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

On Sun, 25 Apr 2004 13:26:00 -0700, Tim Wescott
<tim@wescottnospamdesign.com> wrote:


   (snipped)

Hi Tim,
  Ya' know, I read, often, in the literature of 
DSP that "the DFT summes this about its input", 
or "the DFT assumes that about its input".  
This is all pure nonsense.  The DFT is not alive, 
it has no brain with which to makes assumptions. 
 The DFT has no eyes with which to "view" 
anything.  The DFT is a math algorithm.  The 
question is: "What does the DFT of a time-dmain 
sequence represent?"

Oppenheim & Schafer spend many pages trying to 
explain their answer to that question.  (Even Opp 
& Schafer inappropriately say "The FFT assumes" 
something or another somewhere in their book.
I tried to find that passage again, yesterday, 
but failed.) 

The DFT is a "sampled" version of the continuous 
Fourier transform of a continuous periodic function 
of impulses of varying amplitudes.

[-Rick-] 

 

In article 408da234.1556213937@news.sf.sbcglobal.net, Rick Lyons at
r.lyons@_BOGUS_ieee.org wrote on 04/26/2004 20:02:

> Ya' know, I read, often, in the literature of
> DSP that "the DFT summes this about its input",
> or "the DFT assumes that about its input".
> This is all pure nonsense.  The DFT is not alive,
> it has no brain with which to makes assumptions.
> The DFT has no eyes with which to "view"
> anything.  The DFT is a math algorithm.

oh, c'mon Rick.  there's a time honored tradition of anthropomorphizing
everything from animals or insects to inanimate objects like temperamental
cars or even genetics or diseases.

>  The 
> question is: "What does the DFT of a time-dmain
> sequence represent?"

sure.  and a useful tool for explaining that is to anthropomorphize it a
bit.

> Oppenheim & Schafer spend many pages trying to
> explain their answer to that question.  (Even Opp
> & Schafer inappropriately say "The FFT assumes"
> something or another somewhere in their book.
> I tried to find that passage again, yesterday,
> but failed.) 

i guess i commit the same infraction.  i remember that you could never put
personal pronouns in a technical or scholarly paper.  not even "we", as is
often done in a proof or explanation.  now look at it.

> The DFT is a "sampled" version of the continuous
> Fourier transform of a continuous periodic function
> of impulses of varying amplitudes.

(if it's a periodic function of impulses in the "time" domain, then they are
already impulses in the frequency domain and "sampling" them might be
problematic.)

it sends a set of N (possibly complex) numbers to another set of N possibly
complex numbers.  why must there be *any* concept of continuous Fourier
Transform or (dirac) impulses in that?  how about:

The DFT is a particular linear mapping of an infinite and periodic sequence
of numbers (with period N) to another infinite and periodic sequence of
numbers of the same period defined as:

    (familiar summations)

and has these properties:

    (familiar properties of DFT)


it is perfectly legit to relate that transform to the DTFT which transforms
a possibly non-periodic sequence of numbers to a continuous (and periodic)
spectrum.  then you can show that sampling that spectrum causes an
overlapping periodic extension in the original domain.

whatever,

r b-j

On Mon, 26 Apr 2004 14:03:42 -0700, "Jon Harris"
<goldentully@hotmail.com> wrote:

>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

Hi Jon,

   I think you are correct.  I was too "heavy handed" 
when I submitted my post and used the word "Caution".

The two most incorrect DSP papers I've ever read 
were on the topic of periodic sampling.
I mean wildly incorrect!  So I jumped to a 
conclusion that Dan had fallen into the same 
misunderstanding of the spectral characteristics of 
discrete samples.  I thought Dan was talking about 
discrete sequences when he was really talking about 
analog signals.

I realize my original post upset Dan, and I tried to 
let him know that I wasn't criticizing his 
enginnering skills.  I tried to make him feel 
a little better, but I don't think I was successful.

When he used the words "sampled signal" and 
sampled values", I thought discrete, not analog.
So I misunderstood what he was saying.
I thought, "Oh darn, we should be careful" (cautious) 
when we read his article.

Tim, how about I pass my intended posts here through 
you first, for your review, so that you can keep me 
out of trouble?  :-)
Ya' know, that's not a bad idea at all.
I like that idea. 

[-Rick-]

On Mon, 26 Apr 2004 08:57:04 +0000 (UTC), rhn@mauve.rahul.net (Ronald
H. Nicholson Jr.) wrote:

>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.

Hi,

  your word "disambiguate" tickled the heck 
out of me.  Good goin'.  I like it.

[-Rick-]

Rick Lyons wrote:

> On Sun, 25 Apr 2004 13:26:00 -0700, Tim Wescott
> <tim@wescottnospamdesign.com> wrote:
> 
> 
>    (snipped)
> 
> Hi Tim,
>   Ya' know, I read, often, in the literature of 
> DSP that "the DFT summes this about its input", 
> or "the DFT assumes that about its input".  
> This is all pure nonsense.  The DFT is not alive, 
> it has no brain with which to makes assumptions. 
>  The DFT has no eyes with which to "view" 
> anything.  The DFT is a math algorithm.  The 
> question is: "What does the DFT of a time-dmain 
> sequence represent?"

Presumably one should say "One assumes 'x' about the input when one 
takes a DFT", or "This interpretation of the DFT assumes 'x' about it's 
input" or some such.  Did I imply otherwise?  I reviewed my post & 
didn't see.  While I _do_ find that anthromorphizing systems to be 
useful (particularly with cranky software, but even lawnmowers come 
under the gun sometimes), I've never extended that to algorithms other 
than to curse at them.

