help -- Windzilla (high dynamic range windows)

Fred Marshall  <fmarshall_xremove_the_xs@xacm.org> wrote:

>On 10/28/2010 6:45 PM, dbd wrote:
>Dale,
>I do have a little trouble with your treatment of the sliding DFTs in an 
>analyzer context:

>Let's suppose we have streaming input data samples.
>Let's suppose the samples are at fs=1.024kHz with sampling interval 
>.999023438 msec.
>Let's suppose we will grab 1,024 samples for a DFT.
>So the temporal span of the grab is 0.999023438 seconds and represents 
>1.000... seconds in a DFT/IDFT temporal period context.

>So, the period in time is 1.0000.. seconds and the period in frequency 
>is 1,024 Hz and the frequency sample interval is 1 Hz.

>[snip]

>I don't see how there being a 1Hz temporal period implicit in the DFT 
>nor a 1024 Hz frequency period implicit in the DFT have any impact or 
>bearing on this..... either the 400Hz is there or it isn't and, yes, it 
>may fade and one can generate waterfalls, etc. etc.  but that really has 
>nothing to do with the 1024Hz period here.
>
>But, if the signal component of interest had a period that's longer than 
>1 second then there are all kinds of things going on that go beyond this 
>discussion because of aliasing, etc.  Your examples were sorta like 
>this.  If the *underlying signal period* is longer than the sample 
>sequence interval then there will be aliasing and under-resolution.

This is one of the reasons you use a window in the first place...
to achieve data reduction without (much) aliasing.

>For example, if I had 400.5 Hz above then I wouldn't be able to resolve 
>to 400.5 Hz with the parameters given without computing an estimate.  If 
>noiseless then I might be able to do a reasonable estimate with a single 
>frequency sequence.  If noisy then I might want to have more frequency 
>sequences to work on.  But, the 1Hz limit looms large in this endeavor. 

This goes back to what I explained earlier: if the analysis being
performed is a PSD, then it is well-defined at 400.5 Hz in your example.

You can certainly argue that if you've computed PSD(n) for all integers n
(in units of Hz), there is no _added_ information to be gained by computing
PSD(400.5).  But you can't really argue that PSD(400.5), by itself,
is any less meaningful than PSD(400) or PSD(401).  

Steve

Fred Marshall <fmarshall_xremove_the_xs@xacm.org> wrote:
 
> Well, it's clear that we're each stuck on a particular perspective.
> I don't think there's anything wrong with your perspective but I believe 
> it leads to some unfortunate terminology.
 
> When you say "a period", it appears you're referring to actual periodic 
> content in a signal of some length.  That's very understandable and fine.
 
> When I've been saying "a period" in the context of this thread, I'm 
> referring to the implicit period determined by the number of samples 
> used in the DFT.

Well, you could use a PLL to lock the sampling clock to a 
multiple of the fundamental frequency of a system being sampled.

As an example, to understand the vibrations of a rotating piece
of machinery one could measure the position at the appropriate rate.

-- glen

On 10/29/2010 1:15 PM, glen herrmannsfeldt wrote:
> Fred Marshall<fmarshall_xremove_the_xs@xacm.org>  wrote:
>
>> Well, it's clear that we're each stuck on a particular perspective.
>> I don't think there's anything wrong with your perspective but I believe
>> it leads to some unfortunate terminology.
>
>> When you say "a period", it appears you're referring to actual periodic
>> content in a signal of some length.  That's very understandable and fine.
>
>> When I've been saying "a period" in the context of this thread, I'm
>> referring to the implicit period determined by the number of samples
>> used in the DFT.
>
> Well, you could use a PLL to lock the sampling clock to a
> multiple of the fundamental frequency of a system being sampled.
>
> As an example, to understand the vibrations of a rotating piece
> of machinery one could measure the position at the appropriate rate.
>
> -- glen

glen,

Yes.  I like that trick.  What's it got to do with DFTs?

