glen herrmannsfeldt <gah@ugcs.caltech.edu> writes:
> [...]
> Periodic boundary conditions are there.  

Where?
-- 
Randy Yates                      % "She tells me that she likes me very much,
Digital Signal Labs              %     but when I try to touch, she makes it
mailto://yates@ieee.org          %                            all too clear."
http://www.digitalsignallabs.com %        'Yours Truly, 2095', *Time*, ELO  

On Aug 21, 3:18&#4294967295;pm, glen herrmannsfeldt <g...@ugcs.caltech.edu> wrote:
>
> Periodic boundary conditions are there. &#4294967295;If you don't need to
> test them, that is fine, but watch out for the cases where it matters.
>

but the lesson of the day remains; whether the periodic boundary
conditions matter or not to one's particular application, these
boundary conditions exist in any case.

i haven't been using the language of "boundary conditions" simply
because it's a concept i am more comfortable applying to continuous-
time or continuous-frequency mathematical situations than if i am with
using it for discrete-time problems.  when it's discrete-time there
really isn't any meaning to continuity of the function or its
derivatives at the "boundary".  but it's okay Glen, i think you and i
essentially agree on the scientific POV regarding this, even if we use
a little different language in describing it.

r b-j

robert bristow-johnson <rbj@audioimagination.com> wrote:
(snip)
 
< i know that some lawyers (and their "expert witnesses") are "hired
< liars" who deliberately distort the facts to serve the interests of
< their client.  if someone is on the other side, if their lawyer is any
< good, he/she should be able to cut through the distortions and
< distractions and focus on the facts and the "truth" of the matter and
< the client with the fact supporting their case *should* win the
< decision and deserves to.
 
< but unlike physics and math, in courtrooms and politics, sometimes it
< isn't the facts that matter but perception of the facts allowing two
< different sides with totally unequal support of the facts to both
< claim that their point-of-view is legitimate when, in reality, only
< one is.

You have to be a little careful with this one.  In enough cases
you don't need all the details.  Newtonian mechanics works for
enough problems to keep using it, even when we know about general
relativity and quantum mechanics.  But you have to be sure to 
remember that it is an approximation.
 
(snip on social science)
(snip on spin) 
 
< is it the case that when electron spin makes no difference to his
< experiment (or using your language, his "application") that it would
< mean, in actual fact, that there *is* no electron spin?  so then some
< time later some other physicist does an experiment that *falsifiably*
< demonstrates this property of spin (by "falsifiable", we mean that the
< experiment would have behaved clearly differently if there was no spin
< to measure).  would it be correct for Millikan to tell the later
< physicist that electron spin is only a point-of-view?  

(big snip)

I believe I have previously posted here about general relativity
and GPS.  It was originally designed to account for general relativity,
but the option was turned off.  Fairly quickly it was determined
that it was needed.  When you get stopped by the police, it won't
help to remind them to consider general relativity in computing
your speed.  Most likely it won't help with the judge, either.

Periodic boundary conditions are there.  If you don't need to
test them, that is fine, but watch out for the cases where it matters.

-- glen

On Aug 21, 10:28&#4294967295;am, dbd <d...@ieee.org> wrote:
> I don't think it takes a lawyer to decide whether "several points of
> view can be taken toward the derivation and interpretation of the DFT
> representation of a finite-duration sequence" means that there is more
> than one point of view that can be taken. I don't think that it takes
> a lawyer to decide whether "We will begin by considering the Fourier
> series representation of periodic sequences." means that they are
> presenting the results of the assumption of the application of the DFT
> to periodic sequences.
>
> It has already been pointed out in this thread how interesting it can
> be to go back to the original material to see how things were. If
> anyone &#4294967295;is actually following this thread for the content, I suggest
> that they go to the original material to make their own decisions on
> what was presented and in what context. I also suggest that they take
> even that with a grain (or bag) of salt and decide for themselves what
> is relevant to their own applications (and then test it).

i know that some lawyers (and their "expert witnesses") are "hired
liars" who deliberately distort the facts to serve the interests of
their client.  if someone is on the other side, if their lawyer is any
good, he/she should be able to cut through the distortions and
distractions and focus on the facts and the "truth" of the matter and
the client with the fact supporting their case *should* win the
decision and deserves to.

but unlike physics and math, in courtrooms and politics, sometimes it
isn't the facts that matter but perception of the facts allowing two
different sides with totally unequal support of the facts to both
claim that their point-of-view is legitimate when, in reality, only
one is.

