# Relationship between z and Fourier transforms

Started by July 30, 2013
```On Thu, 1 Aug 2013 22:35:28 +0000 (UTC), glen herrmannsfeldt
<gah@ugcs.caltech.edu> wrote:

>Tim Wescott <tim@seemywebsite.really> wrote:
>> On Thu, 01 Aug 2013 13:42:19 -0700, robert bristow-johnson wrote:
>
>(snip)
>
>>> either way you're periodically extending the data, the math says so, and
>>> it seems to me silly to deny it.  there is no way to get away from it.
>>>   "resistance is futile."
>
>> It works either if you periodically extend the data, or if you say that
>> the data set is finite.
>
>> A pair of examples pertaining to the Fourier series comes from electrical
>> engineering, one involving periodic phenomena, and the other involving
>> bounded phenomena.
>
>> The periodic example is the use of the Fourier series to find the
>> harmonics of a periodic waveform.  This is your "periodic
>> extension" case.
>
>> The aperiodic example (and I'm rustier at this one, but I remember it
>> being done) is in the use of the Fourier series to find the potential
>> variation within a square box that's bounded by conductors.  In this case
>> there is no periodicity -- there could be anything outside the box, the
>> problem stops as soon as you hit that metal wall with its imposed voltage.
>
>OK, but consider a related problem, either in 1D (signal on a coax
>cable) or 3D (EM wave in a metal box). When the wave hits the boundary,
>there is a reflection. The solution looks the same as if the original
>one went through the boundary, and another comes from the outside
>into the problem at the boundary.
>
>Or, consider when you look at the reflection of an object in a
>mirror, it "looks" the same as it would if there was no mirror,
>and an actual object on the other side.

For many of us there are multiple points of view that are valid and
provide the same results.   Personally, I think the windowed or
periodically-extended or boundary-condition points of view are
functionally equivalent and one can use whichever point of view suits
them best.   Personally, I think the fullest undertanding comes from
seeing how all of these points of view work and how they all
reconcile.

The disagreement seems to center only with those who insist that only
a single point of view is correct or mathematically supportable.  I
think that is incorrect.   O&S, in both Digital Signal Processing and
Discrete-Time Signal Processing, take great pains to explain that
there are multiple approaches that work before embarking on one method
to develop the explanations of the transforms.   In Discrete Time
Signal Processing in an early chapter (Ch. 3. in my first edition)
they go almost all the way to deriving the DFT with no required
assumption of periodicity on the input.   I filled in the last small
step in an article here:

http://www.dsprelated.com/showarticle/175.php

In other parts of both texts they use an assumption of periodicity on
the input to finish the development because there are some benefits to
doing so in an explanatory text.   Again, that is done after care is
taken to explain that that is not the only approach.

The various approaches work and can be used to come to the same
results as the others.   I think people should use whichever approach
makes the most sense to them personally, but I think it is incorrect
to say that the other approaches are incorrect or incomplete or not
mathematically supported or sound.

>> The math fits in either case.  So, either view is equally valid.

This.

>-- glen

Eric Jacobsen
Anchor Hill Communications
http://www.anchorhill.com
```
```On 8/1/13 3:57 PM, Eric Jacobsen wrote:
> On Thu, 01 Aug 2013 14:43:37 -0700, robert bristow-johnson
> <rbj@audioimagination.com>  wrote:
>
>> On 8/1/13 2:06 PM, Eric Jacobsen wrote:
>>> On Thu, 01 Aug 2013 13:42:19 -0700, robert bristow-johnson
>>> <rbj@audioimagination.com>   wrote:
>>>
>>>> On 8/1/13 1:25 PM, Tim Wescott wrote:
>> ...
>>>>>    The math just is, and if you're getting the right
>>>>> results you are, ipso facto, using it correctly.
>>>>
>>>> fully agree with that.  and, unless there is no shifting nor convolution
>>>> (in time or frequency) going on (which means the only property or
>>>> theorem of the DFT one can make use of is Linearity) then the *only*
>>>> (and this "only" is me being inflexible) correct mathematics is to
>>>> either explicitly periodically extend the data:
>>>>
>>>>       x[n+N] = x[n]   or   X[k+N] = X[k]
>>>>
>>>> or to implicitly periodically extend the data:
>>>>
>>>>       x[n mod N]    or   X[k mod N]
>>>>
>>>>
>>>> either way you're periodically extending the data, the math says so, and
>>>> it seems to me silly to deny it.  there is no way to get away from it.
>>>>    "resistance is futile."
>>>
>>> Except for the numerous counter-examples that have been pointed out to
>>
>> you have numerous counter-examples which involve shifting the data or
>> convolving the data (or their counterparts in the other domain which
>> means multiplying by something that is not constant) where you can
>> ignore both
>>
>>        x[n+N] = x[n]
>>
>> or
>>
>>        x[n mod N]
>>
>> and the X[k] counterparts?