Red

On 10/28/2010 6:58 PM, dbd wrote:
> On Oct 28, 4:07 pm, Fred Marshall<fmarshall_xremove_the...@xacm.org>
> wrote:
>> On 10/27/2010 10:38 PM, dbd wrote:
>>
>>> No, only the continuous/infinite Fourier Transform (FT) must have a
>>> periodicity in one domain to be non-zero only at discrete values in
>>> the other domain.
>>
>> This confuses me.  Does this mean that the DFT need not have a
>> periodicity in one domain to be non-zero only at discrete values in the
>> other domain or what exactly?
>>
>> I'm not being dense, I just don't get it yet.
>>
>> Fred
>
> If the FT has non-zero values at the sample times and zero values in
> between sample times the continuous time signal must have been
> periodic. Infinitely extending periodicity is what is required for the
> intervals between sample times to be zero. In the DFT there are no
> values between sample times.  There is neither need nor justification
> for a periodic constraint.
>
> Dale B. Dalrymple

Dale,

OK.  I think I understand what you're saying at least.  Sorry it's taken 
so long to ponder.

How about this?

              N-1
X(k+N) =def= sum [x(n)*e^[(-j*2*pi/N)*(k+N)*n]  where k>N-1
              n=0

so
           N-1
X(k+N) =  sum x(n) e^[(-j*2*pi/N)*k*n * d^[-j*2*pi]
           n=0

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

which is a periodicity property.

I guess it depends on how one views the sequences, eh?
Just as in *circular convolution*,  I view the sequences on a circle 
(representing the period).  I think that's a "need".
I find it handy to consider multiple periods on an expanded circle when 
dealing with things like interpolation/decimation.  So, whether I need 
to do this or not, it helps rather tremendously.

The circular representation is just another way of dealing with the 
periodicity.
And, one may "unwrap" a sequence on a circle into a linear, infinite 
sequence.  That's another mapping.

So, that's how I justify it.

But, I can see another viewpoint:
One could envision the sequence being finite on a linear axis.  In that 
case it's just a sequence.  And the DFT or IDFT is just a mapping.
I don't see anything wrong with that but I don't find it as useful as a 
model.  Is it more general and lends to more insight or less so?

Another thing that biases me is this:
One can view the DFT as a special case of the FT.
Start with the FT.
Input a periodic sequence (thus infinite).
Observe: with this constraint, the formulation can simplify to the DFT.
Thus, given a sequence and a DFT one can generalize it back out to a FT 
of a periodic sequence.
Why deny this relationship?

So, there are two apparent models for the DFT and I'm trying to evaluate 
them.  At least I think I'm getting there now.....

Thanks for continuing with me!

Fred

On Oct 31, 6:19 am, Fred Marshall <fmarshall_xremove_the...@xacm.org>
wrote:
>...
>
> Dale,
>
> OK.  I think I understand what you're saying at least.  Sorry it's taken
> so long to ponder.
>
> How about this?
> ...
>
>           N-1
> X(k+N) = sum x(n)*e^[-j*2*pi*k*n/N] = X(k)
>           n=0
>
> which is a periodicity property.

Yes, in the frequency domain, not the time domain which we have been
discussing.

> ...
>
> But, I can see another viewpoint:
> One could envision the sequence being finite on a linear axis.  In that
> case it's just a sequence.  And the DFT or IDFT is just a mapping.
> I don't see anything wrong with that but I don't find it as useful as a
> model.  Is it more general and lends to more insight or less so?
>
> Another thing that biases me is this:
> One can view the DFT as a special case of the FT.

No, the DFT and the FT are not even defined in the same domains.

> Start with the FT.
> Input a periodic sequence (thus infinite).
> Observe: with this constraint, the formulation can simplify to the DFT.
> Thus, given a sequence and a DFT one can generalize it back out to a FT
> of a periodic sequence.
> Why deny this relationship?

Because it isn't a property of the DFT and FT. They are defined in
different domains sepatated by the process of sampling, among other
things. There -are- signals for which it is true that the DFT of a
finite set of samples of the signal will produce a finite set of
samples of the FT of the signal. Those signals are periodic -and- the
set of samples must span one period of the signal. Few real data sets
have this property. Some useful ones are close. Many data sets aren't
even in the neighborhood. When we have enough of the data set, we can
test for the property. For my examples, perform a DFT of the first 8
samples. Calculate a DFT of the next 8 samples and compare. For the
first sequence, the DFTs are the same, so for the region tested, the
signal has the property of periodic extension. The next two sequences
do not have this property. The second sequence may even represent
samples of a signal that continues to grow without limit and does not
even have a FT. Yet we can still calculate a DFT of portions of it.
The DFT is not a "special case" of the FT. There -are- signals that
have properties that can be related across domains.