but in physics and math, reality is much less forgiving nor gullible.
in the hard sciences (so i'm leaving the social sciences out, since
"truth" is still pretty squishy there) the facts are uncaring and
unmoved by any opinions or points-of-view.  the facts are just the
facts and the facts rule the day.

the discovery of electron spin was a later development in physics.  i
can't remember all of the players but i think about a century ago when
Millikan first set out to measure electron charge, his concept was
that of the electron being just a little teeny point that had
properties of mass and charge.  the property of spin was fully
obscured in his experiment and it may never had occurred to him that
these little points weren't little points but had enough size that
spin (and the magnetic moment) was possible.

is it the case that when electron spin makes no difference to his
experiment (or using your language, his "application") that it would
mean, in actual fact, that there *is* no electron spin?  so then some
time later some other physicist does an experiment that *falsifiably*
demonstrates this property of spin (by "falsifiable", we mean that the
experiment would have behaved clearly differently if there was no spin
to measure).  would it be correct for Millikan to tell the later
physicist that electron spin is only a point-of-view?  that it depends
on the application and since in Millikan's oil-drop experiment (that
*does* measure some properties of electrons) there is no apparent
electron spin, that it means that, in fact, there is no electron
spin?  the latter experiment demonstrates the existence of electron
spin, that this spin is a property of the electrons (not the apparatus
of the experiment) and the former experiment does not speak to the
issue.  the factuality of electron spin is not a function of the
experiment, only the observability of that fact is a function of the
experiment.

the "truth" (whatever that is) isn't merely consistent with some of
the facts.  the truth is consistent with all of the facts.  if your
experiment or application of the DFT and mine both demonstrate
consistency with the linearity of the DFT, then we can both *attest*
to the linearity of the DFT.  now if your experiment doesn't even test
for the hypothesized periodicity of the DFT (i.e. the property is
neither confirmed or refuted in your experiment because your
experiment is oblivious to this property if it existed or not), but my
experiment *does* test for that property in a falsifiable manner (that
is, if the property did not exist, the experiment would come out
differently), does that mean that because your experiment failed to
tease that property out of the DFT, that such property does not exist?

you keep clinging to your notion that because this property is not
observed in your experiment (evidently because your experiment does
literally nothing with the DFT output, X[k], other than possibly
scaling by a constant) it does not exist.  when you *eventually*
suggested a non-trivial operation to your DFT output (the magnitude
squared operation) and suggested that there is no evidence of
periodicity, i destroyed your argument.  otherwise, you seem to dare
not propose doing *anything* non-trivial to X[k], because i think by
now, you know i will pounce and spell out to you where the periodicity
*does* have a falsifiable effect.

you continue to keep your head in the sand where you evidently cannot
observe the inherent periodicity of the DFT.  that's only an issue of
your perception (and your deliberate obscuring of it) and does nothing
to change the facts.  besides me, authorities like O&S tell you
explicitly that the DFT has periodicity, that it is inherent to the
DFT, that it is "always there" and you ignore those clear and
unambiguous statements, refer to an introduction that says "well there
are different points-of-view" (they never say that the other points of
view do not show the periodicity, nor even that the other points of
view are valid) and you continue to cling to your personal belief that
the DFT does not necessarily periodically extend your data, when *all*
of the facts where the existence of this periodicity would make a
difference, show that indeed the periodicity is there.

it's not a scientific point-of-view, Dale.  just because you can cook
up a trivial example (or "application") where the periodicity of the
DFT is not tested and not evident, does not negate the existence of
this property that *is* demonstrated in other applications.

for your point-of-view to have *any* correctness, you would have to
produce an experiment that would falsifiably test for this property
and the result of experiment would have to disagree with what would
happen if periodicity was operative.  that means, you would have to
come up with an example or operation that would invoke shifting of
either X[k] or x[n] and something *other* than the periodic extension
of the data (perhaps zero) gets shifted in.  you come up with such an
example, and then you have a plausible case for the notion that the
DFT does not necessarily periodically extend its data.

but you cannot, so your cherished point-of-view is wrong.

r b-j

I don't think it takes a lawyer to decide whether "several points of
view can be taken toward the derivation and interpretation of the DFT
representation of a finite-duration sequence" means that there is more
than one point of view that can be taken. I don't think that it takes
a lawyer to decide whether "We will begin by considering the Fourier
series representation of periodic sequences." means that they are
presenting the results of the assumption of the application of the DFT
to periodic sequences.

