# Relationship between z and Fourier transforms

Started by July 30, 2013
```On 7/31/13 8:25 AM, Bhaskar Bhattacharya wrote:
> You can derive at least one version of the Z-transform from the DFT.
> Compute the inverse DFT and get the samples (of the same length as the
> DFT). Once you know the samples,

what do you know about the samples, x[n] where n is less than 0 or
greater than N-1?

> you know the Z-transform, right?

if you know all of the samples, you know the ZT.  but you haven't gotten
me to the place where i am convinced that you know all of the samples.

--

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

```
```On 7/31/13 8:37 AM, Tim Wescott wrote:
> On Wed, 31 Jul 2013 10:30:52 -0400, Randy Yates wrote:
>
>> Tim Wescott<tim@seemywebsite.please>  writes:
>>
>>> On Wed, 31 Jul 2013 04:39:35 +0000, Eric Jacobsen wrote:
>>>
>>>> On Tue, 30 Jul 2013 20:59:37 -0500, Tim Wescott
>>>> <tim@seemywebsite.please>  wrote:
>>>>
>>>>> On Tue, 30 Jul 2013 19:51:15 -0500, commsignal wrote:
>>>>>
>>>>>> It's sort of a dumb question but I must ask it. If we have access to
>>>>>> the z-transform of a sequence, the DFT is the sampled version of
>>>>>> z-transform evaluated at the unit circle. Consider the opposite
>>>>>> case: if we have the DFT, is there *any* information we can deduce
>>>>>> about the z-transform of the sequence (or the location of poles and
>>>>>> zeroes)?
>>>>>>
>>>>>> _____________________________
>>>>>> Posted through www.DSPRelated.com
>>>>>
>>>>> If it's a truly repetitive signal (or one that is only defined over a
>>>>> finite span) then in theory the DFT tells you everything.
>>>>>
>>>>> If it is not repetitive, then the DFT tells you everything about the
>>>>> signal after it's been truncated and possibly windowed -- but
>>>>> information was lost in the windowing and truncating part.
>>>>
>>>> I disagree on the bit about losing information.   It is not necessary
>>>> to assume periodicity to derive the behavior of the DFT.

that statement, Eric, is disputed.  the properties of a signal is one
thing, the properties of a transformation or mapping is another.

>>>>   You don't
>>>> have any information about the signal outside of the window, but
>>>> that's always true, whether you assume periodicity or not, with
>>>> practical signals.

that statement is true.  it's correct because it's about the *signal*
(what it is or is not outside of the window).  it's not a statement
about the operator (the DFT).

>>> If you're talking z transforms you're talking about time from zero to
>>> infinity, so getting some truncated version of the signal is losing you
>>> information.  This, in turn, means that just knowing a DFT doesn't give
>>> you all the information that's in the signal.
>>>
>>> Compare with the discrete-time Fourier transform which does look at the
>>> signal over all time, and preserves all the information about that
>>> signal.  This means that with a discrete-time Fourier transform in
>>> hand, you can always get a z transform, because in the worst case you
>>> can always take the inverse transform, then take the z transform of
>>> that.
>>>
>>> The reason that I stressed the point where information is lost is
>>> because it is useful to remember that the DFT (actually all the flavors
>>> of the Fourier transform) conserve all the information about the
>>> transformed signal.  If you're going to hang on to that, and you're
>>> going to account for the information that's lost when you truncate,
>>> window, and DFT, then you have to understand that the modification to
>>> the signal is happening before the DFT, not during.

this was precisely the issue that i had been harping about and the
early, early beginning of the argument (sometime in the mid 90's, i
remember Eric and i had our first little brush with it).  then the
articulation that i disputed was something about the DFT inherently
windowing the input data (with a rectangular window of length N).  i was
saying that the windowing happens at a step before the DFT (and the
effects of that windowing must be attributed to that step before the DFT
ever sees the data).  and then i said that what the DFT inherently does
with the data, previously windowed (or otherwise yanked out of a stream
of samples), is periodically extend it.

