glen herrmannsfeldt wrote:

   ...

> A glass of water contains many individual water molecules, so
> shouldn't water be plural?

In many languages, it is. I know little Hebrew, but water in Hebrew is 
"mayim"; the plural form of some word I suspect most Israelis have never 
uttered.

One goes to a one-spring spa to "take the waters". You can probably find 
other vestigial English usages of the plural.

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

Clay wrote:

(snip)

>>It's data are, not data is.  Data is plural.  Datum is singular.  I would
>>have thought that someone as anal retentive as yourself would have known
>>that.

> One difference between American and Imperial English is the treatment
> of collective nouns. We know the "data" is plural, however the use of
> the word is almost always as a collective noun and in American English
> (spoken where RBJ lives), collective nouns are singular. Most of us
> native English speakers understand this and accept this difference
> without the need to try to correct someone on this usage.

I thought this was officially changed.  Consider, though, that in
bit terms data is plural, unless you only have one bit.

A glass of water contains many individual water molecules, so
shouldn't water be plural?

-- glen

Rune Allnor wrote:

(snip)

>>Rune, the DFT doesn't know or care what the "true" data is.  it
>>crunches the numbers the same in any case.  it does not know if the N
>>samples you pass it were N samples of some aperiodic sequence or if
>>those happened to be N samples defining exactly one period of a
>>periodic sequence.  it deals with it the same in either case and (this
>>is what is inexplicably controversial here sometimes) assumes it is the
>>latter.

The question was when is it an approximation.  It is an approximation
if the "true" data isn't periodic with the appropriate period.
I might even say that it isn't an approximation in that case, it
might not even be close.


(snip)

> I disagree with this as being a fundamental property of the DFT.
> I agree that this is a side effect when using the DFT to process
> finite samples of infinite sequences.

In comparing DFT, DST, and DCT the periodic boundary conditions
are fundamental to DFT.  DCT has the boundary condition that the
derivative goes to zero at the end.  DST has the function value
go to zero at the end.  As usual for standing waves, for the DST
and DCT the result has a period twice as long as the transform.

(snip)

> I find it ridiculous to claim that any finite sub sequence of
> an infinite series happens to represent exactly one period
 > of the infinite series.

If the infinite series gets small enough after a while, it
shouldn't be too bad.  In the discussion on bandwidth I was
remembering that an FM signal spectrum has infinite width,
but that doesn't stop placing FM radio stations 200kHz apart.

Most infinite signals aren't very interesting over a
finite amount of time.

-- glen

Ghostie wrote:
> "robert bristow-johnson" <rbj@audioimagination.com> wrote in
> news:1154406448.806363.135510@b28g2000cwb.googlegroups.com:
>
> snip....snip
>
> > Rune, the DFT doesn't know or care what the "true" data is.  it
>
> ship....snip
>
>
> It's data are, not data is.  Data is plural.  Datum is singular.  I would
> have thought that someone as anal retentive as yourself would have known
> that.

One difference between American and Imperial English is the treatment
of collective nouns. We know the "data" is plural, however the use of
the word is almost always as a collective noun and in American English
(spoken where RBJ lives), collective nouns are singular. Most of us
native English speakers understand this and accept this difference
without the need to try to correct someone on this usage.

Clay

Ron N. wrote:

   ...

> It's actually the other way around.  The probability that an aperature
> on a random arbitrary function just happens to be an integer period
> window on a periodic or circular function looks close to zero.  So the
> periodic interpretation might be the exceptional one which requires
> information.

But the DFT doesn't "know" that. Given a set of samples, its output will 
be precisely the same whatever the periodicity of the samples data. 
Since the basis functions into which the sample set is decomposed are 
periodic and related by integers, their sum is periodic as well.

We may use our engineering smarts to apply the results only where they 
make sense, but periodicity is the beginning, and pruning comes later. 
You might call it fudging the result in order to make it consonant with 
reality.

   ...

> If I use a hammer to stir pasta, it's cleanliness and coefficient
> of drag in viscous fluids are probably more relevant than it's swing
> weight and impact resonance.

