# Relationship between z and Fourier transforms

Started by July 30, 2013
```On Tue, 06 Aug 2013 19:01:44 -0500, "commsignal" <58672@dsprelated>
wrote:

>Both Eric's and r-b-j's responses above are very very interesting. Now I
>can see why you guys are caught in an infinite loop regarding this
>discussion. I can think of this concept as an object in an unstable
>equilibrium, like a ball on the top of the hill. With a hint of wind on
>either side, it just rolls down either side and never comes back. I guess
>the two positions taken over this concept must be the same.
>
>So, are there any 'conversions' over this topic in the past decade or so?
>If yes, what argument convinced him/her?
>
>r-b-j, I'm a he, and the reason I don't use my real name is just to avoid
>labelling the kind of stupidity I ask in questions to my real self.
>
>Finally, for the 'other' group, even practically speaking, we admit this
>inherent periodcity in doing DFT in, say, OFDM or multicarrier systems,
>when we repeat the last part of the sequence in *time-domain* before taking
>the DFT to make it *look* periodic. What does that teach us? What is that
>DFT assuming for that time-domain signal?

It doesn't assume, anything, it just processes the numbers.   There's
an easy way to look at the OFDM cyclic extension case:

The cyclic extension isn't trying to make the signal "look" periodic,
it's exploiting the fact that there is circularity, or periodicity, or
whatever wishes to call it, in the output of the transform due to the
input being sampled.  In OFDM the output of the transform is
transmitted and the inverse is taken in the receiver.  It doesn't
matter which is the forward or inverse transform (generally, unless
you're trying to be compatible with an existing system), and many
people think of OFDM as transmitting the frequency-domain of the data
in the time-domain, and from that point of view the transform in the
modulator is a forward transform.

Due to channel memory, the cyclic extension makes the frequency-domain
input to the inverse transform in the demodulator cyclic, as it should
be, since that's what we're expecting in the FD due to the naturally
repeating spectra.  If this *isn't* done, then the channel memory
makes the "spectrum" of the signal non-cyclic, which is not a defined
condition in the analysis of the FT.   Functionally, the effects of
the channel memory are minimized when this is done, so one doesn't
even have to think that it's really done to make all the math work out
for the theorists, it's done because it's a nice trick to minimize the
effect of the channel impairment.   There are systems that stuff zeros
in the prefix instead of a cyclic extension of the tail, and it  works
almost as well as a cyclic prefix and it has some subtle tradeoffs to
other parts of the system.

There are other ways to look at it, including that the time-domain
signal really is a time-domain signal and it can be treated as
periodic over N.   I've never seen anyone here deny that that's a
perfectly valid viewpoint (despite Robert frequently labelling people
that disagree with him on this as "periodicity deniers").   It's just
not the only valid way.  ;)

>
>
>_____________________________
>Posted through www.DSPRelated.com

Eric Jacobsen
Anchor Hill Communications
http://www.anchorhill.com
```
```
or maybe it's a confession.

On 8/6/13 6:14 PM, Eric Jacobsen wrote:
> On Tue, 06 Aug 2013 19:01:44 -0500, "commsignal"<58672@dsprelated>
> wrote:
>
>> Both Eric's and r-b-j's responses above are very very interesting. Now I
>> can see why you guys are caught in an infinite loop regarding this
>> discussion. I can think of this concept as an object in an unstable
>> equilibrium, like a ball on the top of the hill. With a hint of wind on
>> either side, it just rolls down either side and never comes back. I guess
>> the two positions taken over this concept must be the same.
>>
>> So, are there any 'conversions' over this topic in the past decade or so?
>> If yes, what argument convinced him/her?
>>
>> r-b-j, I'm a he, and the reason I don't use my real name is just to avoid
>> labelling the kind of stupidity I ask in questions to my real self.
>>
>> Finally, for the 'other' group, even practically speaking, we admit this
>> inherent periodcity in doing DFT in, say, OFDM or multicarrier systems,
>> when we repeat the last part of the sequence in *time-domain* before taking
>> the DFT to make it *look* periodic. What does that teach us? What is that
>> DFT assuming for that time-domain signal?
>
>
...