There seem to be a number of people who have been mis-educated into
the defective meme that when the DFT has been applied to a set of
samples, the signal has been automagically transmografied into having
the time domain property of periodic extension. Those possessing the
defective meme exhibit an irrepressible knee-jerk reflex to express
the assumed periodicity as fact even for data sets where it is
demonstably invalid. Perhaps their educators came from the era when
the newly advertised DFT was seen as the door to real calculations of
the FT, but before the properties and limitations of the DFT were
commonly understood and applied.

 The real world is full of data sets that approximate the property to
varying degrees. For a signal consisting of the sum of a periodic
component and a stochastic noise component, it is useful for a digital
spectrum analyzer to apply Welch's method (WOSA) to reduce the
variance of the estimate of the power spectrum of the periodic
component. But if the cyclic signal component has a continuing linear
FM, the process will smear the power spectrum across frequency. Useful
or damaging? It depends on the -data- which model applies, not on the
transforms. There are data sets like this where different components
require different models. In the real world we can often collect the
data to test. Much DSP troubleshooting effort has been wasted of the
cases of non-periodic data sets feed to applications that must have
periodicity to function to specs and vice versa.

>
> So, there are two apparent models for the DFT and I'm trying to evaluate
> them.  At least I think I'm getting there now.....
...

There are processes and applications that may assume the models and
there are data sets that consist of components that may or may not
exhibit the characteristics of the models. It is the data that
determines what the relationship between the FT and DFT of the signal
is.

I think that number theorists would suggest that the true number of
useful models for the DFT is greater than two, but they might not
limit their interests to such domains as time. space or frequency.

Dale B. Dalrymple

On 11/1/2010 12:59 AM, dbd wrote:
> On Oct 31, 6:19 am, Fred Marshall<fmarshall_xremove_the...@xacm.org>
> wrote:
>> ...
>>
>> Dale,
>>
>> OK.  I think I understand what you're saying at least.  Sorry it's taken
>> so long to ponder.
>>
>> How about this?
>> ...
>>
>>            N-1
>> X(k+N) = sum x(n)*e^[-j*2*pi*k*n/N] = X(k)
>>            n=0
>>
>> which is a periodicity property.
>
> Yes, in the frequency domain, not the time domain which we have been
> discussing.
>
>> ...
>>
>> But, I can see another viewpoint:
>> One could envision the sequence being finite on a linear axis.  In that
>> case it's just a sequence.  And the DFT or IDFT is just a mapping.
>> I don't see anything wrong with that but I don't find it as useful as a
>> model.  Is it more general and lends to more insight or less so?
>>
>> Another thing that biases me is this:
>> One can view the DFT as a special case of the FT.
>
> No, the DFT and the FT are not even defined in the same domains.

***time/frequency are domains.  Space/wave number are domains.  etc.
I guess you mean continuous vs. discrete.

>
>> Start with the FT.
>> Input a periodic sequence (thus infinite).
>> Observe: with this constraint, the formulation can simplify to the DFT.
>> Thus, given a sequence and a DFT one can generalize it back out to a FT
>> of a periodic sequence.
>> Why deny this relationship?
>
> Because it isn't a property of the DFT and FT. They are defined in
> different domains sepatated by the process of sampling, among other
> things.

***I see that I didn't express the thought intended.  So, I'll try again:
- Start with the FT.
- Input a periodic sequence (thus infinite).
- Observe: with this constraint, the formulation can simplify to the 
Fourier's integral over one period (thus *appearing to be finite* but 
not finite in that sense) to generate the Fourier coefficients.
- Observe: with this constraint, the resulting coefficients (sequence) 
is discrete - perhaps not finite but discrete.
AND, vice versa ... one could start with frequency and go to time or 
from spatial frequency to space, etc. the physical domain isn't the 
point here.
Now, we can take this discrete sequence and reverse the transform using 
the familiar Fourier Series or IDFT both are discrete sums over the 
number of coefficients.  Nothing limits the IDFT formulation to being 
finite except how we usually use it - just like the Fourier Series 
summation.
Thus the pair: periodic/discrete.

But, that's not very convenient for DSP in a couple of ways.  We'd like 
the sequences to be of limited size (I'll avoid using "finite" here just 
to avoid confusion but it would be OK) and we'd like to work with 
discrete samples in both transform domains.