>>
>> would you mind repeating just one of those counter examples?  and in
>> these numerous counter examples you are able to show mathematical
>> correctness while ignoring x[n+N] = x[n] and x[n mod N] (or the same
>> thing in X[k])?
>
> It's been said many times already.   I've long grown tired of
> repeating them over again to you.

many words were said.  and the mathematical case was never made by you
(nor Dale nor anyone else).  not even with that .pdf you sent (after my
.pdf because you didn't like ASCII math on USENET).  i've grown tired of
the whole thing too, but the math doesn't care.

and it was about the same question.  i would like to see *any* shift
operation done in the context of the DFT that isn't circular.  and you
cannot provide such an example because the math won't let you.

>> zero-padding is a questionable counter-example because if you claim that
>> the zero shifted in isn't the same zero that was shifted out the other
>> end, then i would dispute that.
>
> You dispute a lot of things.

perhaps, but about the topic at hand, i only dispute only one thing.
well, maybe two.  i dispute:

1.  that, regarding the DFT mapping, when any operation is done in one
domain that causes shifting in the other domain, that this shifting is
anything other than circular.

2.  that the DFT, itself, is responsible (inherently or in any other
manner) for the effects of windowing.  even rectangular windowing.  the
effects of windowing happens when you actually window the data.  in the
simple case of rectangular windowing, it is when you yank N samples out
if a stream or a longer sequence of samples.  the windowing happens
before the DFT ever sees the data.

other than that, i don't think there is a dispute.

--

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

```
```On 8/1/13 3:26 PM, dbd wrote:
> On Thursday, August 1, 2013 12:48:45 PM UTC-7, robert bristow-johnson wrote:
>> On 8/1/13 12:17 PM, Tim Wescott wrote:
>>>
>>> Your classification system works for me (if not for my newsreader, which
>>> seems to have mangled your nice table).
>>>
>> careful, Tim.  you might inadvertently have taken a side in "_That_
>> argument".  note the upper right corner of the table.
>>
>> r b-j
>
> That you chose to add a notation in a manner that you expected to be hard to notice provides more illumination about your nature than the notation tells us about the nature of transforms.

what do you mean, Dale????  the tilde??  is that the notation that i
"expected to be hard to notice"?  i sure thought that i was explicit and
drawing attention to it (because i was quoting O&S again and i wanted to
regarding other people's words.

> The existence of the DFS in the chart already indicates that the assumptions of periodicity and of period N already apply to signals at that point in the diagram.

can you show me were Kabal has the term "DFS" in that chart?

> A better graphic presentation of some of the transform relationships is available at:
> http://www-mmsp.ece.mcgill.ca/Documents/Reports/2011/KabalR2011c.pdf

wow.  that's a nice graphic.

i can understand when Eric or someone else poo-poos my quoting of O&S
"scripture" as evidence, because quoting anyone is not evidence of
anything except of what these person say.  but i find it hard when i
quote O&S simple as someone else (perhaps someone else of recognized
authority) are saying the same thing as me.  and then you have said
something like "No it isn't, you're misrepresenting O&S."  and that just
blows my mind.

now *here* you are using this (apparently excellent) reference to
support your position and it doesn't.  Kabal and i are saying the same
thing.  he has "DFT" in that chart.  there is no "DFS" on that chart at
all.  now here's the kernel:  [[You cannot even pass x[n] to the DFT in
that chart without first turning it into ~x[n].  There *is* no
non-periodic x[n] going into Kabal's DFT just as there is no
non-periodic x(t) going into his FS.]]

thanks for the reference, Dale.  i haven't read too much of the text
(just looked at the equations), but it appears at first glance to
and Schafer).

note also that *everywhere* that there is sampling in one domain, there
is necessarily periodic extension (the overlap-adding periodic
extension) is the reciprocal domain.  discrete in one domain (assuming
_uniformly_ sampled) always *always* *always* means periodic in the
other domain.

always.

and, Tim, that's what i mean about being inflexible.  but i'm just the
messenger.  it's the _math_ that is being inflexible.

stubborn math.  wish it would loosen up a bit.

--

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

```
```robert bristow-johnson <rbj@audioimagination.com> writes:

> On 8/1/13 3:26 PM, dbd wrote:
> [...]
>> A better graphic presentation of some of the transform relationships is available at:
>> http://www-mmsp.ece.mcgill.ca/Documents/Reports/2011/KabalR2011c.pdf
>
> wow.  that's a nice graphic.

LaTeX does such a nice job!

> i can understand when Eric or someone else poo-poos my quoting of O&S
> "scripture" as evidence, because quoting anyone is not evidence of
> anything except of what these person say.  but i find it hard when i
> quote O&S simple as someone else (perhaps someone else of recognized
> authority) are saying the same thing as me.  and then you have said
> something like "No it isn't, you're misrepresenting O&S."  and that
> just blows my mind.