:-)

You also need to worry that strands of spaghetti don't get wedged in the 
claw. Or had you a ball-peen hammer in mind?

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

robert bristow-johnson wrote:

> say you have a sinusoid that completes precisely 8.5 cycles in the time
> of the N sample aperture.
>
> if you look at it as a sinusoid as a sinusoid of frequency of 8.5/N
> cycles/sample (which is an interpretation and what i meant by if you
> had informal or ad hoc information), then you see a big peak at bins 8
> and 9 (and bins N-9 and N-8) with sinc-like tails coming off of it and
> wrapping around at the boundary between x[N-1] and x[0].  circular
> extension.
>
> or you can look at it another way and say that you have a periodic
> sequence in which in one period it makes 8.5 cycles of a sinusoid, does
> a 180 deg phase jump and continues with another 8.5 sin waves in the
> next period.  then bins x[1] and x[N-1] have the fundamental in it, x[2]
> x[N-2] has the 2nd harmonic, etc.

made a mistake.  meant to say "then bins X[1] and X[N-1] have the
fundamental in it, X[2] and X[N-2] have the 2nd harmonic, etc."

> the data is exactly the same and the DFT output will be exactly the
> same, but it's interpreted differently.  now, given a priori
> information, you might be able to look at the data and surmise (or
> interpret) that it is not likely the latter case, but the former.
> that's a human interpretation.  there is no numerical reason to say it
> is the first vs. the second situation.
>
> two different human interpretations.  doesn't matter to the DFT.
> multiply the DFT output, X[k], by exp(-j*2*pi*m*k/N) and IDFT, and
> you'll get x[(n-m)mod_N] if you don't periodically extend it (but that
> mod operator essentially did the periodic extension for you) or the
> more simple and natural x[n-m] if you do periodically extend it.
>
> still, in any case, it's periodically extended.

that's still the point i stand on, typo nonwithstanding.

r b-j

Ron N. wrote:
> robert bristow-johnson wrote:
> > the DFT *always*, always, always, always, always periodically extends
> > the data that is passed to it, and if you don't keep that in mind, you
> > can get burned.
>
> Funny.  Last DFT I calculated fit in finite memory.

non-sequitur.  but if we do change the subject to that, earlier i said:
"the DFT inherently periodically extends whatever data you give it
(i.e. it assumes the N samples passed to it are representative of
exactly one period)."  it fits in finite memory because you only need
one cycle (N samples) to fully describe a periodic function.

>  Neither did it change the incoming audio sample stream.

windowing off the N samples (buy yanking it out of the audio stream) is
the first thing that changed something (before the DFT even gets it)
and then passing those N samples to the DFT is what periodically
extends them in both directions.  when you windowed off the N samples,
x[n] is undefined outside of [0 .. N) but when those N samples are
passed to the DFT, at that point x[n+N] = x[n] for all n.

>
> > that interpretation (periodic or circular extension) is the only one
> > that makes sense in general.  there are times when you may not need it,
> > but i can't think of any where it doesn't have some effect.  even the
> > wrapping of those sinc-like tales in the spectrum do to windowing in
> > the time domain.
>
> There can be no sinc-like tails if the function is periodic in the
> aperature, since under that assumption, only bin frequencies can
> exist.  Sinc-like tails can only wrap around if you postulate the
> existance of non-bin frequencies, which is inconsistant with the
> assumption that the results must be interpreted as if the data
> were periodically extended.

my point was that either way you look at it, you "virtually" always
have to consider the circular nature of the DFT.

say you have a sinusoid that completes precisely 8.5 cycles in the time
of the N sample aperture.

if you look at it as a sinusoid as a sinusoid of frequency of 8.5/N
cycles/sample (which is an interpretation and what i meant by if you
had informal or ad hoc information), then you see a big peak at bins 8
and 9 (and bins N-9 and N-8) with sinc-like tails coming off of it and
wrapping around at the boundary between x[N-1] and x[0].  circular
extension.