> Oppenheim & Schafer spend many pages trying to 
> explain their answer to that question.  (Even Opp 
> & Schafer inappropriately say "The FFT assumes" 
> something or another somewhere in their book.
> I tried to find that passage again, yesterday, 
> but failed.) 

I've got that book at my elbow right now for a different reason.  I'll 
have to see what they say (and maybe find the "assumes" part).

> The DFT is a "sampled" version of the continuous 
> Fourier transform of a continuous periodic function 
> of impulses of varying amplitudes.

Or perhaps the DFT is a discrete analog of the continuous Fourier 
transform that is most often used with sampled data?  Certainly if you 
take the Fourier transform of a signal that's been sampled by 
multiplying it by an impulse train you get a DFT with a thin wrapper of 
singularities, but there's got to be applications of it that never 
involve sampling (don't ask me what; in spite of accusations to the 
contrary I deal with real-world systems and don't get excited by pretty 
math unless it means something).

> 
> [-Rick-] 
> 
>  


-- 

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

In article ih1jc.37211$IW1.1714703@attbi_s52, glen herrmannsfeldt at
gah@ugcs.caltech.edu wrote on 04/26/2004 01:40:

> 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.

sure it does!

> The sampled signal is a representation of the continuous signal.

it could.  but it wouldn't have to.  once it's sampled, that information is
gone forever.  (at least it is not contained in the retained samples.)

> It could, theoretically, be converted back to a continuous signal as a
> sum of sinc's.

yeah, and those sinc's all have a brickwall filter in the other (frequency)
domain which kills off all of those repeated images.  applying those sinc's
for reconstruction means that you are making an assumption (probably a legit
assumption) about what the continuous signal was before sampling it.

>   Other reconstruction methods generate extraneous
> frequency components which need to be filtered out.

that is if it is already known (or assumed) that the original continuous
signal had no energy where those extraneous components ended up.  what if my
original continuous signal was a piecewise linear with breakpoints exactly
at the sampling instances.  then the correct reconstruction method would use
triangular pulses instead of sinc's.  and those "extraneous" frequency
components belong there.  but i am making an assumption about where those
samples came from in the first place in doing that reconstruction.

r b-j

robert bristow-johnson wrote:

>  a few very respectable folks on comp.dsp (like R. Cain and A.
> Hey, IIRC) have staunchly disagreed with one thing or 'nother.
> 

Robert, you will absolutely not lure me back into saying 
that the DFT merely compiles the correlations of an interval 
of a signal with all the sinusoids and cosinusoids that have 
an integral number of cycles in the same length interval (up 
to the Nyquist one) and that it says nothing whatsoever 
about anything outside that interval.  I absolutely refuse 
to restate my position.  :-)


Bob
-- 

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

                                              A. Einstein

Interesting.  The article appears to be about a point in the
implementation of the sampling process that isn't often discussed.
First a signal (S0) has to be low-pass filtered to removed frequency
components with the potential to alias (S1).  Then it seems for some
implementations one then has to add a lot of high frequency content.
This addition is due to the need to "flatten" the signal so that it
stays sufficiently constant during a finite A/D measurement period
(creating sample-and-hold "stair steps" for signal S2).  Of course
flattening portions of a continuous signal implies steepening other
portions which introduced high frequency content to this intermediate
signal.  It's sort of counter-intuitive to consider that the more high
frequency content introduced during this part of the process the less
this high frequency content is represented in the final samples (as in
the resulting sequence of numbers, S4).

------ Original Message ------
In article <673b149b.0404251141.3b01b925@posting.google.com>,
dan lavry <danlavry@mindspring.com> wrote:
>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


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

In article c6kjod01af0@enews4.newsguy.com, Bob Cain at
arcane@arcanemethods.com wrote on 04/26/2004 23:23:

> 
> 
> robert bristow-johnson wrote:
> 
>> a few very respectable folks on comp.dsp (like R. Cain and A.
>> Hey, IIRC) have staunchly disagreed with one thing or 'nother.
>> 
> 
> Robert, you will absolutely not lure me back into saying
> that the DFT merely compiles the correlations of an interval
> of a signal with all the sinusoids and cosinusoids that have
> an integral number of cycles in the same length interval (up
> to the Nyquist one) and that it says nothing whatsoever
> about anything outside that interval.  I absolutely refuse
> to restate my position.  :-)

okay in the same spirit, i absolutely refuse to decode it either.


???


r b-j

r.lyons@_BOGUS_ieee.org (Rick Lyons) wrote in message news:<408da855.1557782953@news.sf.sbcglobal.net>...
> On Mon, 26 Apr 2004 08:57:04 +0000 (UTC), rhn@mauve.rahul.net (Ronald
> H. Nicholson Jr.) wrote:
> 
> >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.
> 
> Hi,
> 
>   your word "disambiguate" tickled the heck 
> out of me.  Good goin'.  I like it.

What amused you? Its a pretty standard term.

In a lot of defence systems (e.g. multiple PRF radars) aliases are
embraced, and the real thing you are looking for (true range, true
resolved velocity, or whatever) is resolved between the aliases at
multiple sampling rates. This process is often called disambiguating.
It is often called correlation, too, but that gets confusing if the
work involves any mathematical correlation.

Regards,
Steve