>
> now *here* you are using this (apparently excellent) reference to
> support your position and it doesn't.  Kabal and i are saying the same
> thing.  he has "DFT" in that chart.  there is no "DFS" on that chart
> at all.  now here's the kernel:  [[You cannot even pass x[n] to the
> DFT in that chart without first turning it into ~x[n].  There *is* no
> non-periodic x[n] going into Kabal's DFT just as there is no
> non-periodic x(t) going into his FS.]]
>
> thanks for the reference, Dale.  i haven't read too much of the text
> (just looked at the equations), but it appears at first glance to
> Oppenheim and Schafer).
>
> note also that *everywhere* that there is sampling in one domain,
> there is necessarily periodic extension (the overlap-adding periodic
> extension) is the reciprocal domain.  discrete in one domain (assuming
> _uniformly_ sampled) always *always* *always* means periodic in the
> other domain.
>
> always.
>
> and, Tim, that's what i mean about being inflexible.  but i'm just the
> messenger.  it's the _math_ that is being inflexible.
>
> stubborn math.  wish it would loosen up a bit.

Robert, Dale, et al., let me ask a question. In the "loosest" sense (the
one with the least amount of interpretation), the output of the DFT is a
set of complex numbers. It's just a set of stinkin' numbers. Do you
agree? Of course you agree; you HAVE to agree - it's a fact.

In my opinion, the periodicity thing only comes in when you start making
assumptions about what those numbers _mean_, and THAT is open for
interpretation. For example, if interpret them as the points of a
discrete spectrum, then the corresponding signal represented NECESSARILY
has to be periodic.

OK, so we're done, right?
--
Randy Yates
Digital Signal Labs
http://www.digitalsignallabs.com
```
```On Thursday, August 1, 2013 5:26:48 PM UTC-7, robert bristow-johnson wrote:
...
>>> careful, Tim.  you might inadvertently have taken a side in "_That_
>>> argument".  note the upper right corner of the table.
...
>On 8/1/13 3:26 PM, dbd wrote:
>> That you chose to add a notation in a manner that you expected to be hard to notice provides more illumination about your nature than the notation tells us about the nature of transforms.
>
> what do you mean, Dale????  the tilde??

I meant the added remark in the ascii art you suggested that Tim might have missed in the section of your post I quoted. As in, That's why I had just quoted your remark immediately above to provide context. Why are you asking about something many posts away?

>
> > The existence of the DFS in the chart already indicates that the assumptions of periodicity and of period N already apply to signals at that point in the diagram.
>
> can you show me were Kabal has the term "DFS" in that chart?

That "DFS" is in your ascii chart, the one Tim responded to and the one in your (quoted in my post) reply to Tim. The one I had just referred to. I had not introduced the url for the Kabal paper yet. As you can see in my next line that you quoted below.

> > A better graphic presentation of some of the transform relationships is available at:
> > http://www-mmsp.ece.mcgill.ca/Documents/Reports/2011/KabalR2011c.pdf
>
> wow.  that's a nice graphic.
>
> i can understand when Eric or someone else poo-poos my quoting of O&S
> "scripture" as evidence, because quoting anyone is not evidence of
> anything except of what these person say.  but i find it hard when i
> quote O&S simple as someone else (perhaps someone else of recognized
> authority) are saying the same thing as me.  and then you have said
> something like "No it isn't, you're misrepresenting O&S."  and that just
> blows my mind.
>

You have already misinterpreted two simple statements by failure to understand context in just this much of your post. Why should anyone trust you on issues of interpreting context in O&S when you follow context this poorly here?

> now *here* you are using this (apparently excellent) reference to
> support your position and it doesn't.  Kabal and i are saying the same
> thing.  he has "DFT" in that chart.  there is no "DFS" on that chart at
> all.  now here's the kernel:  [[You cannot even pass x[n] to the DFT in
> that chart without first turning it into ~x[n].  There *is* no
> non-periodic x[n] going into Kabal's DFT just as there is no
> non-periodic x(t) going into his FS.]]

I'm glad you are able to see that Kabal doesn't allow the use of an aperiodic signal to generate a discrete Fourier Series (FS). We agree on that. But then you make the suggestion that the aperiodic signal can be "turned" into a periodic signal. How is this done? Kabal says "We refer to this process which forms a time-aliased periodic signal as wrapping." There is an equation for it. It sums all N sample intervals of the non-N-periodic signal. It is not equivalent to the rectangular windowing which we have used to prepare periodic and aperiodic signals for the DFT.

> ...
> note also that *everywhere* that there is sampling in one domain, there
> is necessarily periodic extension (the overlap-adding periodic
> extension) is the reciprocal domain.  discrete in one domain (assuming
> _uniformly_ sampled) always *always* *always* means periodic in the
> other domain.
>
> always.
>

Yes, that means that to get a discrete FS you must transform N samples of an N-periodic signal.

If you start with N samples from an aperiodic or periodic-but-not-in-N signal, you still get a useful N point vector from the DFT calculation, but you don't get a discrete FS.

> ...
