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

r.lyons@_BOGUS_ieee.org (Rick Lyons) wrote in message > 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-]

OK Rick. I know my English is terrible. Regarding the content, I can
see how one can get confused. First we filter, than we do sample and
hold. We learn that pre-filtering is the solution to aliasing, so we
think of it as removal of high frequencies folding into the base band.
True indeed. But the action of sampling itself makes for the high
frequency image energy. Now here is the "split point". I as a real
hand on engineer, have a lot of interest in the sample hold output
signal, and that signal contains a lot of high frequency energy. But
someone else may wish to view it as a series of numbers. That person,
the "numbers" person, will have to encounter that picture - the steps
and "staircases" at the DA side, a place where there is high frequncy
energy typicaly filtered by some anti imaging filter. I guess it is
not trevial to figure what happens first and what is last.

Regards
Dan Lavry

Jerry Avins wrote:
> 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:
>>
>>
>>
>> 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

The sampling process ends up being the mathematical equivalent of 
multiplying the signal by a train of dirac delta functions (impulses). 
In the frequency domain this has the effect of reproducing the frequency 
an infinite amount of times, with each copy centered on the sampling 
rate, so if the original signal x(t) has a frequency-domain 
representation X(w) then the sampled signal looks like:

X_s(w) = sum(X(w + w_s * n)) from x = -infinity to x = infinity, where 
w_s = 2*pi/T_s (radians/sec), T_s = sampling rate.

This means that if X(w) has a bandwidth bigger than 1/T_s (Hertz) parts 
of it will "fold over" the sampling frequency -- this is aliasing.  If 
you don't want aliasing you have to low-pass filter your baseband signal 
to 1/(2T_s) to account for the positive and negative frequency 
components when you translate things into mathemagic land.

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.

-- 

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

Tim Wescott 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
> 
> 
> The sampling process ends up being the mathematical equivalent of 
> multiplying the signal by a train of dirac delta functions (impulses). 
> In the frequency domain this has the effect of reproducing the frequency 
> an infinite amount of times, with each copy centered on the sampling 
> rate, so if the original signal x(t) has a frequency-domain 
> representation X(w) then the sampled signal looks like:
> 
> X_s(w) = sum(X(w + w_s * n)) from x = -infinity to x = infinity, where 
> w_s = 2*pi/T_s (radians/sec), T_s = sampling rate.

Quite.

> This means that if X(w) has a bandwidth bigger than 1/T_s (Hertz) parts 
> of it will "fold over" the sampling frequency -- this is aliasing.  If 
> you don't want aliasing you have to low-pass filter your baseband signal 
> to 1/(2T_s) to account for the positive and negative frequency 
> components when you translate things into mathemagic land.

Sure.

> 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.

That's all true, but it's not what the frequency plots seem to indicate.
They shoe the spectrum of the series of, if not Kroneker deltas, at
least very narrow pulses. The illustration seems to be not of aliasing
of out-of-band signals into the passband, but reflection and repetition
of the signal about the sampling frequency and multiples of it. As I see
it, anti-alias filtering has no connection with that.

Jerry

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

robert bristow-johnson wrote:


> 
> 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.
> 

Despite strong beliefs to the contrary, I swore off that 
arguement.  :-)


Bob
-- 

"Things should be described as simply as possible, but no 
simpler."

                                              A. Einstein

On Sat, 24 Apr 2004 17:39:38 -0700, Tim Wescott
<tim@wescottnospamdesign.com> wrote:

  (snipped)
>
>The sampling process ends up being the mathematical equivalent of 
>multiplying the signal by a train of dirac delta functions (impulses). 
>In the frequency domain this has the effect of reproducing the frequency 
>an infinite amount of times, with each copy centered on the sampling 
>rate, so if the original signal x(t) has a frequency-domain 
>representation X(w) then the sampled signal looks like:
>
>X_s(w) = sum(X(w + w_s * n)) from x = -infinity to x = infinity, where 
>w_s = 2*pi/T_s (radians/sec), T_s = sampling rate.
>
>This means that if X(w) has a bandwidth bigger than 1/T_s (Hertz) parts 
>of it will "fold over" the sampling frequency -- this is aliasing.  If 
>you don't want aliasing you have to low-pass filter your baseband signal 
>to 1/(2T_s) to account for the positive and negative frequency 
>components when you translate things into mathemagic land.
>

Hi,
  
   "fold over" is probably not the best phrase 
to use here.  "Fold over" is a phrase people use 
to describe the freq-domain characteristics of 
discrete signals (sequences).