It has already been pointed out in this thread how interesting it can
be to go back to the original material to see how things were. If
anyone  is actually following this thread for the content, I suggest
that they go to the original material to make their own decisions on
what was presented and in what context. I also suggest that they take
even that with a grain (or bag) of salt and decide for themselves what
is relevant to their own applications (and then test it).

Dale B. Dalrymple

On Aug 20, 2:09&#4294967295;pm, robert bristow-johnson <r...@audioimagination.com>
wrote:
...
> when investigating the recursion formula for the Tchebyshev function:
>
> &#4294967295; &#4294967295; T[n](x) = cos(n*arccos(x))
>
> (we haven't established yet that it's a polynomial), one can *choose*
> to inquire what T[n+1](x) is. &#4294967295;so we look into it and discover by the
> wonders of trigonometry that
>
> &#4294967295; &#4294967295;          T[n+1](x) = 2*T[n](x) - T[n-1](x)

should be:     T[n+1](x) = 2*x*T[n](x) - T[n-1](x)

and with numerous spelling errors that happens when i type, there are
more typos.  i don't worry about typos normally, but i don't like it
when the math is incorrect.

r b-j

On Aug 20, 11:08 am, dbd <d...@ieee.org> wrote:
> On Aug 19, 12:12 pm, robert bristow-johnson
>
> <r...@audioimagination.com> wrote:
> > ...
> > > > BTW, i hope you read the other post where i made it even more clear
> > > > that O&S don't take your position on this.  they don't at all.
>
> > > I read the post and considered your interpretation to be at odds with
> > > the words you quoted from O&S.
>
> The words I refer to are:
>
> On Aug 8, 2:49 pm, robert bristow-johnson <r...@audioimagination.com>
> wrote:
>
> > ...
> > at the beginning of
> > their DFT chapter (ch 8, pp 514 in my edition):
>
> > "Although several points of view can be taken toward the derivation
> > and interpretation of the DFT representation of a finite-duration
> > sequence, we have chosen to base our presentation on the relationship
> > between periodic sequences and finite-length sequences.  We will begin
> > by considering the Fourier series representation of periodic
> > sequences.
> > ...
>
> Having made the -assumption- of periodic sequences, the rest does
> follow, from the assumption.
>
>

it's no assumption.  it's a recognition.  it's a derivation.  we call
that math.


                   N-1
    x[n] = (1/N) * SUM{ X[k] * e^(j*2*pi*n*k/N) }
                   k=0

hmmm, what happens when you plug in n+N in for n?

                     N-1
    x[n+N] = (1/N) * SUM{ X[k] * e^(j*2*pi*(n+N)*k/N) }
                     k=0

                     N-1
           = (1/N) * SUM{ X[k] * e^(j*2*pi*n*k/N) * e^(j*2*pi*N*k/N) }
                     k=0

                     N-1
           = (1/N) * SUM{ X[k] * e^(j*2*pi*n*k/N) * e^(j*2*pi*k) }
                     k=0

                     N-1
           = (1/N) * SUM{ X[k] * e^(j*2*pi*n*k/N) * ! }
                     k=0

                     N-1
           = (1/N) * SUM{ X[k] * e^(j*2*pi*n*k/N) }
                     k=0


           = x[n]
>
>
>
> > O&S: "The inherent periodicity is always present.  Sometimes it causes
> > us difficulty and sometimes we can exploit it, but to totally ignore
> > it is to invite trouble."
>
> > what points do you derive from these O&S words?  certainly not these
> > points:
>
> > 1. There is periodicity regarding the DFT.
>
> > 2. It's inherent to the DFT.
>
> > 3. It's always present with the DFT.
>
> > 4. Sometimes we have to worry about it.
>
> > 5. Sometimes we can exploit it.
>
> > 6. Ignoring it invites trouble.
>
>
> I do agree with those points, they follow directly on their decision
> to consider periodic sequences.

no, you *missed* the point.  that continues to be evident.  they never
said that the facts are true *only* because the investigated the
periodic properties of the DFT.  they are saying that these facts are
true regardless (they never qualified them) and it's *because* they
investigated the potential periodic properties (a fact is a
hypothetical until it's investigated and shown to be true) that they
discover that, in fact, that math says it's true.

don't consider a career in law, Dale.  your clients would be ill
served.