So, as in any practical Fourier Series application, we limit the series 
to some number of terms.  That limits the size and doesn't affect the 
periodicity in the transform domain.

We start with a set of coefficients as in a Fourier Series and are able 
to construct the continuous periodic function.  And, this series may be 
infinite.  But, now we want to sample the constructed function.  So, we 
do that, choosing some desirable sample rate. (And here I'm purposefully 
ignoring aliasing which isn't really to the point right now).  When we 
do this, the Fourier Transform of the samples becomes periodic on fs 
(and whatever aliasing may be presumed to have occurred, occurs).
So, now we have the pair: discrete/periodic
Since we started with periodic/discrete, now we have:
periodic discrete / periodic discrete.
But, so far we haven't dealt with the periodicity in frequency (just to 
keep some context I use "frequency" here) as far as reconstruction goes.

Observing that the frequency samples are now periodic, we can apply a 
summation over a single period which generates a period's worth of 
samples in time - which is the IDFT.

>There -are- signals for which it is true that the DFT of a
> finite set of samples of the signal will produce a finite set of
> samples of the FT of the signal. Those signals are periodic -and- the
> set of samples must span one period of the signal. Few real data sets
> have this property. Some useful ones are close. Many data sets aren't
> even in the neighborhood. When we have enough of the data set, we can
> test for the property. For my examples, perform a DFT of the first 8
> samples. Calculate a DFT of the next 8 samples and compare. For the
> first sequence, the DFTs are the same, so for the region tested, the
> signal has the property of periodic extension. The next two sequences
> do not have this property. The second sequence may even represent
> samples of a signal that continues to grow without limit and does not
> even have a FT. Yet we can still calculate a DFT of portions of it.
> The DFT is not a "special case" of the FT. There -are- signals that
> have properties that can be related across domains.

***Yes.  I agree with that but I don't see where it gets to.

>
> There seem to be a number of people who have been mis-educated into
> the defective meme that when the DFT has been applied to a set of
> samples, the signal has been automagically transmografied into having
> the time domain property of periodic extension. Those possessing the
> defective meme exhibit an irrepressible knee-jerk reflex to express
> the assumed periodicity as fact even for data sets where it is
> demonstably invalid. Perhaps their educators came from the era when
> the newly advertised DFT was seen as the door to real calculations of
> the FT, but before the properties and limitations of the DFT were
> commonly understood and applied.
>
***Ad hominem arguments fall flat.

***Well, I don't think that I'd ever suggested any automagic 
transmogrification.  That a signal may have some periodic character and 
that a transform may introduce some periodic character in an anlytical 
sense are, to me, two entirely different things.  And, I tried to say 
that earlier - that the underlying signal is of no consequence here. 
But, you've suggested the opposite.  Just sample a signal at pi Hz for 1 
second.  The underlying signal has a period of 1/pi and the transform 
has a period of 1 second.  Never the twain shall meet for sample lengths 
that are an integer number of seconds long (except in a practical sense 
if the number of samples is large enough and such that "K seconds" 
nearly meets an integral number of periods of 1/pi).

>   The real world is full of data sets that approximate the property to
> varying degrees. For a signal consisting of the sum of a periodic
> component and a stochastic noise component, it is useful for a digital
> spectrum analyzer to apply Welch's method (WOSA) to reduce the
> variance of the estimate of the power spectrum of the periodic
> component. But if the cyclic signal component has a continuing linear
> FM, the process will smear the power spectrum across frequency. Useful
> or damaging? It depends on the -data- which model applies, not on the
> transforms. There are data sets like this where different components
> require different models. In the real world we can often collect the
> data to test. Much DSP troubleshooting effort has been wasted of the
> cases of non-periodic data sets feed to applications that must have
> periodicity to function to specs and vice versa.
>
>>
>> So, there are two apparent models for the DFT and I'm trying to evaluate
>> them.  At least I think I'm getting there now.....
> ...
>
> There are processes and applications that may assume the models and
> there are data sets that consist of components that may or may not
> exhibit the characteristics of the models. It is the data that
> determines what the relationship between the FT and DFT of the signal
> is.