or you can look at it another way and say that you have a periodic
sequence in which in one period it makes 8.5 cycles of a sinusoid, does
a 180 phase jump and continues with another 8.5 sin waves in the next
period.  then bins x[1] and x[N-1] have the fundamental in it, x[2]
x[N-2] has the 2nd harmonic, etc.

the data is exactly the same and the DFT output will be exactly the
same, but it's interpreted differently.  now, given a priori
information, you might be able to look at the data and surmise (or
interpret) that it is not likely the latter case, but the former.
that's a human interpretation.  there is no numerical reason to say it
is the first vs. the second situation.

two different human interpretations.  doesn't matter to the DFT.
multiply the DFT output, X[k], by exp(-j*2*pi*m*k/N) and IDFT, and
you'll get x[(n-m)mod_N] if you don't periodically extend it (but that
mod operator essentially did the periodic extension for you) or the
more simple and natural x[n-m] if you do periodically extend it.

still, in any case, it's periodically extended.

> In normal use, the data or function outside any finite DFT can
> be either periodic or not.

sure, outside of the DFT the data can be anything, but once you window
some of it off (which has the windowing effects in the spectrum) and
pass it to the DFT, it treats it as periodic with period N.  if it does
it "sometimes" it does it always, because the DFT doesn't know the
difference.

>  Thus the results, interpreted as some
> sort of spectrum, are often underconstrained

that's the first interpretation above.  doesn't matter.  it still was
periodically extended and that will show when you apply shifting.

r b-j

robert bristow-johnson wrote:
> the DFT *always*, always, always, always, always periodically extends
> the data that is passed to it, and if you don't keep that in mind, you
> can get burned.

Funny.  Last DFT I calculated fit in finite memory.  Neither did it
change the incoming audio sample stream.

> that interpretation (periodic or circular extension) is the only one
> that makes sense in general.  there are times when you may not need it,
> but i can't think of any where it doesn't have some effect.  even the
> wrapping of those sinc-like tales in the spectrum do to windowing in
> the time domain.

There can be no sinc-like tails if the function is periodic in the
aperature, since under that assumption, only bin frequencies can
exist.  Sinc-like tails can only wrap around if you postulate the
existance of non-bin frequencies, which is inconsistant with the
assumption that the results must be interpreted as if the data
were periodically extended.

In normal use, the data or function outside any finite DFT can
be either periodic or not.  Thus the results, interpreted as some
sort of spectrum, are often underconstrained (except in many
special cases, such as your synchronous waveform synthesizer).


IMHO. YMMV.
-- 
rhn A.T nicholson d.0.t C-o-M

Ron N. wrote:
> robert bristow-johnson wrote:
> > Ron N. wrote:
> > >
> > > People interpreting the results are the ones making assumptions,
> > > not the DFT operator.  For many purposes, the assumption of
> > > periodicity is obviously false,
> >
> > if you have more information (even informal or ad hoc information) that
> > says that the input data was not likely periodic with period N, fine.
> > if you have no such information, then how can you be sure?
>
> It's actually the other way around.  The probability that an aperature
> on a random arbitrary function just happens to be an integer period
> window on a periodic or circular function looks close to zero.  So the
> periodic interpretation might be the exceptional one which requires
> information.

well that depends on what you're doing.  the OP seems to think that he
can time scale the input so that a period is whatever length he wants.

you and i have both done stuff in audio/music.  in wavetable synthesis
(very similar to NCO or DCO or DDS or whatever), the wavetable
represents precisely one cycle of the waveform.  pass that to your FFT
and you have your harmonic coefficients wrapped in a nice package (no
need to do peak picking, etc.).  not all use of the FFT is "spectral
analysis" in that sense.

