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

robert bristow-johnson wrote:


>>
>>You really didn't understand that?  Granted it's concise but
>>I'm not sure how else to not state it.
> 
> 
> okay, so you make me parse it:
> 
> 
>>>>... you will absolutely not lure me back into saying
> 
> 
> the follow you will not say:

Ah, the intended humor is coming through a little then?

> 
> 
>>>>that the DFT merely compiles the correlations of an interval of a signal
> 
> 
> an "interval of a signal" is going to be a finite length segment of a
> signal, i presume by this time it has been sampled so that finite length is
> a finite number of samples, no?  we'll call that number, N.

Agreed.

> 
> 
>>>>with all the sinusoids and cosinusoids that have an integral number of
>>>>cycles in the same length interval (up to the Nyquist one)
> 
> 
> okay, i got that one (i presume that you're using the Nyquist one since that
> appears in bin N/2 of the DFT if N is even).  but you're not going over the
> Nyquist bin.

Yep.

> 
> 
>>>>and that it says nothing whatsoever about anything outside that interval.
> 
> 
> so you're refuse to say "that it says nothing whatsoever..." or that, if you
> would say it, "that it says nothing whatsoever about anything outside that
> interval."  *that's* a little ambiguous.  from the past, i think you believe
> that it *does* say something about something outside that interval.  no?

No.  It's totally ambiguous as to what might lie outside 
that interval.

> 
> if that *is* the case, we have a disagreement whereas i would say that
> neither the DFT nor those N samples tell you what is going on outside that
> interval.  you are welcome to say that it's, say, zero outside that interval
> (and then the consequence of that *assumption* is that the DFT is the
> sampling of N samples of the DTFT around the unit circle).  

An assumption I make and work with more than any other, but 
it is just one of the possible assumptions.

> i insist that
> it's an assumption that may be reasonable in some contexts, but neither
> those N samples nor the DFT of them give you that information.
> 
> 
>>>> I absolutely refuse to restate my position.  :-)
> 
> 
> let's see where this gets us, Bob.  i promise to be nice (i.e. respectful),
> but, as you might guess, i take no prisoners regarding the content of the
> discussion.

I accept your terms.  This argument that isn't happening can 
proceed.  Or not.  :-)


Bob
-- 

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

                                              A. Einstein

glen herrmannsfeldt wrote:

> 
> For DFT a cyclic rotation of the input points is a phase change in
> the result, but otherwise doesn't affect the result.
> 

It certainly does if that set of points contains a finite 
impulse response and you convolve that rotation with 
something else.


Bob
-- 

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

                                              A. Einstein

In article <H6Ujc.7709$lz5.854606@attbi_s53>,
glen herrmannsfeldt  <gah@ugcs.caltech.edu> wrote:
>The use of the DFT instead of the continuous Fourier transform
>implies the assumption that the function is periodic, and so is
>its transform.

The DFT is merely a transform.  Nothing about periodicity needs to be
assumed about the input function to do a transform.  You can DFT a
pile of random numbers if you want.

However, if you wish to interpret the DFT results as if the input was
periodic, then one should assume the input was periodic for the purposes
of that interpretation of the result.  That's normally what happens when
you assume the result says something about non-windowed frequency content.

>If the function is not periodic, then you should not use a DFT.

Nonsense.  I use DFT's for data compression of non-periodic images
for instance.  Lot's of other uses...


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

In article c6t6go$ljf$1@blue.rahul.net, Ronald H. Nicholson Jr. at
rhn@mauve.rahul.net wrote on 04/30/2004 05:33:

> In article <H6Ujc.7709$lz5.854606@attbi_s53>,
> glen herrmannsfeldt  <gah@ugcs.caltech.edu> wrote:
>> The use of the DFT instead of the continuous Fourier transform
>> implies the assumption that the function is periodic, and so is
>> its transform.
> 
> The DFT is merely a transform.  Nothing about periodicity needs to be
> assumed about the input function to do a transform.

nothing needs be assumed by the *user*.  in fact, the user might have
additional knowledge that these N samples were yanked out of some other
non-periodic sequence.  however (anthropomorphizing), the DFT "assumes" that
those N numbers are one period of an infinite periodic sequence.  the basis
functions that you are fitting to the input are all periodic.

>  You can DFT a pile of random numbers if you want.

and that pile of (i.e. a finite set of) random numbers would be periodically
extended to infinity in both directions by the DFT.

r b-j

In article c6sdf10rdb@enews4.newsguy.com, Bob Cain at
arcane@arcanemethods.com wrote on 04/29/2004 22:25:

> robert bristow-johnson wrote:
>

>> 
>>>>> I absolutely refuse to restate my position.  :-)
>> 
>> 
>> let's see where this gets us, Bob.  i promise to be nice (i.e. respectful),
>> but, as you might guess, i take no prisoners regarding the content of the
>> discussion.
> 
> I accept your terms.  This argument that isn't happening can
> proceed.  Or not.  :-)

well, i'm preparing for a weekend at a camp in upstate NY (downstate for
me).  let's not get back to this argument that isn't happening when i get
back unless you want to not toss the first volley.  (that might be nice, you
can put forth a list of reasons that the DFT is not inherently periodic, but
if you don't wanna, i'll make my list Sunday night.  i plan to google the
thread where were 2 years ago (or whenever) to see where there were loose
ends.)