***OK but I wasn't trying to relate the signal at all.  Just properties 
of the transform.
>
> I think that number theorists would suggest that the true number of
> useful models for the DFT is greater than two, but they might not
> limit their interests to such domains as time. space or frequency.
>
> Dale B. Dalrymple

I'm not saying that there aren't pitfalls in anything we do.  Once I was 
building a sound source that was supposed to sound like a machine.  So, 
we recorded some of the machine sound and put it into one of those "new" 
digital shift-register memories to play it in an endless loop.  All we 
had to do was match up the ends (like matching the time span of samples 
of a periodic signal).  Oh!  But oops.  The length of the memory was too 
short and the resulting spectral analysis looked like a herringbone.

I think we need to use models that are meaningful for each of us.  I'll 
leave it at that.

Fred

On 11/1/2010 12:59 AM, dbd wrote:
> On Oct 31, 6:19 am, Fred Marshall<fmarshall_xremove_the...@xacm.org>
> wrote:
>> ...
>>
>> Dale,
>>
>> OK.  I think I understand what you're saying at least.  Sorry it's taken
>> so long to ponder.
>>
>> How about this?
>> ...
>>
>>            N-1
>> X(k+N) = sum x(n)*e^[-j*2*pi*k*n/N] = X(k)
>>            n=0
>>
>> which is a periodicity property.
>
> Yes, in the frequency domain, not the time domain which we have been
> discussing.
>
>> ...
>>
>> But, I can see another viewpoint:
>> One could envision the sequence being finite on a linear axis.  In that
>> case it's just a sequence.  And the DFT or IDFT is just a mapping.
>> I don't see anything wrong with that but I don't find it as useful as a
>> model.  Is it more general and lends to more insight or less so?
>>
>> Another thing that biases me is this:
>> One can view the DFT as a special case of the FT.
>
> No, the DFT and the FT are not even defined in the same domains.

***time/frequency are domains.  Space/wave number are domains.  etc.
I guess you mean continuous vs. discrete.

>
>> Start with the FT.
>> Input a periodic sequence (thus infinite).
>> Observe: with this constraint, the formulation can simplify to the DFT.
>> Thus, given a sequence and a DFT one can generalize it back out to a FT
>> of a periodic sequence.
>> Why deny this relationship?
>
> Because it isn't a property of the DFT and FT. They are defined in
> different domains sepatated by the process of sampling, among other
> things.

***I see that I didn't express the thought intended.  So, I'll try again:
- Start with the FT.
- Input a periodic sequence (thus infinite).
- Observe: with this constraint, the formulation can simplify to the 
Fourier's integral over one period (thus *appearing to be finite* but 
not finite in that sense) to generate the Fourier coefficients.
- Observe: with this constraint, the resulting coefficients (sequence) 
is discrete - perhaps not finite but discrete.
AND, vice versa ... one could start with frequency and go to time or 
from spatial frequency to space, etc. the physical domain isn't the 
point here.
Now, we can take this discrete sequence and reverse the transform using 
the familiar Fourier Series or IDFT both are discrete sums over the 
number of coefficients.  Nothing limits the IDFT formulation to being 
finite except how we usually use it - just like the Fourier Series 
summation.
Thus the pair: periodic/discrete.

But, that's not very convenient for DSP in a couple of ways.  We'd like 
the sequences to be of limited size (I'll avoid using "finite" here just 
to avoid confusion but it would be OK) and we'd like to work with 
discrete samples in both transform domains.

So, as in any practical Fourier Series application, we limit the series 
to some number of terms.  That limits the size and doesn't affect the 
periodicity in the transform domain.

We start with a set of coefficients as in a Fourier Series and are able 
to construct the continuous periodic function.  And, this series may be 
infinite.  But, now we want to sample the constructed function.  So, we 
do that, choosing some desirable sample rate. (And here I'm purposefully 
ignoring aliasing which isn't really to the point right now).  When we 
do this, the Fourier Transform of the samples becomes periodic on fs 
(and whatever aliasing may be presumed to have occurred, occurs).
So, now we have the pair: discrete/periodic
Since we started with periodic/discrete, now we have:
periodic discrete / periodic discrete.
But, so far we haven't dealt with the periodicity in frequency (just to 
keep some context I use "frequency" here) as far as reconstruction goes.

Observing that the frequency samples are now periodic, we can apply a 
summation over a single period which generates a period's worth of 
samples in time - which is the IDFT.

