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:

  *** Sampling theory ***
  Sampling a value, in the Nyquist sense amounts to 
  "pinning down" signal values at equal time intervals. 
  Let us "cover much of the audio band" by summing four 
  equal amplitude tones (5,10,15 and 20KHz). The time 
  domain plot shows the resulting continues (analog) 
  waveform and it's sampled counterpart (44.1KHz sampling). 
  The frequency domain plot shows the energy concentration 
  of the sampled signal across 0 to 110KHz (tones exist 
  beyond 110KHz). The frequency domain plot shows the 
  energy concentration of the sampled signal across 0 to 
  110KHz (tones exist beyond 110KHz). The sampled signal 
  contains the four tones bellow 22KHz and undesirable 
  energy at frequencies above 22KHz. Recovering the analog 
  signal requires no more then removal of all energy 
  above 22KHz.

          [The misspelled words are Dan's, not mine.]

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.

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.

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.
 
[-Rick-]

Rick Lyons wrote:

> 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:
> 
>   *** Sampling theory ***
>   Sampling a value, in the Nyquist sense amounts to 
>   "pinning down" signal values at equal time intervals. 
>   Let us "cover much of the audio band" by summing four 
>   equal amplitude tones (5,10,15 and 20KHz). The time 
>   domain plot shows the resulting continues (analog) 
>   waveform and it's sampled counterpart (44.1KHz sampling). 
>   The frequency domain plot shows the energy concentration 
>   of the sampled signal across 0 to 110KHz (tones exist 
>   beyond 110KHz). The frequency domain plot shows the 
>   energy concentration of the sampled signal across 0 to 
>   110KHz (tones exist beyond 110KHz). The sampled signal 
>   contains the four tones bellow 22KHz and undesirable 
>   energy at frequencies above 22KHz. Recovering the analog 
>   signal requires no more then removal of all energy 
>   above 22KHz.
> 
>           [The misspelled words are Dan's, not mine.]
> 
> 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.
> 
> 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.
> 
> 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.
>  
> [-Rick-]

Rick,

Isn't it just a matter of interpretation? The process of sampling
converts a continuous signal into a series of pulses (or rectangles)
whose heights matches the original signal at the sample points. The
"tones beyond 110 KHz" that he talks of are actually in the choppy
signal. That's why a reconstruction filter is needed. It seems to me
that, in the passage you question, Lavry wrote about the actual digital
waveform, not about signal that it encodes. "Recovering the analog
signal requires no more than removal of all energy above 22 KHz."
[Lavry's spelling corrected] Your books say much the same.

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

Rick Lyons wrote:
> 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:
> 
>   *** Sampling theory ***
>   Sampling a value, in the Nyquist sense amounts to 
>   "pinning down" signal values at equal time intervals. 
>   Let us "cover much of the audio band" by summing four 
>   equal amplitude tones (5,10,15 and 20KHz). The time 
>   domain plot shows the resulting continues (analog) 
>   waveform and it's sampled counterpart (44.1KHz sampling). 
>   The frequency domain plot shows the energy concentration 
>   of the sampled signal across 0 to 110KHz (tones exist 
>   beyond 110KHz). The frequency domain plot shows the 
>   energy concentration of the sampled signal across 0 to 
>   110KHz (tones exist beyond 110KHz). The sampled signal 
>   contains the four tones bellow 22KHz and undesirable 
>   energy at frequencies above 22KHz. Recovering the analog 
>   signal requires no more then removal of all energy 
>   above 22KHz.
> 
>           [The misspelled words are Dan's, not mine.]
> 
> 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.
> 
> 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.
> 
> 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.
>  
> [-Rick-]
> 

The usual formal method of modeling a sampled signal is to call it the 
continuous signal as multiplied by a train of unit impulses.  With this 
model there _will_ be significant tones stretching out to infinity. 
What _meaning_ you assign to them is up to you.

When you go to reconstruct the signal you'll probably start by running 
it through a DAC to generate rectangular pulses.  Meaningless or not 
there'll still be energy there that needs to be filtered out.

In your excerpt at least his interpretation leads to the correct 
prescription, i.e. a good filter at the output of the DAC.  If this is 
so, how is it a misinterpretation?

