Reply by robert bristow-johnson●December 11, 20112011-12-11
On 12/10/11 3:53 PM, dbd wrote:
> On Dec 10, 10:23 am, robert bristow-johnson <rbj@audioimagination.com> wrote:
...
>> the DFT invertibly maps one discrete and periodic sequence of numbers to another discrete and periodic sequence of numbers of the same period.
>>
>> ... something that the periodicity deniers just don't seem to get.
>>
>
> There are no "periodicity deniers". There are only those deniers who
> know only a single small view of the world and from lack of
> imagination would deny others the use of many other valid useful
> world views.
it's math, not something subjective nor political nor theological nor
philosophical like "world views".
equality signs are pretty much focused in their meaning and unforgiving.
--
r b-j rbj@audioimagination.com
"Imagination is more important than knowledge."
Reply by dbd●December 11, 20112011-12-11
On Dec 10, 1:50�pm, Fred Marshall <fmarshallxremove_th...@acm.org>
wrote:
> ...
> So, we start with a rectangular window of, let's say for illumination, M
> + 1/2 periods of a sinusoid. �I guess it doesn't matter if we sample it
> before or after windowing although if we sample it after windowing then
> the sample rate would be considered to be "too low" because of the sharp
> edges at the ends which suggests an issue with the boundary conditions.
> � But, we accept the aliasing it causes by calling it something else:
> "spectral spreading" or "leakage". �Is it true that spectral spreading,
> being continuous is somehow "more OK" than aliasing of a spectral line
> from one frequency to another? �Seems like it, eh? �And, how "ugly" is
> *that*? �Depends on the eye of the beholder I guess.
>
> But, without the 1/2 period, the boundary conditions are smooth and
> there is no aliasing / spectral spreading.
>
> Fred
There is spreading from windowing, but you can avoid looking at it.
Any time you window you generate spectral spreading, even with a
rectangular window and n-periodic boundary conditions. The convenience
(or "beauty") of this case is that the projection of the N-periodic
signals spread by the rectangular window is zero at the frequency
samples calculated by the N point DFT except at the component
frequency. The spectral spreading is non-zero elsewhere. The time
domain windowing process convolves the FT of the window with the FT of
the signal in the frequency domain. If you take N samples (that is
the rectangular window) of a single N-periodic frequency component,
the DFT calculates samples the delta function, the FT of the single
component, convolved with the sync function that is the FT of the
rectangular window. At the frequencies the N point DFT calculated
samples, the windowed response is non-zero at only one sample. The
zeros are due to the zeros of the sync function. If you sample the
response at any other frequency, the projection is non-zero. An
example of this can be calculated by zero extending the N time domain
samples by N zeros and calculating the 2N point DFT. Alternate samples
of the 2N point DFT output will be the samples calculated by the N
point DFT of the original N samples and the rest of the points will be
non-zero samples of the windowing-spread single frequency component at
the frequencies half way between the frequency of samples calculated
by the N point DFT.
Dale B. Dalrymple
Reply by Fred Marshall●December 10, 20112011-12-10
On 12/10/2011 12:42 PM, dbd wrote:
>> And, of course, we all know that we can select "good" values of the
>> > sample interval T and number of samples N and "bad" values of the same
>> > such that some strong periodic component is grabbed with an integer plus
>> > 1/2 of a period is in the window. That's pretty "aperiodic" and the
>> > underlying (i.e. original) boundary conditions are ugly and the
>> > resulting DFT is ugly too. But calling the DFT "ugly" is a perspective
>> > while saying "it is what it is", I think, is more to the point.
> Ugly is your word, not mine. The DFT coefficients can still be used to
> calculate the frequency, ampltude and phase of the 'plus a 1/2'
> frequency component, which demonstrates that the assumption of N-
> periodicity is not a characteristic of the DFT and the coefficients it
> calculates, but one of many interpretations available to the analyst
> and one that can be usefully ignored when chosing how to analyze DFT
> coefficients. Is that what you mean by ugly?
>> > ...