>There -are- signals for which it is true that the DFT of a
> finite set of samples of the signal will produce a finite set of
> samples of the FT of the signal. Those signals are periodic -and- the
> set of samples must span one period of the signal. Few real data sets
> have this property. Some useful ones are close. Many data sets aren't
> even in the neighborhood. When we have enough of the data set, we can
> test for the property. For my examples, perform a DFT of the first 8
> samples. Calculate a DFT of the next 8 samples and compare. For the
> first sequence, the DFTs are the same, so for the region tested, the
> signal has the property of periodic extension. The next two sequences
> do not have this property. The second sequence may even represent
> samples of a signal that continues to grow without limit and does not
> even have a FT. Yet we can still calculate a DFT of portions of it.
> The DFT is not a "special case" of the FT. There -are- signals that
> have properties that can be related across domains.

***Yes.  I agree with that but I don't see where it gets to.

>
> There seem to be a number of people who have been mis-educated into
> the defective meme that when the DFT has been applied to a set of
> samples, the signal has been automagically transmografied into having
> the time domain property of periodic extension. Those possessing the
> defective meme exhibit an irrepressible knee-jerk reflex to express
> the assumed periodicity as fact even for data sets where it is
> demonstably invalid. Perhaps their educators came from the era when
> the newly advertised DFT was seen as the door to real calculations of
> the FT, but before the properties and limitations of the DFT were
> commonly understood and applied.
>
***Ad hominem arguments fall flat.

***Well, I don't think that I'd ever suggested any automagic 
transmogrification.  That a signal may have some periodic character and 
that a transform may introduce some periodic character in an anlytical 
sense are, to me, two entirely different things.  And, I tried to say 
that earlier - that the underlying signal is of no consequence here. 
But, you've suggested the opposite.  Just sample a signal at pi Hz for 1 
second.  The underlying signal has a period of 1/pi and the transform 
has a period of 1 second.  Never the twain shall meet for sample lengths 
that are an integer number of seconds long (except in a practical sense 
if the number of samples is large enough and such that "K seconds" 
nearly meets an integral number of periods of 1/pi).

>   The real world is full of data sets that approximate the property to
> varying degrees. For a signal consisting of the sum of a periodic
> component and a stochastic noise component, it is useful for a digital
> spectrum analyzer to apply Welch's method (WOSA) to reduce the
> variance of the estimate of the power spectrum of the periodic
> component. But if the cyclic signal component has a continuing linear
> FM, the process will smear the power spectrum across frequency. Useful
> or damaging? It depends on the -data- which model applies, not on the
> transforms. There are data sets like this where different components
> require different models. In the real world we can often collect the
> data to test. Much DSP troubleshooting effort has been wasted of the
> cases of non-periodic data sets feed to applications that must have
> periodicity to function to specs and vice versa.
>
>>
>> So, there are two apparent models for the DFT and I'm trying to evaluate
>> them.  At least I think I'm getting there now.....
> ...
>
> There are processes and applications that may assume the models and
> there are data sets that consist of components that may or may not
> exhibit the characteristics of the models. It is the data that
> determines what the relationship between the FT and DFT of the signal
> is.

***OK but I wasn't trying to relate the signal at all.  Just properties 
of the transform.
>
> I think that number theorists would suggest that the true number of
> useful models for the DFT is greater than two, but they might not
> limit their interests to such domains as time. space or frequency.
>
> Dale B. Dalrymple

I'm not saying that there aren't pitfalls in anything we do.  Once I was 
building a sound source that was supposed to sound like a machine.  So, 
we recorded some of the machine sound and put it into one of those "new" 
digital shift-register memories to play it in an endless loop.  All we 
had to do was match up the ends (like matching the time span of samples 
of a periodic signal).  Oh!  But oops.  The length of the memory was too 
short and the resulting spectral analysis looked like a herringbone.

I think we need to use models that are meaningful for each of us.  I'll 
leave it at that.

Fred

On Nov 1, 10:54&#4294967295;am, Fred Marshall <fmarshall_xremove_the...@xacm.org>
wrote:
> On 11/1/2010 12:59 AM, dbd wrote:
> ...
>
> > There seem to be a number of people who have been mis-educated into
> > the defective meme that when the DFT has been applied to a set of
> > samples, the signal has been automagically transmografied into having
> > the time domain property of periodic extension. Those possessing the
> > defective meme exhibit an irrepressible knee-jerk reflex to express
> > the assumed periodicity as fact even for data sets where it is
> > demonstably invalid. Perhaps their educators came from the era when
> > the newly advertised DFT was seen as the door to real calculations of
> > the FT, but before the properties and limitations of the DFT were
> > commonly understood and applied.
>
> ***Ad hominem arguments fall flat.

