2)*Fs-e and get equally
unambiguous frequency information from the samples, even frequencies
well above the sample rate (given sufficient clock jitter bounds, etc.)


IMHO. YMMV.
-- 
Ron Nicholson   rhn AT nicholson DOT com   http://www.nicholson.com/rhn/ 
#include <canonical.disclaimer>        // only my own opinions, etc.

Jerry Avins wrote:

> Tim Wescott wrote:
> 
>   ...
> 
> 
>> That reflection and repetition of the signal around the sampling 
>> frequency and it's harmonics _is_ aliasing -- that's how a signal at 
>> 5/8 of the sample frequency ends up at 3/8 of the sample frequency 
>> (and 5/8, and 1-3/8, and 1-5/8, etc.).  Since it's usually undesirable 
>> it's what drives the need for an anti-aliasing filter.
> 
> 
> I think you missed the import of the plots. It showed how something at
> 1/4 of the sampling frequency also shows up 3/4, at 1 3/4, at 2 1/4,
> etc. even when a proper anti-alias filter is used. None of that is
> aliasing, and it's all removed by the reconstruction filter.
> 
> Jerry

That's 'cause I was responding to the text of the posts.  Looking at the 
site, yes, he's talking about reconstruction, but since they're more or 
less reciprocal operations most of my comments apply.

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

robert bristow-johnson wrote:

> In article 108o7mfh158eb95@corp.supernews.com, Tim Wescott at
> tim@wescottnospamdesign.com wrote on 04/25/2004 16:26:
> 
> 
>>robert bristow-johnson wrote:
>>
>>
>>>In article N8-dnUQz4eUnxRTdRVn-hA@centurytel.net, Fred Marshall at
>>>fmarshallx@remove_the_x.acm.org wrote on 04/23/2004 13:56:
>>>
>>>...
>>
>>-- snip --
>>
>>>this is very reminiscent of that periodic argument we've have about whether
>>>or not the DFT inherently periodically extends the finite set of input data.
>>>from the purest, simplest mathematical definition, it's pretty clear (to me,
>>>at least, perhaps to Tim, and also to the O&S discussion of the topic) that
>>>the DFT *does* inherently periodically extend the finite length input.  i.e.
>>>the DFT views your input data as one period of a discrete periodic function
>>>of "time" and returns one period of a discrete periodic function in the
>>>"frequency" domain.
>>>
>>>lessee how folks react to that.
> 
> 
>>Oh hell, I'll bite.
> 
> 
> BWAA-HA-HA-HA-HA!
> 
> 
>>The action of sampling a signal (_not_ taking it's DFT) makes your data
>>appear as one period of a theoretically infinite number of periods _in
>>frequency_.
> 
> 
> i agree, but not quite completely.  the language i would use that that the
> action of sampling a continuous-time signal has the effect of periodically
> extending its spectrum in the frequency domain by shifting the spectrum by
> all integer multiples of Fs, overlapping, and summing (this would cause
> aliasing if B is not less than Fs/2).  (there is a scaling issue also, the
> "T" or "1/Fs" factor, but i don't wanna slug that out now since it has been
> *many* times before.)
> 
> where i think i substantively disagree with your semantic is in the
> inclusion of the words "one period of".  i would delete those words in your
> statement to be fully accurate.
> 
> 
>>The action of "sampling" your signal _in frequency_ makes your
>>time-domain data appear to be one period of a theoretically infinite
>>number of periods _in time_.
> 
> 
> same sentiment of agreement and disagreement.
> 
> 
>>In fact, you can prove this mathematically:
>>
>>Take a periodic signal and "window" it down to one period in time.
> 
> 
> okay, fine.  but the DFT (or the plain ol' FT) doesn't do that.  the
> operation of windowing (and what it does to your spectrum) is a separate
> operation done before the DFT or FT sees the data.
> 
> 
>> Now take it's Fourier transform -- you get a result that's continuous
>>in frequency, and extends to infinite frequency in both directions.
> 
> 
> yup.
> 
> 
>> Now 
>>allow the window to be exactly two periods, but divide the signal by 2
>>-- your result is still continuous in frequency, but it's starting to
>>develop peaks.  Now take the Fourier transform of N periods, times 1/N,
>>and take the limit as N goes to infinity -- you get a Foirier transform
>>that's a train of impulses at the harmonics of the fundamental, with
>>zero energy between them.
> 
> 
> sure.  i'll drink to that.
> 
> 
>>You can use this to prove that it's periodic in one domain if and only
>>if it's a train of impulses in the other.  So the DFT is
>>(mathematically) a periodic train of impulses in both domains.
> 
> 
> oh, i thought we had a disagreement brewing here.  i should read the whole
> damn USENET post before i hit "Reply to Newsgroup".
> 
> never mind.
> 
> (actually, i would say the DFT and iDFT works on "numbers", not "impulses".
> the DFT is a discrete-time and discrete-frequency thingie whose domains have
> no impulses.  in these domains, there is no "between them" for there to be
> zero energy.  i think i'm being more anal in my semantics than what you're
> saying, Tim.)
> 
> another way i might word it is that the DFT transforms a periodic discrete
> function (or "periodic sequence") of infinite length and period N (of which
> you need only specify one period or N numbers) to another periodic sequence
> of infinite length and period N and the iDFT transforms it back.
> 
> Tim, multiple times in the past 9 years, we have had this topic tossed
> around and i have taken a fairly rigid stand about it because i think it
> *should* be sorta uncontroversial (sorta like the MATLAB index base issue or
> where the "T" or "1/T" scaling factor belongs in the sampling/reconstruction
> theorem, or the dimension of the dependent variable of a Dirac impulse in
> time).  but i have certainly found out different (about the controversy, not
> my position on this stuff, which is pretty firm) and have the singe marks to
> show for it.  a few very respectable folks on comp.dsp (like R. Cain and A.
> Hey, IIRC) have staunchly disagreed with one thing or 'nother.
> 
> r b-j
> 