> r b-j

As Kabal says on page 9:
" we see that the DFT formula Eq. (40) can be considered to operate on a finite length signal, or alternately, can be considered to operate on one period of a periodic signal. In the latter interpretation, the DFT formula essentially
calculates the Fourier series coefficients of the periodic signal"

Dale B. Dalrymple
```
```On Thu, 01 Aug 2013 23:09:14 GMT, eric.jacobsen@ieee.org (Eric
Jacobsen) wrote:

>On Thu, 1 Aug 2013 22:35:28 +0000 (UTC), glen herrmannsfeldt
><gah@ugcs.caltech.edu> wrote:
>
>>Tim Wescott <tim@seemywebsite.really> wrote:
>>> On Thu, 01 Aug 2013 13:42:19 -0700, robert bristow-johnson wrote:
>>
>>(snip)
>>
>>>> either way you're periodically extending the data, the math says so, and
>>>> it seems to me silly to deny it.  there is no way to get away from it.
>>>>   "resistance is futile."
>>
>>> It works either if you periodically extend the data, or if you say that
>>> the data set is finite.
>>
>>> A pair of examples pertaining to the Fourier series comes from electrical
>>> engineering, one involving periodic phenomena, and the other involving
>>> bounded phenomena.
>>
>>> The periodic example is the use of the Fourier series to find the
>>> harmonics of a periodic waveform.  This is your "periodic
>>> extension" case.
>>
>>> The aperiodic example (and I'm rustier at this one, but I remember it
>>> being done) is in the use of the Fourier series to find the potential
>>> variation within a square box that's bounded by conductors.  In this case
>>> there is no periodicity -- there could be anything outside the box, the
>>> problem stops as soon as you hit that metal wall with its imposed voltage.
>>
>>OK, but consider a related problem, either in 1D (signal on a coax
>>cable) or 3D (EM wave in a metal box). When the wave hits the boundary,
>>there is a reflection. The solution looks the same as if the original
>>one went through the boundary, and another comes from the outside
>>into the problem at the boundary.
>>
>>Or, consider when you look at the reflection of an object in a
>>mirror, it "looks" the same as it would if there was no mirror,
>>and an actual object on the other side.
>
>For many of us there are multiple points of view that are valid and
>provide the same results.   Personally, I think the windowed or
>periodically-extended or boundary-condition points of view are
>functionally equivalent and one can use whichever point of view suits
>them best.   Personally, I think the fullest undertanding comes from
>seeing how all of these points of view work and how they all
>reconcile.
>
>The disagreement seems to center only with those who insist that only
>a single point of view is correct or mathematically supportable.  I
>think that is incorrect.   O&S, in both Digital Signal Processing and
>Discrete-Time Signal Processing, take great pains to explain that
>there are multiple approaches that work before embarking on one method
>to develop the explanations of the transforms.   In Discrete Time
>Signal Processing in an early chapter (Ch. 3. in my first edition)
>they go almost all the way to deriving the DFT with no required
>assumption of periodicity on the input.   I filled in the last small
>step in an article here:
>
>http://www.dsprelated.com/showarticle/175.php
>
>In other parts of both texts they use an assumption of periodicity on
>the input to finish the development because there are some benefits to
>doing so in an explanatory text.   Again, that is done after care is
>taken to explain that that is not the only approach.
>
>The various approaches work and can be used to come to the same
>results as the others.   I think people should use whichever approach
>makes the most sense to them personally, but I think it is incorrect
>to say that the other approaches are incorrect or incomplete or not
>mathematically supported or sound.
>
>>> The math fits in either case.  So, either view is equally valid.
>
>This.
>
>>-- glen
>
>Eric Jacobsen

Hi Eric,
I'm too lazy to read all the posts in this
thread but, for what it's worth, at the beginning of their
chapter titled: "The Discrete Fourier Transform", on page 541 of
their 2nd Edition of "Discrete-Time Signal Processing",
Oppenheim & Schafer state:

"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. While this representation is
important in its own right, we are most often interested
in the application of Fourier series results to the
representation of finite-length sequences. We accomplish
this by constructing a periodic sequence for which each
period is identical to the finite-length sequence. As we
will see. the Fourier series representation of the
periodic sequence corresponds to the DFT of the
finite-length sequence. Thus, our approach is to define
the Fourier series representation for periodic sequences
and to study the properties of such representations.
Then we repeat essentially the same derivations, assuming
that the sequence to be represented is a
finite-length sequence."

Notice their initial words: "Although several points of
view can be taken ..."

[-Rick-]
```
```On Thu, 01 Aug 2013 17:05:11 -0700, robert bristow-johnson
<rbj@audioimagination.com> wrote:

>On 8/1/13 3:57 PM, Eric Jacobsen wrote:
>> On Thu, 01 Aug 2013 14:43:37 -0700, robert bristow-johnson
>> <rbj@audioimagination.com>  wrote:
>>
>>> On 8/1/13 2:06 PM, Eric Jacobsen wrote:
>>>> On Thu, 01 Aug 2013 13:42:19 -0700, robert bristow-johnson
>>>> <rbj@audioimagination.com>   wrote:
>>>>
>>>>> On 8/1/13 1:25 PM, Tim Wescott wrote:
>>> ...