-- 

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

"Tim Wescott" <t...@wescottnospamdesign.com> wrote in message
news:1...@corp.supernews.com...
> Rick Lyons wrote:
> > 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:
> >
> >   *** Sampling theory ***
> >   Sampling a value, in the Nyquist sense amounts to
> >   "pinning down" signal values at equal time intervals.
> >   Let us "cover much of the audio band" by summing four
> >   equal amplitude tones (5,10,15 and 20KHz). The time
> >   domain plot shows the resulting continues (analog)
> >   waveform and it's sampled counterpart (44.1KHz sampling).
> >   The frequency domain plot shows the energy concentration
> >   of the sampled signal across 0 to 110KHz (tones exist
> >   beyond 110KHz). The frequency domain plot shows the
> >   energy concentration of the sampled signal across 0 to
> >   110KHz (tones exist beyond 110KHz). The sampled signal
> >   contains the four tones bellow 22KHz and undesirable
> >   energy at frequencies above 22KHz. Recovering the analog
> >   signal requires no more then removal of all energy
> >   above 22KHz.
> >
> >           [The misspelled words are Dan's, not mine.]
> >
> > 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.
> >
> > 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.
> >
> > 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.
> >
> > [-Rick-]
> >
>
> The usual formal method of modeling a sampled signal is to call it the
> continuous signal as multiplied by a train of unit impulses.  With this
> model there _will_ be significant tones stretching out to infinity.
> What _meaning_ you assign to them is up to you.

Ah!  Sounds like a mixture of viewpoints....

One viewpoint sees the DFT as limited to fs.  This makes sense if we view
the spectrum on a unit circle.  You can only get up to fs which is also
zero.....  I think this is what Rick was using.

The other viewpoint sees the unit circle "unwrapped" to form an infinite
line.  In this case the DFT is periodic at fs and extends to the infinities.
I think this is what Tim was using.

Both are correct / useful.  One for sampled data and one for continuous
signals / functions.  Obviously, dealing with DACs requires our thinking to
transcend between the two.

We have the same sort of thing in antenna patterns.
If an antenna pattern is plotted as a function of the look angle expressed
as the angle itself, then the angle can grow infinitely.  Functions such as
those dealt with by the classical van der Maas and Taylor analyses were done
this way.

If the pattern is plotted as a function of the *cosine* of the look angle,
then the plot is bounded by angles that are a multiple of 2pi and the
remaining part of the pattern is referred to as being "in the invisible
region".  If we *only* look at this region then it's possible to create
situations called "supergaining" where there are huge peaks in the invisible
region.  These are accompanied by large peaks at the edges of the antenna
illumination function.

It's interesting but I don't know that there's a direct analog in signal
processing - just a similarity.  I believe that's partly because the antenna
illumination functions under consideration in those analyses are continuous
and the functions in DSP systems are sampled.

Let's see....
Start with a FIR filter perhaps with very large N due to oversampling - so
that it approximates a continuous function in time.  Then, don't compute a
DFT just compute a continuous FT.  This way we don't view the filter as
being periodic in time.  Now, if we limit our view of the FT to a region
that's below some limit, we can make the FIR filter look really good as long
as we only look below that limiting frequency - that the expense of having
it "blow up" outside that range.

How could this possibly be useful?
Well, I guess one might have this situation:

1) A supergained FIR filter design with the "region of interest" below fs/10
and some huge  peaks above fs/5 let us say - a "don't care" region.
2) A signal with no energy at all above fs/5.

Multiplying the signal Fourier Transform by the filter Fourier Transform
yields a very good filter on the signal below fs/10 without a penalty due to
the big peaks in the filter response above fs/5.

There must be something wrong with this picture taken in the practical
world.  I've not figured out what it is just yet.... maybe the high sampling
rate necessary to give enough room in the spectrum to supergain.  Also, the
danger that there might be signal energy above fs/5.  Also, the numeric
demands of using very small contributions in the filter response.  Probably
doesn't work very well with real arithmetic.