> Dale B. Dalrymple
Dale,
Well, let's see .. Yes, I introduced "ugly" to suggest the nature of the
continous FT of what's in the temporal window .. the spectral spreading
that's evident in that case. I didn't really say that but it's what I
was imagining.
So, we start with a rectangular window of, let's say for illumination, M
+ 1/2 periods of a sinusoid. I guess it doesn't matter if we sample it
before or after windowing although if we sample it after windowing then
the sample rate would be considered to be "too low" because of the sharp
edges at the ends which suggests an issue with the boundary conditions.
But, we accept the aliasing it causes by calling it something else:
"spectral spreading" or "leakage". Is it true that spectral spreading,
being continuous is somehow "more OK" than aliasing of a spectral line
from one frequency to another? Seems like it, eh? And, how "ugly" is
*that*? Depends on the eye of the beholder I guess.
But, without the 1/2 period, the boundary conditions are smooth and
there is no aliasing / spectral spreading.
Fred
Reply by dbd●December 10, 20112011-12-10
On Dec 10, 10:23�am, robert bristow-johnson
<r...@audioimagination.com> wrote:
>...
>
> it's also mathematically correct. �and something that the periodicity
> deniers just don't seem to get.
>
> r b-j r...@audioimagination.com
>
> "Imagination is more important than knowledge."
There are no "periodicity deniers". There are only those deniers who
know only a single small view of the world and from lack of
imagination would deny others the use of many other valid useful
world views.
Dale B. Dalrymple
Reply by dbd●December 10, 20112011-12-10
On Dec 10, 8:13 am, Fred Marshall <fmarshallxremove_th...@acm.org>
wrote:
> On 12/9/2011 9:31 AM, dbd wrote:
> ...
> > On Dec 8, 8:47 am, Fred Marshall<fmarshallxremove_th...@acm.org>
> > wrote:
> >> On 12/7/2011 7:59 PM, glen herrmannsfeldt wrote:
>
> >>> You can consider the DFT as the FT of delta functions at the data
> >>> points, and periodic boundary conditions.
>
> >>> -- glen
>
> >> Glen,
>
> >> What an elegant way to put it! I don't think in all the discussions
> >> that I've seen it expressed this way.
...
> > Fred
>
> > The statement is very succinct,
> > ...
> > Dale B. Dalrymple
>
> Dale,
>
> Yes, I understand and agree to a point. Except for "when?".
>
> Let us say that some signal with aperiodic components (relative to our
> intended DFT) is sampled for a long time.
>
> Now let us select some N or temporal window and do a DFT on those
> samples. The result is a length N discrete transform (complex usually).
>
> Now we have a transform pair. At this stage there are no aperiodic
> components .. even though the original aperiodic components may have
> affected the original samples. In effect what we have is a *new*
> periodic sequence which deviates from the original *underlying* periodic
> components. Actually this occurred when we selected N. When we select
> the sample rate and N, we are one way or another asserting its
> periodicity. Well, I should add I guess "if we are going to compute a
> DFT". I suppose there are other applications of those N samples that
> wouldn't suggest such a thing.
Yes, there are other applications of not just the N samples but the
very same DFT coefficients. The algorithms know as 'frequency
reassignment' calculate the amplitude, center frequency and phase of
linear FM components from the same DFT coefficients. The N-periodic
components are a subset of those linear FMs. So are periodic, but not
N-periodic components. So, while you, as the analyst may chose to
interpret via an N-periodic assumption, the use of DFT coefficients
and selection of N and Fs and the signals to apply them to, do not
need to depend on any prefered periodic assumption.
>
> Perhaps another way to put it is:
> - once you've sampled there's no going back .. perfect reconstruction
> being the only counter example that I can think of.
> - once you've windowed i.e. selected N samples .. except for perhaps
> things like concatenation of sequences with their neighboring samples ..
> there's no going back.
> - once you've DFT'd, you've put things in a context where everything is
> periodic and there's no going back.