>>>>>>    The math just is, and if you're getting the right
>>>>>> results you are, ipso facto, using it correctly.
>>>>>
>>>>> fully agree with that.  and, unless there is no shifting nor convolution
>>>>> (in time or frequency) going on (which means the only property or
>>>>> theorem of the DFT one can make use of is Linearity) then the *only*
>>>>> (and this "only" is me being inflexible) correct mathematics is to
>>>>> either explicitly periodically extend the data:
>>>>>
>>>>>       x[n+N] = x[n]   or   X[k+N] = X[k]
>>>>>
>>>>> or to implicitly periodically extend the data:
>>>>>
>>>>>       x[n mod N]    or   X[k mod N]
>>>>>
>>>>>
>>>>> either way you're periodically extending the data, the math says so, and
>>>>> it seems to me silly to deny it.  there is no way to get away from it.
>>>>>    "resistance is futile."
>>>>
>>>> Except for the numerous counter-examples that have been pointed out to
>>>
>>> you have numerous counter-examples which involve shifting the data or
>>> convolving the data (or their counterparts in the other domain which
>>> means multiplying by something that is not constant) where you can
>>> ignore both
>>>
>>>        x[n+N] = x[n]
>>>
>>> or
>>>
>>>        x[n mod N]
>>>
>>> and the X[k] counterparts?
>>>
>>> would you mind repeating just one of those counter examples?  and in
>>> these numerous counter examples you are able to show mathematical
>>> correctness while ignoring x[n+N] = x[n] and x[n mod N] (or the same
>>> thing in X[k])?
>>
>> It's been said many times already.   I've long grown tired of
>> repeating them over again to you.
>
>many words were said.  and the mathematical case was never made by you
>(nor Dale nor anyone else).  not even with that .pdf you sent (after my
>.pdf because you didn't like ASCII math on USENET).  i've grown tired of
>the whole thing too, but the math doesn't care.

>and it was about the same question.  i would like to see *any* shift
>operation done in the context of the DFT that isn't circular.  and you
>cannot provide such an example because the math won't let you.

You just cannot recognize that this has all been done before by so
many other people.  I cannot help that you continually, over decades,
refuse to see arguments other than your own.  It does no harm to
anyone else that you do this other than the periodic pest you make of
yourself over it, so its no big deal.

>>> zero-padding is a questionable counter-example because if you claim that
>>> the zero shifted in isn't the same zero that was shifted out the other
>>> end, then i would dispute that.
>>
>> You dispute a lot of things.
>
>perhaps, but about the topic at hand, i only dispute only one thing.
>well, maybe two.  i dispute:
>
>1.  that, regarding the DFT mapping, when any operation is done in one
>domain that causes shifting in the other domain, that this shifting is
>anything other than circular.
>
>2.  that the DFT, itself, is responsible (inherently or in any other
>manner) for the effects of windowing.  even rectangular windowing.

The DFT has no responsibilities.   It is not an entity.

the
>effects of windowing happens when you actually window the data.  in the
>simple case of rectangular windowing, it is when you yank N samples out
>if a stream or a longer sequence of samples.  the windowing happens
>before the DFT ever sees the data.

The DFT cannot "see" anything.

Such anthropomorphisms come from one's own interpretation and
assignment and not from any actual properties of the transform.
Other people have equally valid interpretations other than your own.
I do not know why you think everyone should have to see it only the
way you do, and that they are wrong if they do not.

>other than that, i don't think there is a dispute.
>
>
>--
>
>r b-j                  rbj@audioimagination.com
>
>"Imagination is more important than knowledge."
>
>

Eric Jacobsen
Anchor Hill Communications
http://www.anchorhill.com
```
```On Fri, 02 Aug 2013 05:30:21 -0700, Rick Lyons
<R.Lyons@_BOGUS_ieee.org> wrote:

>On Thu, 01 Aug 2013 23:09:14 GMT, eric.jacobsen@ieee.org (Eric
>Jacobsen) wrote:
>
>>On Thu, 1 Aug 2013 22:35:28 +0000 (UTC), glen herrmannsfeldt
>><gah@ugcs.caltech.edu> wrote:
>>
>>>Tim Wescott <tim@seemywebsite.really> wrote:
>>>> On Thu, 01 Aug 2013 13:42:19 -0700, robert bristow-johnson wrote:
>>>
>>>(snip)
>>>
>>>>> either way you're periodically extending the data, the math says so, and
>>>>> it seems to me silly to deny it.  there is no way to get away from it.
>>>>>   "resistance is futile."
>>>
>>>> It works either if you periodically extend the data, or if you say that
>>>> the data set is finite.
>>>
>>>> A pair of examples pertaining to the Fourier series comes from electrical
>>>> engineering, one involving periodic phenomena, and the other involving
>>>> bounded phenomena.