> There are other ways to look at it, including that the time-domain
> signal really is a time-domain signal and it can be treated as
> periodic over N.   I've never seen anyone here deny that that's a
> perfectly valid viewpoint (despite Robert frequently labeling people
> that disagree with him on this as "periodicity deniers").   It's just
> not the only valid way.  ;)
>

when we talk about DSP in general, it's only time that quantized.  e.g.
IIR filters don't have any frequency quantization that i'm aware of
(however knowledge of the IIR filter's order N and 2N+1 discrete
frequency response points should suffice to tell you everything you need
to know about the transfer function).

so discrete-time of infinite extent and continuous and periodic
frequency (omega in the DTFT) is the deal with DSP.  it's the dual of
good old-fashioned Fourier series with continuous time and discrete
frequency.

so there's a sense of equivalency of points of view in that them
"periodicity deniers" (who the hell said *that*?) are saying that the
DFT only evaluates

w^(n k)    where w = e^(-j 2 pi/N)

for n and k that are integers:  0 <= n < N, 0 <= k < N .

now the "periodicity believers" are affirming the Sacred Integerness of
the divine arguments, n and k.  eternally always and forevermore shall
they remain Integers.  but them periodicity believers don't have any
problem allowing n and k to be unbounded in all the heavenly realm of
real integers Z.  Z is an ikon in the Church of the Periodicity of the DFT.

and the periodicity agnostics are saying "we don't know a thing about
the nature of n and k.  hell, they could be anything, real, complex, or
matrices for all we know."

now it seems to me that there are some periodicity deniers who are a
little bit of a cafeteria denier.  one entr&#2013265929;e is k need be neither an
integer nor be limited in range.  that is imparting the attributes of
the DTFT onto the DFT.  but to compare to the DTFT (k = N*omega/(2pi) is
continuous), then they must define x[n] = 0 for all n<0 or n>=N and x[n]
is not periodically extended. (there, i said it.)  and X(k) (it's not
"X[k]" because k need not be an integer) is continuous and periodic with
period N.

and there could just as well be the periodicity deniers that can just as
well say the same thing, but reversing the roles of n and k.  now it's
x(n) that is continuous and periodic and X[k] that is discrete and zero
for all k outside of the 0 <= k < N limits.  hmmm, just like Fourier series.

anyone ever see this two-part South Park episode "Go, God, Go!"?  it's
just wonderful.  because Cartman doesn't have patience to wait for the
Wii to come out, he gets Butters to assist him in getting frozen out in
the Colorado mountains, then an avalance happens, frozen Cartman is
buried and 500 years later Cartman is thawed out by ultra-modern humans.

but in the future, religion has been disposed of, Richard Dawkins is the
ancient Prophet of Science, and the future atheists were fighting among
themselves about what the true Science teaches.  Cartman would say
"Science Dammit!" similarly to how he used to say "goddammit".  the
future atheists would say "Science be praised!"  and "Science be with you".

so i wonder how one chooses which denomination of periodicity atheism.
which one?  the DFT=DTFT branch where n is discrete and x[n] is zero
unless 0 <= n < N?  or shall it be the DFT=FS branch that says the exact
same thing about k and X[k]?  Catholic vs. Protestant.  Sunni vs. Shia.

the nasty right-wing conservative periodicity believers who want to
foist their beliefs upon everyone else**, them hypocrites just say
smugly, "n and k are equivalent and have equal standing before the god
of DFT mathematics".  and they also say that the DFT is neither the DTFT
nor the Fourier Series.  (but they *do* say the DFT is the same as the
DFS, for all intents and purchases.)

**("our nation was founded as a periodicity-believing nation."  "the
founding fathers were periodicity-believers, except Jefferson who was
more of a periodicity-deist."  "so we ain't votin' for no periodicity
atheist.")

--

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

```
```>On Tue, 06 Aug 2013 19:01:44 -0500, "commsignal" <58672@dsprelated>
>wrote:
>
>>Both Eric's and r-b-j's responses above are very very interesting. Now I
>>can see why you guys are caught in an infinite loop regarding this
>>discussion. I can think of this concept as an object in an unstable
>>equilibrium, like a ball on the top of the hill. With a hint of wind on
>>either side, it just rolls down either side and never comes back. I
guess
>>the two positions taken over this concept must be the same.