>> And there are 5 angels on a pin head...
>>
>> What is it about this topic? It reduces us to moths banging ourselves
>> onto a bright light...
>
> Nah, that's the debate over whether or not the input to the DFT is of a
> cyclical nature.

the input to the DFT is the input.  there is no additional qualifier
that i add to that.  the issue that we fight about is what the DFT
*does* to that input.  the conservative observation that i make that is
apparently controversial is: "The DFT periodically extends the data
passed to it because the DFT fits a basis to the data that is periodic."

--

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

```
```Tim Wescott <tim@seemywebsite.please> wrote:

(snip, someone wrote)
>>>> It's sort of a dumb question but I must ask it. If we have access to
>>>> the z-transform of a sequence, the DFT is the sampled version of
>>>> z-transform evaluated at the unit circle. Consider the opposite case:
>>>> if we have the DFT, is there *any* information we can deduce about the
>>>> z-transform of the sequence (or the location of poles and zeroes)?

(snip)

> If you're talking z transforms you're talking about time from zero to
> infinity, so getting some truncated version of the signal is losing you
> information.  This, in turn, means that just knowing a DFT doesn't give
> you all the information that's in the signal.

Infinite signals and series are fine in books, but not useful
in computers.

A book will tell you the Fourier series for a square wave or
triangle wave as an infinite sum of sines. For DSP, we only have
a finite number of samples. Could be a large number, though.
An 80 minute CD holds 80*3600*44100 samples.

As the data gets longer (more samples) the DFT has more terms,
in proportion to the length of the signal.

Seems to me that Nyquist sampling theory requires an infinite
number of samples, but we seem to be doing just fine using a
finite approximation to it.

> Compare with the discrete-time Fourier transform which does look at the
> signal over all time, and preserves all the information about that
> signal.  This means that with a discrete-time Fourier transform in hand,
> you can always get a z transform, because in the worst case you can
> always take the inverse transform, then take the z transform of that.

> The reason that I stressed the point where information is lost is because
> it is useful to remember that the DFT (actually all the flavors of the
> Fourier transform) conserve all the information about the transformed
> signal.  If you're going to hang on to that, and you're going to account
> for the information that's lost when you truncate, window, and DFT, then
> you have to understand that the modification to the signal is happening
> before the DFT, not during.

Yes.

-- glen
```
```On Wed, 31 Jul 2013 11:13:42 -0700, robert bristow-johnson wrote:

> On 7/31/13 8:37 AM, Tim Wescott wrote:
>> On Wed, 31 Jul 2013 10:30:52 -0400, Randy Yates wrote:
>>
>>> Tim Wescott<tim@seemywebsite.please>  writes:
>>>
>>>> On Wed, 31 Jul 2013 04:39:35 +0000, Eric Jacobsen wrote:
>>>>
>>>>> On Tue, 30 Jul 2013 20:59:37 -0500, Tim Wescott
>>>>> <tim@seemywebsite.please>  wrote:
>>>>>
>>>>>> On Tue, 30 Jul 2013 19:51:15 -0500, commsignal wrote:
>>>>>>
>>>>>>> It's sort of a dumb question but I must ask it. If we have access
>>>>>>> to the z-transform of a sequence, the DFT is the sampled version
>>>>>>> of z-transform evaluated at the unit circle. Consider the opposite
>>>>>>> case: if we have the DFT, is there *any* information we can deduce
>>>>>>> about the z-transform of the sequence (or the location of poles
>>>>>>> and zeroes)?
>>>>>>>
>>>>>>> _____________________________
>>>>>>> Posted through www.DSPRelated.com
>>>>>>
>>>>>> If it's a truly repetitive signal (or one that is only defined over
>>>>>> a finite span) then in theory the DFT tells you everything.