>>>
>>>> The periodic example is the use of the Fourier series to find the
>>>> harmonics of a periodic waveform.  This is your "periodic
>>>> extension" case.
>>>
>>>> The aperiodic example (and I'm rustier at this one, but I remember it
>>>> being done) is in the use of the Fourier series to find the potential
>>>> variation within a square box that's bounded by conductors.  In this case
>>>> there is no periodicity -- there could be anything outside the box, the
>>>> problem stops as soon as you hit that metal wall with its imposed voltage.
>>>
>>>OK, but consider a related problem, either in 1D (signal on a coax
>>>cable) or 3D (EM wave in a metal box). When the wave hits the boundary,
>>>there is a reflection. The solution looks the same as if the original
>>>one went through the boundary, and another comes from the outside
>>>into the problem at the boundary.
>>>
>>>Or, consider when you look at the reflection of an object in a
>>>mirror, it "looks" the same as it would if there was no mirror,
>>>and an actual object on the other side.
>>
>>For many of us there are multiple points of view that are valid and
>>provide the same results.   Personally, I think the windowed or
>>periodically-extended or boundary-condition points of view are
>>functionally equivalent and one can use whichever point of view suits
>>them best.   Personally, I think the fullest undertanding comes from
>>seeing how all of these points of view work and how they all
>>reconcile.
>>
>>The disagreement seems to center only with those who insist that only
>>a single point of view is correct or mathematically supportable.  I
>>think that is incorrect.   O&S, in both Digital Signal Processing and
>>Discrete-Time Signal Processing, take great pains to explain that
>>there are multiple approaches that work before embarking on one method
>>to develop the explanations of the transforms.   In Discrete Time
>>Signal Processing in an early chapter (Ch. 3. in my first edition)
>>they go almost all the way to deriving the DFT with no required
>>assumption of periodicity on the input.   I filled in the last small
>>step in an article here:
>>
>>http://www.dsprelated.com/showarticle/175.php
>>
>>In other parts of both texts they use an assumption of periodicity on
>>the input to finish the development because there are some benefits to
>>doing so in an explanatory text.   Again, that is done after care is
>>taken to explain that that is not the only approach.
>>
>>The various approaches work and can be used to come to the same
>>results as the others.   I think people should use whichever approach
>>makes the most sense to them personally, but I think it is incorrect
>>to say that the other approaches are incorrect or incomplete or not
>>mathematically supported or sound.
>>
>>>> The math fits in either case.  So, either view is equally valid.
>>
>>This.
>>
>>>-- glen
>>
>>Eric Jacobsen
>
>Hi Eric,
>   I'm too lazy to read all the posts in this
>thread but, for what it's worth, at the beginning of their
>chapter titled: "The Discrete Fourier Transform", on page 541 of
>their 2nd Edition of "Discrete-Time Signal Processing",
>Oppenheim & Schafer state:
>
>  "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. While this representation is
>  important in its own right, we are most often interested
>  in the application of Fourier series results to the
>  representation of finite-length sequences. We accomplish
>  this by constructing a periodic sequence for which each
>  period is identical to the finite-length sequence. As we
>  will see. the Fourier series representation of the
>  periodic sequence corresponds to the DFT of the
>  finite-length sequence. Thus, our approach is to define
>  the Fourier series representation for periodic sequences
>  and to study the properties of such representations.
>  Then we repeat essentially the same derivations, assuming
>  that the sequence to be represented is a
>  finite-length sequence."
>
>
>Notice their initial words: "Although several points of
>view can be taken ..."
>
>[-Rick-]

That's the text I was referring to.   There is similar text in O&S's
"Digital Signal Processing".

Eric Jacobsen
Anchor Hill Communications
http://www.anchorhill.com
```
```On Thu, 01 Aug 2013 20:47:23 -0400, Randy Yates
<yates@digitalsignallabs.com> wrote:

>robert bristow-johnson <rbj@audioimagination.com> writes:
>
>> On 8/1/13 3:26 PM, dbd wrote:
>> [...]
>>> A better graphic presentation of some of the transform relationships is available at:
>>> http://www-mmsp.ece.mcgill.ca/Documents/Reports/2011/KabalR2011c.pdf
>>
>> wow.  that's a nice graphic.
>
>LaTeX does such a nice job!
>
>> i can understand when Eric or someone else poo-poos my quoting of O&S
>> "scripture" as evidence, because quoting anyone is not evidence of
>> anything except of what these person say.  but i find it hard when i
>> quote O&S simple as someone else (perhaps someone else of recognized
>> authority) are saying the same thing as me.  and then you have said
>> something like "No it isn't, you're misrepresenting O&S."  and that
>> just blows my mind.
>>
>> now *here* you are using this (apparently excellent) reference to
>> support your position and it doesn't.  Kabal and i are saying the same
>> thing.  he has "DFT" in that chart.  there is no "DFS" on that chart
>> at all.  now here's the kernel:  [[You cannot even pass x[n] to the
>> DFT in that chart without first turning it into ~x[n].  There *is* no
>> non-periodic x[n] going into Kabal's DFT just as there is no
>> non-periodic x(t) going into his FS.]]