This is an observation on the nature of some discussions on usenet and
elsewhere. The fact that you recognize that there exists a linear
arrangement of data that might be of some use indicates that it
doesn't apply to you.

>
> ***Well, I don't think that I'd ever suggested any automagic
> transmogrification.

You haven't.

>&#4294967295;That a signal may have some periodic character and
> that a transform may introduce some periodic character in an anlytical
> sense are, to me, two entirely different things. &#4294967295;

That is a rare distinction and much appreciated. I was pointing out
the difference between you interpretation and another common one.

Dale B. Dalrymple

On Nov 1, 11:13 am, Fred Marshall <fmarshall_xremove_the...@xacm.org>
wrote:
> On 11/1/2010 12:59 AM, dbd wrote:
>
> > On Oct 31, 6:19 am, Fred Marshall<fmarshall_xremove_the...@xacm.org>
> > wrote:
> >> ...
> ...

> >> Another thing that biases me is this:
> >> One can view the DFT as a special case of the FT.
>
> > No, the DFT and the FT are not even defined in the same domains.
> ...
> I guess you mean continuous vs. discrete.

Yes.

>
> >> Start with the FT.
> >> Input a periodic sequence (thus infinite).

This is the key point, you must chose a periodic sequence to
demonstrate the relationship.

> >> Observe: with this constraint, the formulation can simplify to the DFT.
> >> Thus, given a sequence and a DFT one can generalize it back out to a FT
> >> of a periodic sequence.
> >> Why deny this relationship?
>
> > Because it isn't a property of the DFT and FT. They are defined in
> > different domains separated by the process of sampling, among other
> > things.

I see that I didn't express the thought intended. I agree with the
property of periodic signals. I deny that the relationship is a
characteristic of the DFT and FT. It is necessary for you to assume
signals that have a special relationship between discrete and
continuous domains to even present the argument.

>
> ***I see that I didn't express the thought intended.  So, I'll try again:
> - Start with the FT.
> - Input a periodic sequence (thus infinite).

Again, as you correctly state, you must use a periodic sequence.

> ...
> Observing that the frequency samples are now periodic, we can apply a
> summation over a single period which generates a period's worth of
> samples in time - which is the IDFT.

What do you mean "now periodic"?!!? You assumed the signal periodic in
the first place. Your reasoning is circular. You have demonstrated
your assumption.

> > It is the data that
> > determines what the relationship between the FT and DFT of the signal
> > is.
>
> ***OK but I wasn't trying to relate the signal at all.  Just properties
> of the transform.
>

But in each case you do "relate to the signal" by specifying a
periodic signal. Properties of the transform are independent of the
signal. The property we are discussing is not independent of the
signal, so it is not a property of the transform.

I'm not denying the property (of periodic signals), I'm claiming that
the describing the property as a characteristic of the transforms has
mislead many people who fail to carry along the necessary initial
assumption of periodicity. It is the people who have forgotten, or
never been taught, about the limitation of the property to the
periodic assumption who have been mis-educated. It is these people
that I pointed out the existence of in my first post on Nov. 1.