And as I realized, Dan's discussion is strictly
about continuous (analog) signals.
Instead of "fold over", I'd call it "overlapped 
spectral replications".

I'm just being nit picky Tim.

[-Rick-]

 

Jerry Avins <jya@ieee.org> wrote in message news:<408b2d86$0$28910> > 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.
> 
> That's all true, but it's not what the frequency plots seem to indicate.
> They shoe the spectrum of the series of, if not Kroneker deltas, at
> least very narrow pulses. The illustration seems to be not of aliasing
> of out-of-band signals into the passband, but reflection and repetition
> of the signal about the sampling frequency and multiples of it. As I see
> it, anti-alias filtering has no connection with that.
> 
> Jerry

I agree with Jerry. In my paper, I just started with 4 inband (under
Nyquist) sine. Of corse I could have included an antialiasing filter,
but that is not where the high frequency source is. It is the sample
hold. One may choose to view it as numbers, but a "hardware guy" needs
to view it as a physical entity - a signal that can be measured with a
scope (time domain), a spectrum analyzer (frquency waves) and other
means. Say you take a 1KHz sine wave and do a sample hold operation at
44.1KHz. Each sampled 1KHz cycles is made out of 441 little steps. A
step is a "sharp edge", a "sharp corner" on a time domain plot, and
"those things" require a signal infinite bandwidth.
Of corse one does not get perfect square waves. The rise time of each
step is not vertical, it is usualy exponetial (capacitor charing or
discharging). The exponent does not ever start as an abrupt corner,
because real hardware does not have infinit bandwidth (it is sort of
rounded where the corner should be in theory). But the point is - the
real actual signal has high frequency content due to the sample hold
action.
Is it importent to make that point? It is if you design hardware or
measure hardware. The front end sample hold is a very imortant
circuit. This is where timing jitter has its bad impact. This is where
your cap value seems too small from leakage (holding charge) and feed
through (unwanted signals coupled in capacitivly) the point of view,
but raising the capacitance will prevent it from charging fully to the
new valu (under maximim slew rate conditions).
So a hardware guy can not view the final (charged capacitor) per
sample as a "number", or "sample value".
I guess a DSP person may choose to view it as a number. What is the
relationship between a series of numbers to frequncy content? Well,
the DSP code person may choose to "not think about it", but the DA
desiger will have to deal with real signals with high frequency
content, thus scopes, spectrum analyzers analog filters...

BR
Dan Lavry

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.
> 
> r b-j
> 
> 

Oh hell, I'll bite.

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_.

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_.

In fact, you can prove this mathematically:

Take a periodic signal and "window" it down to one period in time.  Now 
take it's Fourier transform -- you get a result that's continuous in 
frequency, and extends to infinite frequency in both directions.  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.

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.

-- 

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

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.

-- glen

Jerry Avins wrote:

> Tim Wescott 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
>>
>>
>>
>> The sampling process ends up being the mathematical equivalent of 
>> multiplying the signal by a train of dirac delta functions (impulses). 
>> In the frequency domain this has the effect of reproducing the 
>> frequency an infinite amount of times, with each copy centered on the 
>> sampling rate, so if the original signal x(t) has a frequency-domain 
>> representation X(w) then the sampled signal looks like:
>>
>> X_s(w) = sum(X(w + w_s * n)) from x = -infinity to x = infinity, where 
>> w_s = 2*pi/T_s (radians/sec), T_s = sampling rate.
> 
> 
> Quite.
> 
>> This means that if X(w) has a bandwidth bigger than 1/T_s (Hertz) 
>> parts of it will "fold over" the sampling frequency -- this is 
>> aliasing.  If you don't want aliasing you have to low-pass filter your 
>> baseband signal to 1/(2T_s) to account for the positive and negative 
>> frequency components when you translate things into mathemagic land.
> 
> 
> Sure.
> 
>> 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.
> 
> 
> That's all true, but it's not what the frequency plots seem to indicate.
> They shoe the spectrum of the series of, if not Kroneker deltas, at
> least very narrow pulses. The illustration seems to be not of aliasing
> of out-of-band signals into the passband, but reflection and repetition
> of the signal about the sampling frequency and multiples of it. As I see
> it, anti-alias filtering has no connection with that.
> 
> Jerry
> 

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.

-- 

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

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