>>>>>>
>>>>>> If it is not repetitive, then the DFT tells you everything about
>>>>>> the signal after it's been truncated and possibly windowed -- but
>>>>>> information was lost in the windowing and truncating part.
>>>>>
>>>>> I disagree on the bit about losing information.   It is not
>>>>> necessary to assume periodicity to derive the behavior of the DFT.
>
> that statement, Eric, is disputed.  the properties of a signal is one
> thing, the properties of a transformation or mapping is another.
>
>>>>>   You don't
>>>>> have any information about the signal outside of the window, but
>>>>> that's always true, whether you assume periodicity or not, with
>>>>> practical signals.
>
> that statement is true.  it's correct because it's about the *signal*
> (what it is or is not outside of the window).  it's not a statement
> about the operator (the DFT).
>
>>>> If you're talking z transforms you're talking about time from zero to
>>>> infinity, so getting some truncated version of the signal is losing
>>>> you information.  This, in turn, means that just knowing a DFT
>>>> doesn't give you all the information that's in the signal.
>>>>
>>>> Compare with the discrete-time Fourier transform which does look at
>>>> the signal over all time, and preserves all the information about
>>>> that signal.  This means that with a discrete-time Fourier transform
>>>> in hand, you can always get a z transform, because in the worst case
>>>> you can always take the inverse transform, then take the z transform
>>>> of that.
>>>>
>>>> The reason that I stressed the point where information is lost is
>>>> because it is useful to remember that the DFT (actually all the
>>>> flavors of the Fourier transform) conserve all the information about
>>>> the transformed signal.  If you're going to hang on to that, and
>>>> you're going to account for the information that's lost when you
>>>> truncate, window, and DFT, then you have to understand that the
>>>> modification to the signal is happening before the DFT, not during.
>
> this was precisely the issue that i had been harping about and the
> early, early beginning of the argument (sometime in the mid 90's, i
> remember Eric and i had our first little brush with it).  then the
> articulation that i disputed was something about the DFT inherently
> windowing the input data (with a rectangular window of length N).  i was
> saying that the windowing happens at a step before the DFT (and the
> effects of that windowing must be attributed to that step before the DFT
> ever sees the data).  and then i said that what the DFT inherently does
> with the data, previously windowed (or otherwise yanked out of a stream
> of samples), is periodically extend it.
>
>>> And there are 5 angels on a pin head...
>>>
>>> What is it about this topic? It reduces us to moths banging ourselves
>>> onto a bright light...
>>
>> Nah, that's the debate over whether or not the input to the DFT is of a
>> cyclical nature.
>
> the input to the DFT is the input.  there is no additional qualifier
> that i add to that.  the issue that we fight about is what the DFT
> *does* to that input.  the conservative observation that i make that is
> apparently controversial is: "The DFT periodically extends the data
> passed to it because the DFT fits a basis to the data that is periodic."

Oh.  _That_ argument.

You know, you can call a truce by saying that the DFT only operates on
the data passed to it, but it does so in a way that is _ambiguous_ about
whether the result (or the input) is periodic.

Then you can just stand back and smirk while everyone else argues.

--

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

```
```On Wed, 31 Jul 2013 12:21:42 -0500, Tim Wescott
<tim@seemywebsite.really> wrote:

>On Wed, 31 Jul 2013 16:03:47 +0000, Eric Jacobsen wrote:
>
>> On Wed, 31 Jul 2013 09:23:19 -0500, Tim Wescott
>> <tim@seemywebsite.please> wrote:
>>
>>>On Wed, 31 Jul 2013 04:39:35 +0000, Eric Jacobsen wrote:
>>>
>>>> On Tue, 30 Jul 2013 20:59:37 -0500, Tim Wescott
>>>> <tim@seemywebsite.please> wrote:
>>>>
>>>>>On Tue, 30 Jul 2013 19:51:15 -0500, commsignal wrote:
>>>>>
>>>>>> It's sort of a dumb question but I must ask it. If we have access to
>>>>>> the z-transform of a sequence, the DFT is the sampled version of
>>>>>> z-transform evaluated at the unit circle. Consider the opposite
>>>>>> case: if we have the DFT, is there *any* information we can deduce
>>>>>> about the z-transform of the sequence (or the location of poles and
>>>>>> zeroes)?