> But they don't say that you must or
> can only deal with periodic sequences,

but they say in an unqualified manner that if you ignore it, you're
heading for trouble.  your inability to face the facts, even when they
are slapping your face, reveals in you this common malady of
"cognitive dissonance".  you just cannot admit, either publically, or
likely even to yourself in the privacy of your thoughts, that *maybe*,
just *maybe* you missed something.  but the fantasy that you're
completely on top of this was destroyed (by me) several times.  i
pressed you for specifics about what, if anything, you're doing with
your DFT outputs X[k].  you continued to be evasive and i continued to
say that "if you do nothing with your DFT output X[k] (beyond
multiplication by a constant), the periodicity will not be
superficially eveident.  so then, finally, you suggested that if you
magnitude squared the DFT output, that such was an example where there
is no periodic extension evident.  but that was objective incorrect.
the iDFT{|X[k]|^2} is the *circular* (or "periodic", choose your
poison) convolution of x[n] with x[-n]

i'll bet money, Dale, that you're a Bushie Republican.  those people
*never* admit they're wrong, no matter how high the body count gets.

> but if you do consider periodic
> sequences, those are their conclusions. That's their choice of
> application to discuss, not their characterization of the DFT.

that's the dumbest (mis)application of logic.  the choice to
investigate the behavior of x[n] outside of the domain of n originally
supplied is just a choice.  but the fact remains whether you make that
choice or not.  that is the way it is with *any* mathematical
derivation.

when investigating the recursion formula for the Tchebyshev function:


    T[n](x) = cos(n*arccos(x))


(we haven't established yet that it's a polynomial), one can *choose*
to inquire what T[n+1](x) is.  so we look into it and discover by the
wonders of trigonometry that

    T[n+1](x) = 2*T[n](x) - T[n-1](x)

that combined with the obvious initial conditions for n=0 and n=1
leaves us with the *conclusion* (not the assumption) that the
Tchebyshev function is really an nth order polynomial.  it's math.
not an assumption.

in fact, even though the original definition of T[n](x) doesn't have
apparent meaning for |x| > 1, the recognition of T[n](x) as a
polynomial cause mathematicians (and electrical engineers that don't
have their head in the sand) to understand that the original
definition implies an extension *outside* of the original domain for
x, even if the arccos(x) function has no definition outside.  then,
mathematicians (and electrical engineers that don't have their head in
the sand), investigate how that extension would appear and we get:


              { cos(n*arccos(x))                 |x| <= 1
    T[n](x) = {
              { (sgn(x))^n * cosh(n*arccosh|x|)  |x| >= 1


now, you're free to leave your head in the sand and deny the math.
perhaps in your fantasy land, the Tchebyshev function (which the rest
of us know is a polynomial) only has definition for |x| <= 1.  but
it's not true.  if you're smart and intellectually curious and say to
yourself, "gee, with cos(n*arccos(x)) definition that doesn't work for
|x|>1, we find out that it's really a polynomical, i wonder if my
synapses would fry if i considered the possibility that this
polynomial will still work for |x|>1?"

you are free to recognize or not recognize what the math tells us.
tho "choice" only exposes a mathematical fact that has existed
eternally.  that fact remains despite what the choice was.  it may
remain undiscovered, but a mathematical truth (or "theorem" if you
don't like "truth" language) remains whether some of us fail to
discover it or not.

this is not religion.  this is about objective, falsifiable fact.
before it got to be politically incorrect, what we used to call "the
truth."  but in today's politically correctness, we seem to have
relegated all notions of truth to one's own belief system ("Your
truths may be different than mine.").  so now, when materialists try
to discuss what really happens in reality, have to use language such
as "objective fact", "falsifiablity" and the like, otherwize the
Intelligent Design people will slip in their *beliefs* and claim that
they have every claim to "truth" as do the biologists and
palentologists.

Dale, the DFT transforms one periodic sequence of period N to another
periodic sequence also with period N.  that is an objective and
falsifiable fact.  because it's falsifiable, that means if we
hypothesize that maybe it isn't true, we should be able to come up
with a contradiction.  and we do.  we've been doing this math over and
over again.

you deny the periodicity of the DFT and rely on that denial to do
*anything* non-trivial with the results of the DFT, you *will* run
into trouble.  that is because e^(j*2*pi*k) = 1 for any integer k.
you cannot avoid that trouble unless you do *nothing* to your DFT
results.

the Young Earth Creationists insist that the earth was created about
6000 years ago.  when confronted by evidence that there is stuff
laying around that is older, they counter in a manner just like you
do, Dale.  they say, well, when the earth was created, it was created
with a history.  so the fossils with Carbon-14 evidence that are
billions of years old suddenly appeared on the landscape some 6000
years ago.