Why does this have any chance of working anyway?
Think of the filter frequency response as being made up of a series of sincs
(OK, Dirichlet kernels) that correspond with the length of the FIR filter.
Just like reconstruction in time but flipped into the frequency domain by
virtue of duality.  (The FIR filter is "time-limited" to compare with
"bandlimited")

If the signal frequency range of interest is only fs/10 then we can insert
frequency sincs above fs/10 such that their "tails" contribute to improved
response below fs/10.  Because the tails are relatively small, the sincs
often have to be pretty large to accomplish this.  One result of doing this
is to get narrower transition regions in filter response below fs/10.
[fs/10 is just a point picked as an example - it's not a magic number].

Interesting.  Not really useful.

Fred

Fred Marshall wrote:
> "Tim Wescott" <t...@wescottnospamdesign.com> wrote in message
> news:1...@corp.supernews.com...
> 
>>Rick Lyons wrote:
--snip--
> 
> Ah!  Sounds like a mixture of viewpoints....
> 
> One viewpoint sees the DFT as limited to fs.  This makes sense if we view
> the spectrum on a unit circle.  You can only get up to fs which is also
> zero.....  I think this is what Rick was using.
> 
> The other viewpoint sees the unit circle "unwrapped" to form an infinite
> line.  In this case the DFT is periodic at fs and extends to the infinities.
> I think this is what Tim was using.
> 
> Both are correct / useful.  -- snip --
> 
> 

If you're young enough to have watched Sesame Street: "That's about the 
size, where you put your eyes ..."  Really irritating tune to have 
bouncing around in your head, but a very good lesson when you're trying 
to figure out that peculiar problem that _might_ be easy if you look at 
it from a different direction.

-- 

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

r...@_BOGUS_ieee.org (Rick Lyons) wrote in message
news:<4...@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:
> 
>   *** Sampling theory ***
>   Sampling a value, in the Nyquist sense amounts to 
>   "pinning down" signal values at equal time intervals. 
>   Let us "cover much of the audio band" by summing four 
>   equal amplitude tones (5,10,15 and 20KHz). The time 
>   domain plot shows the resulting continues (analog) 
>   waveform and it's sampled counterpart (44.1KHz sampling). 
>   The frequency domain plot shows the energy concentration 
>   of the sampled signal across 0 to 110KHz (tones exist 
>   beyond 110KHz). The frequency domain plot shows the 
>   energy concentration of the sampled signal across 0 to 
>   110KHz (tones exist beyond 110KHz). The sampled signal 
>   contains the four tones bellow 22KHz and undesirable 
>   energy at frequencies above 22KHz. Recovering the analog 
>   signal requires no more then removal of all energy 
>   above 22KHz.
> 
>           [The misspelled words are Dan's, not mine.]
> 
> 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.
> 
> 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.
> 
> 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.
>  
> [-Rick-]


Rick,

First the disclaimer: I haven't read Dan's paper.  Now the comment:
Would you agree that if I sample at a 44.1 KHz rate, that the sampled
spectrum is replicated every 44.i KHz?


Maurice

r...@_BOGUS_ieee.org (Rick Lyons) wrote in message
news:<4...@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:

I am showing what happens when you you sample without antialiasing
filter, and when you do so, you get high frequency images. I do so
INTENTIONALY to show the need for anti aliasing filters ahead of the
AD converters. I later explain that the enrgy folds into the base
band. I meant it to be a stepping stone, and sorry if it ended up
being confusing.

> 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.

NOT FAIR!!!

THE ENERGY CONTENT IN THE NARROW IMPULSES DOES CONTAIN HIGH
FREQUENCIES AS SHOWN. WHEN DOES THE ALIASING TAKE PLACE? AT THE SAMPLE
 AND HOLD? AT THE AD? AT THE DA? WHAT I SAID IS CORRECT. A NARROW
IMPULSE DOES CONTAIN A LOT OF HARMONICS AND THAT IS WHAT I SHOWED. THE
NRZ ALSO CONTAINES ALOT OF HOGH FREQUENCIES.

I AGREE THAT BY THE TIME YOU ARE DONE WITH THE WHOLE PROCESS OF
SAMPLING AND RETRIVAL, YOU CAN NOT HAVE ENERGY ABOVE NYQUIST. BUT A
NARROW IMPULSE DOES CONTAIN THOSE HARMONICS. THEY ARE MEASURABLE AND
REAL.