>>
>> thanks for the reference, Dale.  i haven't read too much of the text
>> (just looked at the equations), but it appears at first glance to
>> Oppenheim and Schafer).
>>
>> note also that *everywhere* that there is sampling in one domain,
>> there is necessarily periodic extension (the overlap-adding periodic
>> extension) is the reciprocal domain.  discrete in one domain (assuming
>> _uniformly_ sampled) always *always* *always* means periodic in the
>> other domain.
>>
>> always.
>>
>> and, Tim, that's what i mean about being inflexible.  but i'm just the
>> messenger.  it's the _math_ that is being inflexible.
>>
>> stubborn math.  wish it would loosen up a bit.
>
>Robert, Dale, et al., let me ask a question. In the "loosest" sense (the
>one with the least amount of interpretation), the output of the DFT is a
>set of complex numbers. It's just a set of stinkin' numbers. Do you
>agree? Of course you agree; you HAVE to agree - it's a fact.
>
>In my opinion, the periodicity thing only comes in when you start making
>assumptions about what those numbers _mean_, and THAT is open for
>interpretation. For example, if interpret them as the points of a
>discrete spectrum, then the corresponding signal represented NECESSARILY
>has to be periodic.
>
>OK, so we're done, right?

Are they DFS or DFT coefficients?  ;)

Application of a window to the input gives the same result for a DFS
or DFT, regardless of whether the input was periodic over N.  Hence

A DFT over N points gives the same result regardless of whether a
length-N window was applied previously to the data or not or whether
the data is periodic over N or not.

So whether it even matters is completely up to interpretation.  Many
of us think that since the various points of view are mathematically
supportable (despite the inability of some to see this) and arrive at
the same results that it doesnt matter.   The dissenting argument is
that only one point of view is correct and everyone else who does not
share that point of view is wrong.   At least that's been my
interpretation over the last way-too-many-years that this has been
going on.

I have yet to see any examples of someone getting an incorrect result
or coming to a wrong conclusion because of which point of view that
they used, so I don't think it matters.

>--
>Randy Yates
>Digital Signal Labs
>http://www.digitalsignallabs.com

Eric Jacobsen
Anchor Hill Communications
http://www.anchorhill.com
```
```On 8/2/13 8:50 AM, Eric Jacobsen wrote:
> On Thu, 01 Aug 2013 17:05:11 -0700, robert bristow-johnson
> <rbj@audioimagination.com>  wrote:
>
>> On 8/1/13 3:57 PM, Eric Jacobsen wrote:
>>> On Thu, 01 Aug 2013 14:43:37 -0700, robert bristow-johnson
>>> <rbj@audioimagination.com>   wrote:
>>>
>>>> On 8/1/13 2:06 PM, Eric Jacobsen wrote:
>>>>> On Thu, 01 Aug 2013 13:42:19 -0700, robert bristow-johnson
>>>>> <rbj@audioimagination.com>    wrote:
>>>>>
>>>>>> On 8/1/13 1:25 PM, Tim Wescott wrote:
>>>> ...
>>>>>>>     The math just is, and if you're getting the right
>>>>>>> results you are, ipso facto, using it correctly.
>>>>>>
>>>>>> fully agree with that.  and, unless there is no shifting nor convolution
>>>>>> (in time or frequency) going on (which means the only property or
>>>>>> theorem of the DFT one can make use of is Linearity) then the *only*
>>>>>> (and this "only" is me being inflexible) correct mathematics is to
>>>>>> either explicitly periodically extend the data:
>>>>>>
>>>>>>        x[n+N] = x[n]   or   X[k+N] = X[k]
>>>>>>
>>>>>> or to implicitly periodically extend the data:
>>>>>>
>>>>>>        x[n mod N]    or   X[k mod N]
>>>>>>
>>>>>>
>>>>>> either way you're periodically extending the data, the math says so, and
>>>>>> it seems to me silly to deny it.  there is no way to get away from it.
>>>>>>     "resistance is futile."
>>>>>
>>>>> Except for the numerous counter-examples that have been pointed out to
>>>>
>>>> you have numerous counter-examples which involve shifting the data or
>>>> convolving the data (or their counterparts in the other domain which
>>>> means multiplying by something that is not constant) where you can
>>>> ignore both
>>>>
>>>>         x[n+N] = x[n]
>>>>
>>>> or
>>>>
>>>>         x[n mod N]
>>>>
>>>> and the X[k] counterparts?
>>>>
>>>> would you mind repeating just one of those counter examples?  and in
>>>> these numerous counter examples you are able to show mathematical
>>>> correctness while ignoring x[n+N] = x[n] and x[n mod N] (or the same
>>>> thing in X[k])?
>>>
>>> It's been said many times already.   I've long grown tired of
>>> repeating them over again to you.