they hold up the fossil and draw a completely different conclusion
about it that the palentologist does.  their truth seems just as
plausible as the palentologist's truth, *until* they try to do
anything with their version of science.  then it fails.  that's what
falsifiability is.

when you try to do anything non-trivial with your version of the DFT
(that does not periodically extend the data passed to it), you fail.
you get errors or contradictions.  you made a falsifiable claim and it
didn't survive the test.  because for falsifiability, *you* don't get
to choose the test, the devil hands you the test and if your theory is
correct, it continues to be true, without contradiction, despite
whatever test the devil hands to you.  that's how you prove things in
mathematics.


r b-j

On Aug 19, 12:12 pm, robert bristow-johnson
<r...@audioimagination.com> wrote:

> ...
> > > BTW, i hope you read the other post where i made it even more clear
> > > that O&S don't take your position on this.  they don't at all.
>
> > I read the post and considered your interpretation to be at odds with
> > the words you quoted from O&S.

The words I refer to are:

On Aug 8, 2:49 pm, robert bristow-johnson <r...@audioimagination.com>
wrote:
> ...
> at the beginning of
> their DFT chapter (ch 8, pp 514 in my edition):
>
> "Although several points of view can be taken toward the derivation
> and interpretation of the DFT representation of a finite-duration
> sequence, we have chosen to base our presentation on the relationship
> between periodic sequences and finite-length sequences.  We will begin
> by considering the Fourier series representation of periodic
> sequences.
> ...

Having made the -assumption- of periodic sequences, the rest does
follow, from the assumption.

>
> O&S: "The inherent periodicity is always present.  Sometimes it causes
> us difficulty and sometimes we can exploit it, but to totally ignore
> it is to invite trouble."
>
> what points do you derive from these O&S words?  certainly not these
> points:
>
> 1. There is periodicity regarding the DFT.
>
> 2. It's inherent to the DFT.
>
> 3. It's always present with the DFT.
>
> 4. Sometimes we have to worry about it.
>
> 5. Sometimes we can exploit it.
>
> 6. Ignoring it invites trouble.
>
> r b-j

I do agree with those points, they follow directly on their decision
to consider periodic sequences. But they don't say that you must or
can only deal with periodic sequences, but if you do consider periodic
sequences, those are their conclusions. That's their choice of
application to discuss, not their characterization of the DFT.

Dale B. Dalrymple

On Aug 20, 5:22 am, Randy Yates <ya...@ieee.org> wrote:
> dbd <d...@ieee.org> writes:
> > [...]
> > To find periodic extension it must be assumed
>
> If you assume the output of the DFT represents a discrete set of samples
> in the frequency domain, then the input must necessarily be periodic,
> because any frequency domain signal evenly-spaced "spikes" is
> necessarily periodic.

The frequency domain signal can only be "spikes" if you are describing
a domain that has values at places the DFT does not yield values. You
can only identify the presence of "spikes by the existence of values
between the values of the peaks of the "spikes". This isn't in the
domain of the DFT, but there are applications where it is a useful
pedagogical construct.

>
> If you want to make no assumptions on the meaning of the DFT output,
> then it's just a mathematical operation that transforms N complex
> numbers to N complex numbers. In this case, no such "periodic extension"
> needs to be made.

My point exactly.

>
> I'll stick with the former - it's a lot more useful to me.

You are always welcome to apply the assumptions that are useful to
your applications. Some people insist on enforcing particular
assumptions without consideration of applications. You understand the
choice available and the nature your applications and seem to have
chosen on a purposeful basis.

> --
> Randy Yates                      % "Maybe one day I'll feel her cold embrace,
> Digital Signal Labs              %                    and kiss her interface,
> mailto://ya...@ieee.org          %            til then, I'll leave her alone."http://www.digitalsignallabs.com%        'Yours Truly, 2095', *Time*, ELO

Randy Yates <yates@ieee.org> writes:

> dbd <dbd@ieee.org> writes:
>> [...]
>> To find periodic extension it must be assumed 
>
> If you assume the output of the DFT represents a discrete set of samples
> in the frequency domain, 

Should read, "...a discrete set of evenly-spaced samples in the frequency
domain..."
-- 
Randy Yates                      % "Bird, on the wing,
Digital Signal Labs              %   goes floating by
mailto://yates@ieee.org          %   but there's a teardrop in his eye..."
http://www.digitalsignallabs.com % 'One Summer Dream', *Face The Music*, ELO