>>
>>So, are there any 'conversions' over this topic in the past decade or
so?
>>If yes, what argument convinced him/her?
>>
>>r-b-j, I'm a he, and the reason I don't use my real name is just to
avoid
>>labelling the kind of stupidity I ask in questions to my real self.
>>
>>Finally, for the 'other' group, even practically speaking, we admit this
>>inherent periodcity in doing DFT in, say, OFDM or multicarrier systems,
>>when we repeat the last part of the sequence in *time-domain* before
taking
>>the DFT to make it *look* periodic. What does that teach us? What is
that
>>DFT assuming for that time-domain signal?
>
>It doesn't assume, anything, it just processes the numbers.   There's
>an easy way to look at the OFDM cyclic extension case:
>
>The cyclic extension isn't trying to make the signal "look" periodic,
>it's exploiting the fact that there is circularity, or periodicity, or
>whatever wishes to call it, in the output of the transform due to the
>input being sampled.  In OFDM the output of the transform is
>transmitted and the inverse is taken in the receiver.  It doesn't
>matter which is the forward or inverse transform (generally, unless
>you're trying to be compatible with an existing system), and many
>people think of OFDM as transmitting the frequency-domain of the data
>in the time-domain, and from that point of view the transform in the
>modulator is a forward transform.
>
>Due to channel memory, the cyclic extension makes the frequency-domain
>input to the inverse transform in the demodulator cyclic, as it should
>be, since that's what we're expecting in the FD due to the naturally
>repeating spectra.  If this *isn't* done, then the channel memory
>makes the "spectrum" of the signal non-cyclic, which is not a defined
>condition in the analysis of the FT.   Functionally, the effects of
>the channel memory are minimized when this is done, so one doesn't
>even have to think that it's really done to make all the math work out
>for the theorists, it's done because it's a nice trick to minimize the
>effect of the channel impairment.   There are systems that stuff zeros
>in the prefix instead of a cyclic extension of the tail, and it  works
>almost as well as a cyclic prefix and it has some subtle tradeoffs to
>other parts of the system.
>
>There are other ways to look at it, including that the time-domain
>signal really is a time-domain signal and it can be treated as
>periodic over N.   I've never seen anyone here deny that that's a
>perfectly valid viewpoint (despite Robert frequently labelling people
>that disagree with him on this as "periodicity deniers").   It's just
>not the only valid way.  ;)

the last L (equal to channel memory) samples from the end of the block back
to the start before DFT processing. If that's not a proof of DFT assuming
periodic input, I wonder what is.

Also, in simple single-carrier frequency domain equalization too, the
cyclic prefix is used, although there is no transform taken at the
transmitter. That is definitely done to make the input to the *first*
transform at the receiver look periodic.

I agree with your last paragraph :)    However somehow, if you read your
own above post from the 'other' point of view, it surprisingly seems to
support that view in every sentence, seriously :) "If this *isn't* done,
then the channel memory makes the "spectrum" of the signal non-cyclic,
which is not a defined condition in the analysis of the FT." You're saying
it yourself.

I don't mean to unnecessarily waste your time though.

>
>>
>>
>>_____________________________
>>Posted through www.DSPRelated.com
>
>Eric Jacobsen
>Anchor Hill Communications
>http://www.anchorhill.com
>

_____________________________
Posted through www.DSPRelated.com
```
```On Wed, 07 Aug 2013 17:39:32 -0500, "commsignal" <58672@dsprelated>
wrote:

>>On Tue, 06 Aug 2013 19:01:44 -0500, "commsignal" <58672@dsprelated>
>>wrote:
>>
>>>Both Eric's and r-b-j's responses above are very very interesting. Now I
>>>can see why you guys are caught in an infinite loop regarding this
>>>discussion. I can think of this concept as an object in an unstable
>>>equilibrium, like a ball on the top of the hill. With a hint of wind on
>>>either side, it just rolls down either side and never comes back. I
>guess
>>>the two positions taken over this concept must be the same.
>>>
>>>So, are there any 'conversions' over this topic in the past decade or
>so?
>>>If yes, what argument convinced him/her?
>>>
>>>r-b-j, I'm a he, and the reason I don't use my real name is just to
>avoid
>>>labelling the kind of stupidity I ask in questions to my real self.