Probably because some of the nuances of the Fourier transform are highly 
counter-intuitive (or at least a-intuitive) so you have to talk yourself 
into believing them.  You can get yourself into a rut that way.

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

rhn@mauve.rahul.net (Ronald H. Nicholson Jr.) wrote in message news:<c6iit0$6b7> Normally one low pass filters stuff below Fs/2-e before sampling to
> disambiguate these aliased frequencies, but one could just as easily
> bandpass filter stuff between N*Fs+e and (N+1/2)*Fs-e and get equally
> unambiguous frequency information from the samples, even frequencies
> well above the sample rate (given sufficient clock jitter bounds, etc.)
> 
> 
> IMHO. YMMV.

See message 16 (Jerry) and mine message 17. It is not about the antialias filter...

BR
Dan Lavry

In article <108m263idq1hr13@corp.supernews.com>,
Tim Wescott  <tim@wescottnospamdesign.com> wrote:
>The sampling process ends up being the mathematical equivalent of 
>multiplying the signal by a train of dirac delta functions (impulses). 

Didn't some UK ham radio type recently use this argument to annoy this
newsgroup with some completely bizarre DSP theory?

The conclusion seemed to be that there is no train of Dirac delta
functions in the sampling process, only a train of numbers, each number
of which could be considered as the result of the integral of a single
impulse (approaching the Dirac delta as a limit) with some input signal
or function.

And talking about the continuous spectra of a finite array of numbers
seems to only make sense in the context of a definition of some (either
mathematical or implementable) continuous reconstruction process.


IMHO. YMMV.
-- 
Ron Nicholson   rhn AT nicholson DOT com   http://www.nicholson.com/rhn/ 
#include <canonical.disclaimer>        // only my own opinions, etc.

Perhaps a private e-mail question to Dan, or even a post here questioning that
particular point would have been a more appropriate first step than a "CAUTION"
message that calls his entire paper into doubt.  IMHO.

-Jon

"Rick Lyons" <r.lyons@_BOGUS_ieee.org> wrote in message
news:4089c891.1303891562@news.sf.sbcglobal.net...
> On 23 Apr 2004 12:35:12 -0700, danlavry@mindspring.com (dan lavry)
> wrote:
>
> >r.lyons@_BOGUS_ieee.org (Rick Lyons) wrote in message
news:<408922bf.1261440968@news.sf.sbcglobal.net>...
> >> Hi Guys,
> >>   in a recent thread, mention was made of a "Sampling"
> >> paper by Dan Lavry.  At the following web site
> >>
> >> http://www.lavryengineering.com/pdfs/sample.pdf
> >>
> >> you can see Dan's 1997 paper:
> >> "Sampling, Oversampling, Imaging and
> >> Aliasing - a basic tutorial".
> >>
> >>
> >> I recommend caution if you decide to
> >> read that paper.  In the second paragraph Dan wrote:
> >
> >> It appears to me that Dan did what we've all done
> >> at one time or another.  That is, we model some
> >> process with software and then we completely
> >> misinterpret our results.
> >
> >BTW, It was all written in Mathcad.
> >
> >> I'm not bad-mouthing Dan Lavry.  He's seems to be
> >> a *highly-skilled* audio engineer.  I'm just suggesting
> >> caution when reading his sampling paper.
> >
> >Well, maybe not really bad mouthing me. I wish you did not write it
> >all off so quickly. I do my best to to provide free education to folks
> >in audio, (many of them that don't know much math). Yes English is not
> >my first langauge and I do make a lot of mistakes. I am trying harder,
> >and have folks read it and correct my English as best they can.
> >
> >I do not think I desrved that comment warning folks to be carfull.
> >
> >Dan Lavry
>
>
> Hi,
>
>    I meant no offense Dan.  Really.
> I'm not kidding.
>
> I thought I detected a misinterpretation
> of the spectral effects of "periodic sampling".
>
> So Dan, please don't be upset.  And please know
> that anyone who takes the time and trouble to
> write tutorials for their colleagues has our
> admiration and gratitude.  That means you Dan.