>>>>>>
>>>>>> _____________________________
>>>>>> Posted through www.DSPRelated.com
>>>>>
>>>>>If it's a truly repetitive signal (or one that is only defined over a
>>>>>finite span) then in theory the DFT tells you everything.
>>>>>
>>>>>If it is not repetitive, then the DFT tells you everything about the
>>>>>signal after it's been truncated and possibly windowed -- but
>>>>>information was lost in the windowing and truncating part.
>>>>
>>>> I disagree on the bit about losing information.   It is not necessary
>>>> to assume periodicity to derive the behavior of the DFT.   You don't
>>>> have any information about the signal outside of the window, but
>>>> that's always true, whether you assume periodicity or not, with
>>>> practical signals.
>>>
>>>If you're talking z transforms you're talking about time from zero to
>>>infinity, so getting some truncated version of the signal is losing you
>>>information.  This, in turn, means that just knowing a DFT doesn't give
>>>you all the information that's in the signal.
>>>
>>>Compare with the discrete-time Fourier transform which does look at the
>>>signal over all time, and preserves all the information about that
>>>signal.  This means that with a discrete-time Fourier transform in hand,
>>>you can always get a z transform, because in the worst case you can
>>>always take the inverse transform, then take the z transform of that.
>>>
>>>The reason that I stressed the point where information is lost is
>>>because it is useful to remember that the DFT (actually all the flavors
>>>of the Fourier transform) conserve all the information about the
>>>transformed signal.  If you're going to hang on to that, and you're
>>>going to account for the information that's lost when you truncate,
>>>window, and DFT, then you have to understand that the modification to
>>>the signal is happening before the DFT, not during.
>>
>> We may be in violent agreement.   My perspective was just that nothing
>> can be said about "losing" information that one didn't have, and if it
>> wasn't in the window, as far as the DFT analysis is concerned it isn't
>> lost, it was never there in the first place.   Whether it was zero
>> outside the window, periodically extended, or, more likely, something
>> else entirely, makes no difference whatsoever.
>
>Yup.  Where it makes a difference is if you have some signal that you are
>truncating for the purposes of feeding it into a DFT.  In that case there
>may be some value in knowing where the information loss is occurring.
>
>The two most practical applications of this somewhat esoteric point of
>fact would be either because you're taking the DFT of cyclical (or finite-
>length) data or because you're choosing your length and windowing.  In
>the cyclical case you want to capture exactly one cycle and not window
>(or average many cycles point by point).  In the "choose your length"
>case the analysis of the effects of windowing and truncation makes more
>sense if you know that it is, indeed, windowing and truncation that you
>should be looking at and not the effect of the DFT itself.

They're indistinguishable when the window is length-N.   One can zero
pad to less than N, so that the rectangular window is of length N-M,
and the sinx/x frequency response is consistent with the N-M length of
the resulting window.   A one reduces M the main lobe of the sinx/x
narrows consistently and predictably and when M = 0 the sinx/x
response of the "externally applied" window then becomes consistent
with a length N window.

It doesn't matter whether one thinks of the window as happening before
the DFT or as an inherent part of it.    If one has data of length M*N
and truncates the DFT basis functions to length N, the sinx/x output
frequency response will be consistent with the length N window, even
though the data is longer and the basis functions have had the window
applied inside the DFT instead.   It doesn't matter the point of view,
the results are the same.    Those that insist that only one point of
view is valid are mistaken.

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

Eric Jacobsen
Anchor Hill Communications
http://www.anchorhill.com
```
```Yes. That is straightforward. I was talking about any smart deductions that
avoid this inverse and forward transforms.

>You can derive at least one version of the Z-transform from the DFT.