> > > and thus will produce incorrect interpretations of the those DFT results.
> >
> > the difference between a correct and incorrect interpretation is purely
> > dependent on what the assumptions we make about what the input looked
> > like outside of the segment of samples passed to the DFT.
> >
> > > The math using inifnite sinusoids would work equally well with
> > > truncated sinusoids for many of the properties of the FFT/DFT,
> > > although not for others.
> >
> > yet the math using infinite sinusoids works correctly with *every* one
> > of the properties of the DFT.  so why use the truncated sinusoid model
> > when it doesn't work with everything (namely shifting and convolution)?
> >
> > >  One only needs to extend the truncated
> > > sinusoids for uses which need those properties.
> >
> > why bother to have the truncated sinusoids in the first place that you
> > may need to extend?
>
> I agree.  If you need to extend them, then you shouldn't use them
> in that particular analysis or operation.

again, my point is that you cannot find a situation where that periodic
extension gives you the wrong answer, but i can find plenty
(multiplying one domain by anything that results in shifting or
convolution in the other) where leaving the periodic extension out
*does* create an error.

> > for some reason, the message i get from you, Ron, is that the truncated
> > sinusoid model or definition is the "natural" one and that this is the
> > one to use until you come upon a need to periodically extend it (which
> > is what you would need to do for shifting and convolution).
>
> Not at all.  The natural definition is the one which fits the
> assumptions about the data and the analysis needed.
> If you need to do shifting or circular convolution on data
> which might be periodic, then one definition is most useful.
> If it is highly improbable that the data is periodic, then
> other assumptions about the basis might require less "undoing"
> of properties which are inconsistant with the analysis desired.

i disagree.  even if the N samples are drawn from something that is not
periodic (or periodic with period N), this periodic extension happens
anyway, whether you like it or not.  doesn't matter where you draw if
from, if after the DFT you multiply the data by something and IDFT, you
will get circular convolution as if the data was originally from a
periodic sequence of period N.  if you don't like it, then you need to
deal with it and put in a work around (which is precisely what the
overlap-add and overlap-save schemes to do fast convolution are about).

that's the problem:  people decide that their input data is not
periodic (that's fine) and then they send it to an FFT thinking that it
remains aperiodic (not fine) and start complaining when they see or
hear glitches in the output because their impulse response straddled
the boundary between x[N-1] and x[0].

the DFT *always*, always, always, always, always periodically extends
the data that is passed to it, and if you don't keep that in mind, you
can get burned.

> > where i
> > (and Oppenhiem and Schafer and some other texts) are trying to tell you
> > that there is no operational difference between the DFT and the
> > Discrete Fourier Series and that the natural definition is the one with
> > infinite length sinusoids and you never need to "extend" that model
> > because it is always valid.
>
> The natural definition makes sense only if the interpretation that
> the underlying data is periodic or circular also makes sense.

that interpretation (periodic or circular extension) is the only one
that makes sense in general.  there are times when you may not need it,
but i can't think of any where it doesn't have some effect.  even the
wrapping of those sinc-like tales in the spectrum do to windowing in
the time domain.  there is virtually always some place where you have
to  keep it in mind.

> The DFT can be used that way (for convolution, etc.), but is
> often not (usually for lack of a more appropriate tool anywhere
> near as computationally efficient as the FFT), especially for
> non-synchronous spectral analysis.

i don't get what you're saying.  besides speed (and numerical round-off
errors), what's the difference between the DFT and FFT?

> If I use a hammer to stir pasta, it's cleanliness and coefficient
> of drag in viscous fluids are probably more relevant than it's swing
> weight and impact resonance.

i don't get the analogy.