Dale B. Dalrymple

On 11/2/2010 10:59 AM, dbd wrote:
> On Nov 1, 11:13 am, Fred Marshall<fmarshall_xremove_the...@xacm.org>
> wrote:
>> On 11/1/2010 12:59 AM, dbd wrote:
>>
>>> On Oct 31, 6:19 am, Fred Marshall<fmarshall_xremove_the...@xacm.org>
>>> wrote:
>>>> ...
>> ...
>
>>>> Another thing that biases me is this:
>>>> One can view the DFT as a special case of the FT.
>>
>>> No, the DFT and the FT are not even defined in the same domains.
>> ...
>> I guess you mean continuous vs. discrete.
>
> Yes.
>
>>
>>>> Start with the FT.
>>>> Input a periodic sequence (thus infinite).
>
> This is the key point, you must chose a periodic sequence to
> demonstrate the relationship.
>
>>>> Observe: with this constraint, the formulation can simplify to the DFT.
>>>> Thus, given a sequence and a DFT one can generalize it back out to a FT
>>>> of a periodic sequence.
>>>> Why deny this relationship?
>>
>>> Because it isn't a property of the DFT and FT. They are defined in
>>> different domains separated by the process of sampling, among other
>>> things.
>
> I see that I didn't express the thought intended. I agree with the
> property of periodic signals. I deny that the relationship is a
> characteristic of the DFT and FT. It is necessary for you to assume
> signals that have a special relationship between discrete and
> continuous domains to even present the argument.
>
>>
>> ***I see that I didn't express the thought intended.  So, I'll try again:
>> - Start with the FT.
>> - Input a periodic sequence (thus infinite).
>
> Again, as you correctly state, you must use a periodic sequence.
>
>> ...
>> Observing that the frequency samples are now periodic, we can apply a
>> summation over a single period which generates a period's worth of
>> samples in time - which is the IDFT.
>
> What do you mean "now periodic"?!!? You assumed the signal periodic in
> the first place. Your reasoning is circular. You have demonstrated
> your assumption.
>
>>> It is the data that
>>> determines what the relationship between the FT and DFT of the signal
>>> is.
>>
>> ***OK but I wasn't trying to relate the signal at all.  Just properties
>> of the transform.
>>
>
> But in each case you do "relate to the signal" by specifying a
> periodic signal. Properties of the transform are independent of the
> signal. The property we are discussing is not independent of the
> signal, so it is not a property of the transform.
>
> I'm not denying the property (of periodic signals), I'm claiming that
> the describing the property as a characteristic of the transforms has
> mislead many people who fail to carry along the necessary initial
> assumption of periodicity. It is the people who have forgotten, or
> never been taught, about the limitation of the property to the
> periodic assumption who have been mis-educated. It is these people
> that I pointed out the existence of in my first post on Nov. 1.
>
> Dale B. Dalrymple
>

Dale,

You make some great points!  I'll have to ponder this further.

I sure hope this isn't leading us into one of those silly arguments 
about delta functions!!  I'm a believer.
Maybe that's why I relate the FT and DFT so readily?
I do view them rather as "the same thing", if you will, because the 
integrals become sums going from continous to discrete and one can 
approach this in a "limit" sense, no?
Is that agreeable at least?

If not, then one would have to be able to argue that the discrete 
versions with sums have some special properties - is that reasonable?

But, OK.  I'll try to paraphrase what we're saying:

I've said that discrete finite sequences (and I'll say now "arbitrary 
sequences" to get away from any underlying signal periodicity issue), 
connected by a DFT, have a periodicity property that is assumed or 
implied or ....  (Well, of course, only the initial sequence can be 
arbitrary and the other follows from the transform - chicken and egg).

You've said that the periodicity assumption is unnecessary and 
"harmful"?  I don't want to put words in your mouth here.

I guess I'm biased by the z transform approach to things where frequency 
is represented on a circle rather than a straight line.  And, I carry 
that notion into DFTs as well.  And, I find it handy to do so.

I agree that one can view the sequence pair associated by a DFT as a 
pair of strictly finite sequences.  That seems an abstraction to me but 
OK - I can't say the perspective is invalid for some reason.  But, I 
don't view it as being very useful.

On the other hand, viewing each sequence in the pair as representing a 
single period of an infinite, periodic sequence does seem useful. 
Again, this has nothing to do with the particular sequences which can be 
arbitrary.

And, the two can be tied together by a simple mapping.
If I have a function or sequence in x on a linear scale, I can map it to 
r on a circular scale using    r=sin(x).
And, as long as we're clear that the linear scale contains only finite 
sequences such that [0 <= x <= 2pi)  so that (-1 <+ r <= +1] (or some 
scaled versions thereof) then dealing with the finite sequences on a 
linear scale or dealing with them on a circular scale amounts to the 
same thing.

Then, if one wants to do things like sample rate conversion, either one 
visualization (er... mapping) or the other might be handy.

So, I guess I've been stuck on the circular version in this thread (even 
though in my mind's eye it's projected onto a linear scale) and that's 
about all there is to it.  I shouldn't have been so pedantic to insist 
that it is the only way to look at it.

Fred