rbj@surfglobal.net (robert bristow-johnson) wrote in message news:<4cbb922e.0405142205.3def6089@posting.google.com>...
> i *did* mean to imply that "merely" referring to
> the linear mapping of the DFT was not sufficient to justify that it
> has no inherent periodicity and the context that i was reacting to was
> here:
Who has said that the periodicity is INVALID?
Invalid != unnecessary.
I don't think anyone has said that the historical interpretation is
wrong. Did someone? Did I miss that?
It's just not the ONLY one, and it is an unnecessarily limiting
interpretation.
> back to the current post:
> > The historical meaning of the DFT is quite clear.
>
> fine. but i never made any appeal to the historical. my only issues
> here are conceptual (or foundational) and practical.
You have not shown why there is any need whatsoever to appeal to the
periodic nature of the DFT.
> again, repeating a fallacy doesn't sell it here. dunno why you say i
> am putting words in your mouth. these words (above) are plenty to
> take issue with.
Please show where insisting on infinite periodicity is NECESSARY in a
DFT.
You have never done that.
Taken as a linear-algebraic transform there is no, zero need to insist
on any thing outside the time limits of the DFT. The fact that, yes,
you CAN have another interpretation is sometimes useful, I agree, but
you have done nothing to show that it's necessary. You can't, because
it's not.
You demand I prove the negative, and you refer to what I am saying as
a fallacy.
But you offer no positive proof that what you insist is necessary is
necessary.
The obligation to your extraordinary claim is entirely upon you. Since
I can show that the DFT is orthonormal, and since you have stipulated
that, I have fully, absolutely, and incontrovertibly proven my point
already, and you have so stipulated.
But yet you claim that when I say that, I'm repeating a "fallacy".
Your position is inconsistant, and your use of the word "fallacy" a
professional insult of the worst kind, yet you have stipulated the
truth of my position already.
> > There is no need to bring up the periodicity issue. The DFT stands on
> > its own as a linear algebraic transform.
>
> *merely* saying that "The DFT stands on its own as a linear algebraic
> transform" is insufficient to support saying "There is no need to
> bring up the periodicity issue." it is a straw man. a red herring.
> a distraction. non-sequitur.
Is it, or is it not, true, that the DFT stands on its own as a
linear-algebraic transform"?
If the answer is "yes" then I can use it not caring at all about the
periodicity issue, whether or not it exists. That's a fact.
There is no insufficiency whatsoever. I can use the theorems that
apply, etc, without ever considering the fact that there is a
periodicity involved. That is true, you know.
I may understand something more, sometimes, by understanding the
periodic implications, certainly, I haven't ever said anything else.
> do you now understand why i used the word "merely"? i didn't just
> pull it out of the air.
Because there is no "mere" to showing that something is an orthonormal
transform. Once one shows that, it brings up a whole host of very
desirable properties that one can depend on, etc.
> i *did* qualify it with "apparently". you appear to deny that the DFT
> inherently periodically extends the data given it when you say it's
> "NOT NECESSARY" or "NO NEED".
I do nothing of the sort. That is a pure straw man, based on something
I simply can't fathom. I am saying that once we know it's an
orthonormal transform, we know a lot of important things. The fact it
diagonalizes real signals very well is also very important. None of
that relates to its periodic interpretation at all.
> the fact that it is valid remains even in contexts where you insist
> that it's "NOT NECESSARY".
Of course. Many valid things are not necessary. What's the deal?
> repeating a fallacy does not make it less fallacious.
And repeating a false professional accusation does not make it any
less objectionable, either.
> so, i'm saying that you can't get away from it (this periodic
> extension of data inherent to the DFT) even if you try.
Why must I care about that? I need not, and in fact, when I cease to
worry about that, now I can much more easily formulate an OBT, or MLT,
in fact, than I can if I stick to this one-block-periodicity you keep
pointing out. It's just one example out of an infinite set of cases...
Repeat over one block, 2 blocks, n blocks, when formulated correctly,
all the same.
> once aliasing (be it frequency domain or time domain) has occurred,
> you do not know if the alias was from wrap-around or a periodic copy
> or if the "alias" was there in the first place. in any case i think
> this is non-sequitur.
That would appear to argue that a QMF does not work. Does it, or does
it not, work?
Does, or does not, an MDCT work? You just suggested that, as well.
> you don't have to participate which makes me curious why you would if
> there was no point.
Because I think, and strongly continue to think, given your obvious
knowledge and your continued obtuseness, that the periodic
interpretation "sticks" you into a set of unnecessary constraints.
> The DFT maps one infinite and periodic discrete "time" sequence
> of period N, of which one only specifies N contiguous samples
> of it, to another infinite and periodic discrete "frequency"
> sequence of the same period and the iDFT maps it back. The DFT
> "understands" or "assumes" (if anthropomorphism is allowed) the
> N samples supplied to it to be one period of this periodic
> discrete sequence and likewise for the iDFT. In this manner,
> the DFT (and iDFT) periodically extends the data supplied to it
> which has salience where those N samples were extracted from a
> longer, possibly non-periodic sequence. One may know those N
> samples x[n] to not be from a periodic context, but when they
> go into the DFT, that fact is lost to it, its output X[k], and
> all further operations done on it. All further operations that
> cause shifting to either x[n] or X[k] (including convolution)
> *must* *necessarily* be considered to be operating on periodic
> extensions of the original x[n] or X[k] to get correct results.
> The DFT and the DFS are, in any meaningful way, one and the same.
And, I repeat, you're limiting your own understanding by including
unnecessary constraints in your definition. How about something that
is periodic over TWO analysis blocks. What does that get you?
Etc, etc.
> > The DFT is different, because of the environment it lives in. It need
> > not have an infinite-energy formulation. It need not worry at all
> > about its validity, unless the input data contains infinities, unlike
> > a Fourier Integral. It has a different set of properties, a set of
> > properties that extend beyond the meaning of the Fourier Integral.
How dare you insist that a basic difference in properties is a
non-sequiter? Your sheer arrogance in denying these basic,
mathematically obvious FACTS are germane is simply astonishing.
> let's see if it's necessary if linearity is applied:
>
> DFT{ x[n] + y[n] } = X[k] + Y[k]
>
> DFT{ A*x[n] } = A*X[k]
>
> ya know, i guess in this case, it ain't necessary.
Yes?
> likewise, in this case, what does one do with x[n-m] when m>n ? we
> gotta do something to x[n] to make it work for those indices, which
> is necessarily an extension, and the only correct extension is the
> periodic extension, x[n+N] = x[n].
Why do you ignore the idea of aliasing here, only to demonstrate it?
Your very obtuseness shows, I think, the harmful nature of the
too-limited interpretation.