Choosing to analyze and reason from only a single block of N samples
is sometimes a necessity and sometimes just the analyst's choice, but
it is a position of ignorance with respect to the properties of the
original sequence whether voluntary or not. Analysts are free to chose
assumptions to deal with that ignorance. You have your favorite
assumption and I have often used it as well. It has the advantage of
requiring the least additional processing after the DFT and is often
good enough for applications.
>
> I'm not trolling for arguments or counter examples. Maybe these
> assertions are just a "mind set". Without "proof" I think it's a useful
> framework.
We often try to pick N and Fs so that the assumption of periodicity is
valid. Some instruments generate Fs to be synchronous with the
periodicities of the signal. There is literature on resampling to make
the Fs synchronous. It is often useful to make the assumption of
periodiciy whether it is valid or not. My point is that the assumption
is one possible interpretation (among others) by the analyst to cope
with the consequences of examining only the analysis of one block of N
samples. At the cost of further analysis of the N DFT coefficients,
the assumption may not be necessary, useful, or appropriate.
>
> So, anyway, that's how I deal with the question of aperiodic components
> .. which are only evident before N is selected. And, N, conceptually at
> least, becomes a period of something that was originally not periodic
> when we consider that we allow a discrete version of the FT of those
> samples.
In addition to the algorithms already suggested for application to the
N DFT coefficients, the aperiodic components are also accesible to the
analyst after N is selected by analyzing adjacent blocks of N samples
and comparing magnitude and phase. Components that differ between the
blocks are not N-periodic. (If the magnitudes are the same, the
component may be periodic with period other than N.) If the components
are identical, the components are candidates for periodicity. In
theory, it would be necesary to analyze all blocks to prove
periodicity. In practice it is often adequate to make the distinction
between aperiodic or close-enough to periodic by comparing two or few
blocks. In fact, until you do this there is no basis other than
assumption, as stated by Glen's formulation or construction of the
original time sequence by periodic extension by the analyst, as O&S do
in their examples, for the periodic assumption. Well, in the
construction case, the original sequence is defined by periodic
extension, so the DFT and assumptions have nothing to do with it. But
the assumption is convenient and often useful so it is often made
without any basis.
> Obviously we can compute the FT of the N samples and get a continuous
> transform. Then the temporal periodicity wouldn't come up. But that's
> not what we do. So, I call "what we do" a context. Then there are more
> rigorous mathematical treatments to say the same thing but I rather
> think that this somewhat philosophical treatment is worthwhile.
>
It a choice available to the analyst. It is both a choice of context
and a choice of interpretation within that context.
> And, of course, we all know that we can select "good" values of the
> sample interval T and number of samples N and "bad" values of the same
> such that some strong periodic component is grabbed with an integer plus
> 1/2 of a period is in the window. That's pretty "aperiodic" and the
> underlying (i.e. original) boundary conditions are ugly and the
> resulting DFT is ugly too. But calling the DFT "ugly" is a perspective
> while saying "it is what it is", I think, is more to the point.
Ugly is your word, not mine. The DFT coefficients can still be used to
calculate the frequency, ampltude and phase of the 'plus a 1/2'
frequency component, which demonstrates that the assumption of N-
periodicity is not a characteristic of the DFT and the coefficients it
calculates, but one of many interpretations available to the analyst
and one that can be usefully ignored when chosing how to analyze DFT
coefficients. Is that what you mean by ugly?
> ...
Dale B. Dalrymple
Reply by robert bristow-johnson●December 10, 20112011-12-10
[[forgot to add a point i had intended.]]
On 12/10/11 11:13 AM, Fred Marshall wrote:
> On 12/9/2011 9:31 AM, dbd wrote:
>>> On 12/7/2011 7:59 PM, glen herrmannsfeldt wrote:
>>>
>>>> You can consider the DFT as the FT of delta functions at the data
>>>> points, and periodic boundary conditions.
>>
>> The statement is very succinct, anyway. How many readers of comp.dsp
>> do you think correctly convert "periodic boundary conditions" to the
>> required assumption that the delta functions can only consist of
>> samples of signals that are sums of components at the frequencies of
>> the basis functions of the DFT?
i certainly don't make that assumption. how have you determined that it
is "required"?