r b-j

robert bristow-johnson wrote:
> Ron N. wrote:
> > robert bristow-johnson wrote:
> > > the DFT doesn't know or care what the "true" data is.  it
> > > crunches the numbers the same in any case.  it does not know if the N
> > > samples you pass it were N samples of some aperiodic sequence or if
> > > those happened to be N samples defining exactly one period of a
> > > periodic sequence.
> >
> > The above contradicts the following:
> >
> > > it deals with it the same in either case and (this
> > > is what is inexplicably controversial here sometimes) assumes it is the
> > > latter.
> > ...
> > > when you pass those samples to the DFT, it inherently periodically
> > > extends that.
> >
> > If the DFT inherently does something, then it does know or care.
> > But it doesn't.
>
> the one ("it does know or care") does not follow the other ("the DFT
> inherently does something").  large planets in space inherently become
> spherical, but i doubt they have a consciousness that knows are cares
> about it.  some inanimate things in reality have inherent properties.
> i am saying (as does the math) that the DFT has an inherent property of
> periodically extending the data passed to it.
>
> > > i know Rick and other often object to anthropomorphizing
> > > the DFT,
>
> i do that, BTW, for the same reason some computer
> scientists/programmers do when they describe how a complicated
> algorithm might work.  do those Expert Systems with Artificial
> Intelligence really think?  can we sometimes describe their operation
> with anthropomorphisms?
>
> > Because the DFT doesn't make assumptions.  It just operates
> > on data.  You could feed it periodic data, infinitely non-periodic
> > data, or ASCII poetry, and it would still transform a segment of
> > that data into a different basis vector space.
>
> what are the basis functions that are being fit to the input data?  the
> DFT is, by its very definition, fitting a bunch of discrete and
> periodic basis functions to the ordered set of data passed to it.  to
> anthropomorphize again, it is saying "what linear combination of these
> sinusoids fit this input data?"  all of those sinusoids are inherently
> periodic with period N.  the linear combination of those sinusoids are
> also.
>
> >
> > People interpreting the results are the ones making assumptions,
> > not the DFT operator.  For many purposes, the assumption of
> > periodicity is obviously false,
>
> if you have more information (even informal or ad hoc information) that
> says that the input data was not likely periodic with period N, fine.
> if you have no such information, then how can you be sure?

It's actually the other way around.  The probability that an aperature
on a random arbitrary function just happens to be an integer period
window on a periodic or circular function looks close to zero.  So the
periodic interpretation might be the exceptional one which requires
information.

> > and thus will produce incorrect
> > interpretations of the those DFT results.
>
> the difference between a correct and incorrect interpretation is purely
> dependent on what the assumptions we make about what the input looked
> like outside of the segment of samples passed to the DFT.
>
> > The math using inifnite sinusoids would work equally well with
> > truncated sinusoids for many of the properties of the FFT/DFT,
> > although not for others.
>
> yet the math using infinite sinusoids works correctly with *every* one
> of the properties of the DFT.  so why use the truncated sinusoid model
> when it doesn't work with everything (namely shifting and convolution)?
>
> >  One only needs to extend the truncated
> > sinusoids for uses which need those properties.
>
> why bother to have the truncated sinusoids in the first place that you
> may need to extend?

I agree.  If you need to extend them, then you shouldn't use them
in that particular analysis or operation.

> for some reason, the message i get from you, Ron, is that the truncated
> sinusoid model or definition is the "natural" one and that this is the
> one to use until you come upon a need to periodically extend it (which
> is what you would need to do for shifting and convolution).

Not at all.  The natural definition is the one which fits the
assumptions about the data and the analysis needed.
If you need to do shifting or circular convolution on data
which might be periodic, then one definition is most useful.
If it is highly improbable that the data is periodic, then
other assumptions about the basis might require less "undoing"
of properties which are inconsistant with the analysis desired.

> where i
> (and Oppenhiem and Schafer and some other texts) are trying to tell you
> that there is no operational difference between the DFT and the
> Discrete Fourier Series and that the natural definition is the one with
> infinite length sinusoids and you never need to "extend" that model
> because it is always valid.

The natural definition makes sense only if the interpretation that
the underlying data is periodic or circular also makes sense.
The DFT can be used that way (for convolution, etc.), but is
often not (usually for lack of a more appropriate tool anywhere
near as computationally efficient as the FFT), especially for
non-synchronous spectral analysis.

If I use a hammer to stir pasta, it's cleanliness and coefficient
of drag in viscous fluids are probably more relevant than it's swing
weight and impact resonance.


IMHO. YMMV.
-- 
rhn A.T nicholson d.0.t C-o-M