>>
>> many words were said.  and the mathematical case was never made by you
>> (nor Dale nor anyone else).  not even with that .pdf you sent (after my
>> .pdf because you didn't like ASCII math on USENET).  i've grown tired of
>> the whole thing too, but the math doesn't care.
>
>> and it was about the same question.  i would like to see *any* shift
>> operation done in the context of the DFT that isn't circular.  and you
>> cannot provide such an example because the math won't let you.
>
> You just cannot recognize that this has all been done before by so
> many other people.  I cannot help that you continually, over decades,
> refuse to see arguments other than your own.

not seeing an argument isn't the same as rejecting it.  and presenting
an argument is not the same as defending it.

>  It does no harm to
> anyone else that you do this other than the periodic pest you make of
> yourself over it, so its no big deal.
>
>>>> zero-padding is a questionable counter-example because if you claim that
>>>> the zero shifted in isn't the same zero that was shifted out the other
>>>> end, then i would dispute that.
>>>
>>> You dispute a lot of things.
>>
>> perhaps, but about the topic at hand, i only dispute only one thing.
>> well, maybe two.  i dispute:
>>
>> 1.  that, regarding the DFT mapping, when any operation is done in one
>> domain that causes shifting in the other domain, that this shifting is
>> anything other than circular.

so are you stipulating to number 1?  (evidently not to number 2.)

>>
>> 2.  that the DFT, itself, is responsible (inherently or in any other
>> manner) for the effects of windowing.  even rectangular windowing.
>
> The DFT has no responsibilities.   It is not an entity.

you sure about that?  the DFT is a "thing" and at least when i look up
"entity", it pretty much means "thing" and the thing need not be
conscious nor sentient nor animate.  every online source (which was only
Wikipedia and google) i looked at seems to agree.

the DFT is an algorithm, right?  can we agree that algorithms perform
if we agree on those 3 stipulations, do you say that the word
"responsibility" cannot be applied to a "thing" (whether you count it as
an "entity" or not) that performs some task to some set of specifications?

i mean, what is the meaning of something like an API or even simply a
"plan" (involving multiple "entities") without a concept of spec'ing out
"responsibility".  different tasks and algs do different things and
expect (there i go anthropomorphizing them algs again) other things
(like their input) to be a certain way in order to work.

and another sense of "responsibility" vis-a-vis inanimate things is that
of *cause* (the first half of "cause and effect").

*something* is causing that scalloping in the spectrum and that
something is windowing.  and that windowing happens to the data before
the data goes to the DFT.  the DFT just tells you what you got in the
discrete-frequency domain.  (and that's doing pretty good for a non-entity.)

>>    the
>> effects of windowing happens when you actually window the data.  in the
>> simple case of rectangular windowing, it is when you yank N samples out
>> if a stream or a longer sequence of samples.  the windowing happens
>> before the DFT ever sees the data.
>
> The DFT cannot "see" anything.

well, for that matter than neither do we.  we're just biochemical
mechanisms that just react (oh, that's too anthropomorphic) errr,
respond to stimulus.

the DFT sees its input.  the DFT sees x[0], x[1], x[2] ... x[N-1] which
came from samples  x[m+0], x[m+1], x[m+2] ... x[m+N-1] from some stream
or longer sequence of data.  the DFT does *not* see x[m-1] nor x[m-2]
nor x[m+N] nor x[m+N+1].

> Such anthropomorphisms come from one's own interpretation and
> assignment and not from any actual properties of the transform.

they're just words.  they're words that start with one meaning, and from
usage get applied metaphorically to similar situations involving objects
of a different group than in the original use.

are you saying that C++ (or pick your OOP) classes don't inherit because
they are not living things?  don't have DNA?

> Other people have equally valid interpretations other than your own.
> I do not know why you think everyone should have to see it only the
> way you do, and that they are wrong if they do not.

if they (or you) say that, when using the DFT, if they apply an
operation in one domain (e.g. multiply by something non-constant) that
causes shifting in the other domain, and that their interpretation of
the DFT allows them to consider that this shifting is not circular, then
they (or you) are wrong.  that's a mathematical fact and is no mere
interpretation.

to mathematically depict this circular shifting, i have seen *two*
notations and no others (but if you see another, we can talk about it).

either you must say x[n+N] = x[n]  or you must use  x[n mod_N] (or the
only property of the DFT that counts is linearity because any other
property depends explicitly on this periodic extension).  now the first
means explicitly periodic extension and the latter means the same damn
thing.  how can you say that this modulo index addressing is not one and
the same as periodic extension?  even if you have another notation for
it, because of mathematical necessity, it must mean the same thing as
one of these two notations.  it must also mean periodically extended.

Eric, i can try to restate the issues without anthropomorphizing (and,
in my opinion, the language would get more clunky *not* better), but i
really think you're tossing up a strawman to hide behind when you point
to such illustration as defective and then use that to avoid answering
the question that you cannot for substantive reasons.

--

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

```