Compute
>the inverse DFT and get the samples (of the same length as the DFT). Once

>you know the samples, you know the Z-transform, right?
>
>- Bhaskar Bhattacharya
>
>"commsignal" <58672@dsprelated> wrote in message
>news:PqadnVGJ5roewmXMnZ2dnUVZ_uydnZ2d@giganews.com...
>> It's sort of a dumb question but I must ask it. If we have access to
the
>> z-transform of a sequence, the DFT is the sampled version of
z-transform
>> evaluated at the unit circle. Consider the opposite case: if we have
the
>> DFT, is there *any* information we can deduce about the z-transform of
the
>> sequence (or the location of poles and zeroes)?
>>
>> _____________________________
>> Posted through www.DSPRelated.com
>
>

_____________________________
Posted through www.DSPRelated.com
```
```On Wednesday, July 31, 2013 12:48:34 PM UTC-7, Tim Wescott wrote:
> ...
> Oh.  _That_ argument.
>
> You know, you can call a truce by saying that the DFT only operates on
> the data passed to it, but it does so in a way that is _ambiguous_ about
> whether the result (or the input) is periodic.
>
> Then you can just stand back and smirk while everyone else argues.
>
> --
>
> Tim Wescott

Tim

I think you are exactly right... well.. except that "_ambiguous_" should really be "#ignorant#". Few people understand the true depth and meaningfulness of this distinction but I think I can rant at you about it until you should finally be allowed to post again in only a few weeks. Ready?

Dale B. Dalrymple
```
```On Wed, 31 Jul 2013 21:10:18 -0700, dbd wrote:

> On Wednesday, July 31, 2013 12:48:34 PM UTC-7, Tim Wescott wrote:
>> ...
>> Oh.  _That_ argument.
>>
>> You know, you can call a truce by saying that the DFT only operates on
>> the data passed to it, but it does so in a way that is _ambiguous_
>> about whether the result (or the input) is periodic.
>>
>> Then you can just stand back and smirk while everyone else argues.
>>
>> --
>>
>> Tim Wescott
>
> Tim
>
> I think you are exactly right... well.. except that "_ambiguous_" should
> really be "#ignorant#". Few people understand the true depth and
> meaningfulness of this distinction but I think I can rant at you about
> it until you should finally be allowed to post again in only a few
> weeks. Ready?

I detect either a minor semantic difference or some really deep
philosophical difference.

To me, "ignorant" implies cognition, which I do not impute to any
algorithms.

So either you use the word differently from me (minor semantic
difference), or you think that mathematics thinks (major philosophical
difference).

I wonder if the OP ever got his question answered?

--

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

```
```On 7/31/13 9:51 PM, Tim Wescott wrote:
> >
> So either you use the word differently from me (minor semantic
> difference), or you think that mathematics thinks (major philosophical
> difference).

well, for a materialist (or a physicalist) we don't really think either.
just biochemical responses.

sometimes i like to anthropomorphize an algorithm.  like what does the
alg know (from its input) and what doesn't it know?  this is sorta about
what an alg "assumes".

--

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

```
```On Wed, 31 Jul 2013 22:06:16 -0700, robert bristow-johnson wrote:

> On 7/31/13 9:51 PM, Tim Wescott wrote:
>> >
>> So either you use the word differently from me (minor semantic
>> difference), or you think that mathematics thinks (major philosophical
>> difference).
>
> well, for a materialist (or a physicalist) we don't really think either.
>   just biochemical responses.
>
> sometimes i like to anthropomorphize an algorithm.  like what does the
> alg know (from its input) and what doesn't it know?  this is sorta about
> what an alg "assumes".

When I anthropomorphize an algorithm I bear in mind that I'm doing it.

Granted, sometimes I forget that I'm just a mass of biochemical stimulus
and response, and I anthropomorphize myself -- I hope that I might be
excused for this.

--

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

```