>> Aperiodic components and components
>> periodic at other frequencies than the DFT's basis functions don't
>> meet the "periodic boundary conditions".
no kidding. that's why the DTFT and the DFT ain't the same thing
(whereas the DFS and DFT *are* the same thing).
>
> Let us say that some signal with aperiodic components (relative to our
> intended DFT) is sampled for a long time.
>
> Now let us select some N or temporal window and do a DFT on those
> samples. The result is a length N discrete transform (complex usually).
>
> Now we have a transform pair. At this stage there are no aperiodic
> components .. even though the original aperiodic components may have
> affected the original samples. In effect what we have is a *new*
> periodic sequence which deviates from the original *underlying* periodic
> components.
we need to be specific about what "we have" and what is deviating from
the original. as i can observe it, *nothing* is deviating from the
original if your entire universe is only those N samples. but if the
entire universe is only those N samples, then it doesn't make any sense
to talk of those "other" components, be they aperiodic or having a
period other than N. in a universe of only N samples, there is no (and
have never been) any meaning to any other components.
but, if you think of these N samples as a sorta "pocket universe" (sorry
to borrow from cosmology, Glen and Clay will probably wince) surrounded
by an infinite sea of zeros, then (if you FT) you have the DTFT. [[and,
at this point there likely *is* some deviation of what "we have" and the
original underlying periodic (of some different period than N) or
aperiodic components.]] the spectrum is continuous, but repeats every
2*pi. the spectrum is not zero outside of the [-pi +pi) interval, but
if you make it so (in the mind of your brain or some other
mathematician's brain), then those N samples are no longer attached to N
dirac deltas, but are attached to N sinc() functions that go on forever
and there is no periodicity in that domain. not yet.
NOW (picking up on Fred's "when"), whether you zeroed the spectrum
outside of [-pi +pi) or not (i don't care if you do or not), if you
uniformly sample that spectrum with N samples from -pi to just under +pi
(or from 0 to just under 2*pi, i don't really care), you have the DFT.
and the act of sampling that spectrum *does* *necessarily* cause the
periodic extension of the original data, the N samples.
this is how you go from the one valid concept that "the DTFT is what you
get when you attach N delta functions to the N original data points
(uniformly spaced) and the DFT is what you get when you sample the DTFT
result" to the other equally valid (but i say is *more* fundamental)
that "the DFT invertibly maps one discrete and periodic sequence of
numbers to another discrete and periodic sequence of numbers of the same
period."
...
> So, anyway, that's how I deal with the question of aperiodic components
> .. which are only evident before N is selected. And, N, conceptually at
> least, becomes a period of something that was originally not periodic
> when we consider that we allow a discrete version of the FT of those
> samples.
>
> Obviously we can compute the FT of the N samples and get a continuous
> transform. Then the temporal periodicity wouldn't come up.
not yet...
> But that's not what we do.
... that's right. what we do is *sample* the DTFT and the undeniable
effect of that is the periodic extension of the
> So, I call "what we do" a context. Then there are more
> rigorous mathematical treatments to say the same thing but I rather
> think that this somewhat philosophical treatment is worthwhile.
it's also mathematically correct. and something that the periodicity
deniers just don't seem to get.
--
r b-j rbj@audioimagination.com
"Imagination is more important than knowledge."
Reply by robert bristow-johnson●December 10, 20112011-12-10
On 12/10/11 11:13 AM, Fred Marshall wrote:
> On 12/9/2011 9:31 AM, dbd wrote:
>>> On 12/7/2011 7:59 PM, glen herrmannsfeldt wrote:
>>>
>>>> You can consider the DFT as the FT of delta functions at the data
>>>> points, and periodic boundary conditions.
>>
>> The statement is very succinct, anyway. How many readers of comp.dsp
>> do you think correctly convert "periodic boundary conditions" to the
>> required assumption that the delta functions can only consist of
>> samples of signals that are sums of components at the frequencies of
>> the basis functions of the DFT?
i certainly don't make that assumption. how have you determined that it
is "required"?