>>>
>>>Finally, for the 'other' group, even practically speaking, we admit this
>>>inherent periodcity in doing DFT in, say, OFDM or multicarrier systems,
>>>when we repeat the last part of the sequence in *time-domain* before
>taking
>>>the DFT to make it *look* periodic. What does that teach us? What is
>that
>>>DFT assuming for that time-domain signal?
>>
>>It doesn't assume, anything, it just processes the numbers.   There's
>>an easy way to look at the OFDM cyclic extension case:
>>
>>The cyclic extension isn't trying to make the signal "look" periodic,
>>it's exploiting the fact that there is circularity, or periodicity, or
>>whatever wishes to call it, in the output of the transform due to the
>>input being sampled.  In OFDM the output of the transform is
>>transmitted and the inverse is taken in the receiver.  It doesn't
>>matter which is the forward or inverse transform (generally, unless
>>you're trying to be compatible with an existing system), and many
>>people think of OFDM as transmitting the frequency-domain of the data
>>in the time-domain, and from that point of view the transform in the
>>modulator is a forward transform.
>>
>>Due to channel memory, the cyclic extension makes the frequency-domain
>>input to the inverse transform in the demodulator cyclic, as it should
>>be, since that's what we're expecting in the FD due to the naturally
>>repeating spectra.  If this *isn't* done, then the channel memory
>>makes the "spectrum" of the signal non-cyclic, which is not a defined
>>condition in the analysis of the FT.   Functionally, the effects of
>>the channel memory are minimized when this is done, so one doesn't
>>even have to think that it's really done to make all the math work out
>>for the theorists, it's done because it's a nice trick to minimize the
>>effect of the channel impairment.   There are systems that stuff zeros
>>in the prefix instead of a cyclic extension of the tail, and it  works
>>almost as well as a cyclic prefix and it has some subtle tradeoffs to
>>other parts of the system.
>>
>>There are other ways to look at it, including that the time-domain
>>signal really is a time-domain signal and it can be treated as
>>periodic over N.   I've never seen anyone here deny that that's a
>>perfectly valid viewpoint (despite Robert frequently labelling people
>>that disagree with him on this as "periodicity deniers").   It's just
>>not the only valid way.  ;)
>
>the last L (equal to channel memory) samples from the end of the block back
>to the start before DFT processing. If that's not a proof of DFT assuming
>periodic input, I wonder what is.

It isn't necessarily.   It does recognize that convolution with
transforms is circular rather than linear, though.   It can be
confusing with OFDM since the choice of "domain" gets mixed up pretty
easily depending on the perspective.

The cyclic prefix just makes the convolution with the channel response
behave as a circular convolution rather than a linear convolution.

Remember, though, that perhaps the more important reason for the
cyclic prefix is to prevent ISI from the previous symbol.   This is
why ZP-OFDM works, too, since the ZP prevents ISI but doesn't make the
channel convolution appear circular.   It's generally more important
to prevent the ISI than to make the convolution circular.   Since
there is great benefit  to having a guard interval between the
symbols, using the cyclic extension as the guard interval has the
benefit of making the channel convolution circular and also helping
tradeoffs with amplifiers and spectral control (compared to a ZP).

>Also, in simple single-carrier frequency domain equalization too, the
>cyclic prefix is used, although there is no transform taken at the
>transmitter. That is definitely done to make the input to the *first*
>transform at the receiver look periodic.

Again, that's mostly to prevent interference from the previous block
and to make the convolution with the channel appear to be circular
rather than linear.   FD equalization can do ZP as well, but with the

>I agree with your last paragraph :)    However somehow, if you read your
>own above post from the 'other' point of view, it surprisingly seems to
>support that view in every sentence, seriously :) "If this *isn't* done,
>then the channel memory makes the "spectrum" of the signal non-cyclic,
>which is not a defined condition in the analysis of the FT." You're saying
>it yourself.

Don't mistake acknowledging the periodicity properties of FTs, and
explointing them effectively as believing that time-domain
N-periodicity is a *necessary* analytical point-of-view.

>I don't mean to unnecessarily waste your time though.

It's not a waste if it helps you understand.

