Relationship between z and Fourier transforms

Started by July 30, 2013
```Tim Wescott <tim@seemywebsite.please> writes:
> [...]
> I'm not sure that the concept of poles and zeros has a great deal of
> meaning in terms of a signal.  While I think you could make a case for
> saying it's still valid to talk about them, I feel that the concept of
> poles and zeros more appropriately belong to signals, not systems.

Tim,

Didn't you mean to write "systems, not signals?"
--
Randy Yates
Digital Signal Labs
http://www.digitalsignallabs.com
```
```On Wed, 31 Jul 2013 06:20:15 -0400, Randy Yates wrote:

>> [...]
>> I'm not sure that the concept of poles and zeros has a great deal of
>> meaning in terms of a signal.  While I think you could make a case for
>> saying it's still valid to talk about them, I feel that the concept of
>> poles and zeros more appropriately belong to signals, not systems.
>
> Tim,
>
> Didn't you mean to write "systems, not signals?"

Course of meant I that do to.

I just sometimes type things in reverse order.

Thanks for picking that up.

--
Tim Wescott
Control system and signal processing consulting
www.wescottdesign.com
```
```Tim Wescott <tim@seemywebsite.please> writes:

> On Wed, 31 Jul 2013 06:20:15 -0400, Randy Yates wrote:
>
>>> [...]
>>> I'm not sure that the concept of poles and zeros has a great deal of
>>> meaning in terms of a signal.  While I think you could make a case for
>>> saying it's still valid to talk about them, I feel that the concept of
>>> poles and zeros more appropriately belong to signals, not systems.
>>
>> Tim,
>>
>> Didn't you mean to write "systems, not signals?"
>
> Course of meant I that do to.
>
> I just sometimes type things in reverse order.
>
> Thanks for picking that up.

I had a wonderful, wonderful math teacher in sixth grade (Mrs. Melvin)
who, in addition to teaching us math, taught us little kernels of
wisdom, such as, "Say what you mean and MEAN what you say!"

This woman, with her good teaching and kind but firm demeanor, is
probably responsible to a significant degree for my career in math and
engineering.
--
Randy Yates
Digital Signal Labs
http://www.digitalsignallabs.com
```
```On Wed, 31 Jul 2013 04:39:35 +0000, Eric Jacobsen wrote:

> On Tue, 30 Jul 2013 20:59:37 -0500, Tim Wescott
>
>>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.

--
Tim Wescott
Control system and signal processing consulting
www.wescottdesign.com
```
```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
>>
>>>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.

And there are 5 angels on a pin head...

onto a bright light...

(No offense, Tim or Eric.)
--
Randy Yates
Digital Signal Labs
http://www.digitalsignallabs.com
```
```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?

"commsignal" <58672@dsprelated> wrote in message
> 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

```
```On Wed, 31 Jul 2013 10:30:52 -0400, Randy Yates wrote:

>
>> On Wed, 31 Jul 2013 04:39:35 +0000, Eric Jacobsen wrote:
>>
>>> On Tue, 30 Jul 2013 20:59:37 -0500, Tim Wescott
>>>
>>>>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.
>
> And there are 5 angels on a pin head...
>
> onto a bright light...

Nah, that's the debate over whether or not the input to the DFT is of a
cyclical nature.

If you pay attention to what's actually happening as you follow each step
in the recipe, then you equip yourself to step outside the recipe when
the situation demands it, and succeed.

--

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

```
```On Wed, 31 Jul 2013 09:23:19 -0500, Tim Wescott

>On Wed, 31 Jul 2013 04:39:35 +0000, Eric Jacobsen wrote:
>
>> On Tue, 30 Jul 2013 20:59:37 -0500, Tim Wescott
>>
>>>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.

>--
>Tim Wescott
>Control system and signal processing consulting
>www.wescottdesign.com

Eric Jacobsen
Anchor Hill Communications
http://www.anchorhill.com
```
```On Wed, 31 Jul 2013 09:12:04 -0500, Tim Wescott

>On Wed, 31 Jul 2013 06:20:15 -0400, Randy Yates wrote:
>
>>> [...]
>>> I'm not sure that the concept of poles and zeros has a great deal of
>>> meaning in terms of a signal.  While I think you could make a case for
>>> saying it's still valid to talk about them, I feel that the concept of
>>> poles and zeros more appropriately belong to signals, not systems.
>>
>> Tim,
>>
>> Didn't you mean to write "systems, not signals?"
>
>Course of meant I that do to.
>
>I just sometimes type things in reverse order.
>
>Thanks for picking that up.
>
>--
>Tim Wescott
>Control system and signal processing consulting
>www.wescottdesign.com

That's funny, because I read it the way intended and didn't even
notice the error.

Eric Jacobsen
Anchor Hill Communications
http://www.anchorhill.com
```
```On Wed, 31 Jul 2013 16:03:47 +0000, Eric Jacobsen wrote:

> On Wed, 31 Jul 2013 09:23:19 -0500, Tim Wescott
>
>>On Wed, 31 Jul 2013 04:39:35 +0000, Eric Jacobsen wrote:
>>
>>> On Tue, 30 Jul 2013 20:59:37 -0500, Tim Wescott
>>>
>>>>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.

--

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

```