> 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

dan lavry wrote:

> r...@_BOGUS_ieee.org (Rick Lyons) wrote in message
news:<4...@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:
> 
> 
> I am showing what happens when you you sample without antialiasing
> filter, and when you do so, you get high frequency images. I do so
> INTENTIONALY to show the need for anti aliasing filters ahead of the
> AD converters. I later explain that the enrgy folds into the base
> band. I meant it to be a stepping stone, and sorry if it ended up
> being confusing.

  ...

Until now, I thought I understood. I saw a properly sampled signal
reflected around the sampling frequency and the whole thing repeated up
and up ...

How does an anti-alias filter before the ADC come into that?

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

On 23 Apr 2004 12:35:12 -0700, d...@mindspring.com (dan lavry)
wrote:

>r...@_BOGUS_ieee.org (Rick Lyons) wrote in message
news:<4...@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:
>
>I am showing what happens when you you sample without antialiasing
>filter, and when you do so, you get high frequency images. I do so
>INTENTIONALY to show the need for anti aliasing filters ahead of the
>AD converters. I later explain that the enrgy folds into the base
>band. I meant it to be a stepping stone, and sorry if it ended up
>being confusing.
>
>> 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.
>
>NOT FAIR!!!
>
>THE ENERGY CONTENT IN THE NARROW IMPULSES DOES CONTAIN HIGH
>FREQUENCIES AS SHOWN. WHEN DOES THE ALIASING TAKE PLACE? AT THE SAMPLE
> AND HOLD? AT THE AD? AT THE DA? WHAT I SAID IS CORRECT. A NARROW
>IMPULSE DOES CONTAIN A LOT OF HARMONICS AND THAT IS WHAT I SHOWED. THE
>NRZ ALSO CONTAINES ALOT OF HOGH FREQUENCIES.
>
>I AGREE THAT BY THE TIME YOU ARE DONE WITH THE WHOLE PROCESS OF
>SAMPLING AND RETRIVAL, YOU CAN NOT HAVE ENERGY ABOVE NYQUIST. BUT A
>NARROW IMPULSE DOES CONTAIN THOSE HARMONICS. THEY ARE MEASURABLE AND
>REAL.
>
>> 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".

When the first sentence used the phrase 
"Sampling a value, in the Nyquist sense, ...", 
and I saw those spectral replications (images) 
of equal amplitude my brain 
interpreted it all to mean "A/D conversion".
And the last sentence's phrase " Recovering the 
analog signal ..." reinforced my notion 
that we're talking about A/D conversion. 

Your remark that "tones exist up to 110 kHz" 
threw me off balance.  I thought, "Oh shoot, 
something doesn't seem right here.

In your above text (capitalized) 
you used the phrase "narrow impulse".  But in your 
article you didn't use that "narrow impulse" 
terminology, you used phrases like "sampled signal", 
and "sampled values".
So when I read those words I thought you meant 
discrete (binary) samples.  (A/D conversion.)
I misunderstood.  

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.

Gosh don't worry about your English, your English 
is fine.  And I can see that you and I have 
about equal skill in spelling.  :-)

Regards,
[-Rick-]

In article N...@centurytel.net, Fred Marshall at
f...@remove_the_x.acm.org wrote on 04/23/2004 13:56:

...

> Ah!  Sounds like a mixture of viewpoints....
> 
> One viewpoint sees the DFT as limited to fs.  This makes sense if we view
> the spectrum on a unit circle.  You can only get up to fs which is also
> zero.....  I think this is what Rick was using.
> 
> The other viewpoint sees the unit circle "unwrapped" to form an infinite
> line.  In this case the DFT is periodic at fs and extends to the infinities.
> I think this is what Tim was using.
> 
> Both are correct / useful.

i don't think they are equally correct.  (i agree with Tim, if i understand
the positions correctly.)

>  One for sampled data and one for continuous
> signals / functions.  Obviously, dealing with DACs requires our thinking to
> transcend between the two.

...

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.

r b-j