>>
>>>
>>>
>>>_____________________________
>>>Posted through www.DSPRelated.com
>>
>>Eric Jacobsen
>>Anchor Hill Communications
>>http://www.anchorhill.com
>>
>
>_____________________________
>Posted through www.DSPRelated.com

Eric Jacobsen
Anchor Hill Communications
http://www.anchorhill.com
```
```it is not a dumb question all i know it is imp............
```
```On Tue, 25 Feb 2014 18:12:16 -0800 (PST), vasavi27101992@gmail.com
wrote:

>it is not a dumb question all i know it is imp............

At this point I imagine a young student interrupted in his post,
violently pulled away from his keyboard and taken away, hooded, in
chains, by any of a variety of internet etiquette enforcement
extremists.

Regardless, the answer to the presumed complete question is easily
found by searching "Fourier Transform z-transform relation" and
examining any of the first few links.

Synopsis:

The DTFT is the Z-transform evaluated on the unit circle.

Eric Jacobsen
Anchor Hill Communications
http://www.anchorhill.com
```
```On Wed, 26 Feb 2014 03:54:08 +0000, Eric Jacobsen wrote:

> On Tue, 25 Feb 2014 18:12:16 -0800 (PST), vasavi27101992@gmail.com
> wrote:
>
>>it is not a dumb question all i know it is imp............
>
> At this point I imagine a young student interrupted in his post,
> violently pulled away from his keyboard and taken away, hooded, in
> chains, by any of a variety of internet etiquette enforcement
> extremists

It's not polite to tell someone they're not being polite.

Neener.

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

```
```On 2/26/14 3:58 PM, Tim Wescott wrote:
> On Wed, 26 Feb 2014 03:54:08 +0000, Eric Jacobsen wrote:
>
>> On Tue, 25 Feb 2014 18:12:16 -0800 (PST), vasavi27101992@gmail.com
>> wrote:
>>
>>> it is not a dumb question all i know it is imp............
>>
>> At this point I imagine a young student interrupted in his post,
>> violently pulled away from his keyboard and taken away, hooded, in
>> chains, by any of a variety of internet etiquette enforcement
>> extremists
>
> It's not polite to tell someone they're not being polite.
>
> Neener.
>

--

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

```
```On Wed, 26 Feb 2014 22:33:11 -0500, robert bristow-johnson
<rbj@audioimagination.com> wrote:

>On 2/26/14 3:58 PM, Tim Wescott wrote:
>> On Wed, 26 Feb 2014 03:54:08 +0000, Eric Jacobsen wrote:
>>
>>> On Tue, 25 Feb 2014 18:12:16 -0800 (PST), vasavi27101992@gmail.com
>>> wrote:
>>>
>>>> it is not a dumb question all i know it is imp............
>>>
>>> At this point I imagine a young student interrupted in his post,
>>> violently pulled away from his keyboard and taken away, hooded, in
>>> chains, by any of a variety of internet etiquette enforcement
>>> extremists
>>
>> It's not polite to tell someone they're not being polite.
>>
>> Neener.

Nyah, nyah... so there!

--
John
```
```On Wed, 26 Feb 2014 23:01:14 -0600, quiasmox@yahoo.com wrote:

>On Wed, 26 Feb 2014 22:33:11 -0500, robert bristow-johnson
><rbj@audioimagination.com> wrote:
>
>>On 2/26/14 3:58 PM, Tim Wescott wrote:
>>> On Wed, 26 Feb 2014 03:54:08 +0000, Eric Jacobsen wrote:
>>>
>>>> On Tue, 25 Feb 2014 18:12:16 -0800 (PST), vasavi27101992@gmail.com
>>>> wrote:
>>>>
>>>>> it is not a dumb question all i know it is imp............
>>>>
>>>> At this point I imagine a young student interrupted in his post,
>>>> violently pulled away from his keyboard and taken away, hooded, in
>>>> chains, by any of a variety of internet etiquette enforcement
>>>> extremists
>>>
>>> It's not polite to tell someone they're not being polite.
>>>
>>> Neener.
>
>Nyah, nyah... so there!
>
>--
>John

Neener neener.

Eric Jacobsen
Anchor Hill Communications
http://www.anchorhill.com
```