L8r,

r b-j

robert bristow-johnson wrote:


>> You can DFT a pile of random numbers if you want.
> 
> 
> and that pile of (i.e. a finite set of) random numbers would be periodically
> extended to infinity in both directions by the DFT.
> 

And this is the crux of our disagreement.  You get the 
original result if you assume that all the sinusoids and 
cosinusoids involved in the inner products of the DFT (the 
basis functions) are tone bursts that only last for the 
duration of the interval.  Under that assumption you can 
perform the DFT integral from negative to positive infinity 
and the inverse will give back the original signal interval 
with it extended by zeros in both directions outside the 
interval.

This is how I think about impulse response transforms.


Bob
-- 

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

                                              A. Einstein

robert bristow-johnson wrote:


> well, i'm preparing for a weekend at a camp in upstate NY (downstate for
> me).  let's not get back to this argument that isn't happening when i get
> back unless you want to not toss the first volley.  (that might be nice, you
> can put forth a list of reasons that the DFT is not inherently periodic, but
> if you don't wanna, i'll make my list Sunday night.  i plan to google the
> thread where were 2 years ago (or whenever) to see where there were loose
> ends.)
> 

I think I've just done part of that in my response to 
another post of yours in this thread.

You once challenged me with the periodicity (or wrap) of 
convolution by frequency domain multiplication and that 
stumped me.  I now wonder if that isn't a property of the 
convolution theorem rather than a property of the DFT but 
haven't a proof of that theorem in front of me to check out 
its assumptions.

In any event, I see that property more as a kind of aliasing 
problem wherein the length of the container is insufficient 
to contain the length of the result so it mathematically 
wraps.  If one merely extends the original sequences with 
zeros and then does the FFT, multiply, IFFT then one has 
satisfied the constraint that the container be long enough 
to contain the result and no periodicity is evident.

This is the underlying basis of the overlap add/save 
algorithms for convolving and if you treat them as black 
boxes mathematically then no periodicity is evident anywhere.

I believe that this periodicity assumption is the root cause 
of considerable confusion among those getting on board all 
this stuff and should be purged.

I look forward to your return.


Bob
-- 

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

                                              A. Einstein

Bob Cain <arcane@arcanemethods.com> writes:

> robert bristow-johnson wrote:
> 
> 
> >> You can DFT a pile of random numbers if you want.
> > and that pile of (i.e. a finite set of) random numbers would be
> > periodically
> 
> > extended to infinity in both directions by the DFT.
> >
> 
> 
> And this is the crux of our disagreement.  You get the original result
> if you assume that all the sinusoids and cosinusoids involved in the
> inner products of the DFT (the basis functions) are tone bursts that
> only last for the duration of the interval.  Under that assumption you
> can perform the DFT integral from negative to positive infinity and
> the inverse will give back the original signal interval with it
> extended by zeros in both directions outside the interval.
> 
> 
> This is how I think about impulse response transforms.

Bob,

You're wrong. I don't mean to be disrespectul, and Lord knows
I not tactful, but you are flat-out wrong. Here's why.

The DFT, BY DEFINITION, produces a *FINITE* set of values as output.
These are, BY DEFINITION, frequency-domain output samples. It is
an absolute IRREFUTABLE property that discrete-time samples in the
frequency REQUIRE a periodic time-domain function. Period. End of
argument.
-- 
Randy Yates
Sony Ericsson Mobile Communications
Research Triangle Park, NC, USA
randy.yates@sonyericsson.com, 919-472-1124

In article <BCB80384.AD77%rbj@surfglobal.net>,
robert bristow-johnson  <rbj@surfglobal.net> wrote:
>>  You can DFT a pile of random numbers if you want.
>
>and that pile of (i.e. a finite set of) random numbers would be
>periodically extended to infinity in both directions by the DFT.

DFT's cannot alter your input data, periodic, infinite or not.  Only your
possibly false assumptions about the meaning of particular DFT results
can do that.

One can assume the DFT results to be many other things other than
orthonormal functions of infinite extent (statistical pattern data for
example; a neural-net classifier "knows" nothing about sin/cos functions).


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

In article <xxpu0z1mgfo.fsf@usrts005.corpusers.net>,
Randy Yates  <randy.yates@sonyericsson.com> wrote:
>The DFT, BY DEFINITION, produces a *FINITE* set of values as output.
>These are, BY DEFINITION, frequency-domain output samples. 

Only if you make certain assumptions about the input.  You've gone
in a complete circle.

A DFT is only an operator which munges one finite set of bits into
another.  


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