>> Aperiodic components and components
>> periodic at other frequencies than the DFT's basis functions don't
>> meet the "periodic boundary conditions".
no kidding. that's why the DTFT and the DFT ain't the same thing
(whereas the DFS and DFT *are* the same thing).
>
> Let us say that some signal with aperiodic components (relative to our
> intended DFT) is sampled for a long time.
>
> Now let us select some N or temporal window and do a DFT on those
> samples. The result is a length N discrete transform (complex usually).
>
> Now we have a transform pair. At this stage there are no aperiodic
> components .. even though the original aperiodic components may have
> affected the original samples. In effect what we have is a *new*
> periodic sequence which deviates from the original *underlying* periodic
> components.
we need to be specific about what "we have" and what is deviating from
the original. as i can observe it, *nothing* is deviating from the
original if your entire universe is only those N samples. but if the
entire universe is only those N samples, then it doesn't make any sense
to talk of those "other" components, be they aperiodic or having a
period other than N. in a universe of only N samples, there is no (and
have never been) any meaning to any other components.
but, if you think of these N samples as a sorta "pocket universe" (sorry
to borrow from cosmology, Glen and Clay will probably wince) surrounded
by an infinite sea of zeros, then (if you FT) you have the DTFT. the
spectrum is continuous, but repeats every 2*pi. the spectrum is not
zero outside of the [-pi +pi) interval, but if you make it so (in the
mind of your brain or some other mathematician's brain), then those N
samples are no longer attached to N dirac deltas, but are attached to N
sinc() functions that go on forever and there is no periodicity in that
domain. not yet.
NOW (picking up on Fred's "when"), whether you zeroed the spectrum
outside of [-pi +pi) or not (i don't care if you do or not), if you
uniformly sample that spectrum with N samples from -pi to just under +pi
(or from 0 to just under 2*pi, i don't really care), you have the DFT.
and the act of sampling that spectrum *does* *necessarily* cause the
periodic extension of the original data, the N samples.
this is how you go from the one valid concept that "the DTFT is what you
get when you attach N delta functions to the N original data points
(uniformly spaced) and the DFT is what you get when you sample the DTFT
result" to the other equally valid (but i say is *more* fundamental)
that "the DFT invertibly maps one discrete and periodic sequence of
numbers to another discrete and periodic sequence of numbers of the same
period."
...
> So, anyway, that's how I deal with the question of aperiodic components
> .. which are only evident before N is selected. And, N, conceptually at
> least, becomes a period of something that was originally not periodic
> when we consider that we allow a discrete version of the FT of those
> samples.
>
> Obviously we can compute the FT of the N samples and get a continuous
> transform. Then the temporal periodicity wouldn't come up.
not yet...
> But that's not what we do.
... that's right. what we do is *sample* the DTFT and the undeniable
effect of that is the periodic extension of the
> So, I call "what we do" a context. Then there are more
> rigorous mathematical treatments to say the same thing but I rather
> think that this somewhat philosophical treatment is worthwhile.
it's also mathematically correct. and something that the periodicity
deniers just don't seem to get.
--
r b-j rbj@audioimagination.com
"Imagination is more important than knowledge."
Reply by Fred Marshall●December 10, 20112011-12-10
On 12/9/2011 9:31 AM, dbd wrote:
> On Dec 8, 8:47 am, Fred Marshall<fmarshallxremove_th...@acm.org>
> wrote:
>> On 12/7/2011 7:59 PM, glen herrmannsfeldt wrote:
>>
>>> You can consider the DFT as the FT of delta functions at the data
>>> points, and periodic boundary conditions.
>>
>>> -- glen
>>
>> Glen,
>>
>> What an elegant way to put it! I don't think in all the discussions
>> that I've seen it expressed this way.
>>
>> I've always just thought: this is how it is .. period.
>>
>> What does this imply about "and if you don't" [consider the DFT as the
>> FT of delta functions....etc.]?
>>
>> Fred
>
> Fred
>
> The statement is very succinct, anyway. How many readers of comp.dsp
> do you think correctly convert "periodic boundary conditions" to the
> required assumption that the delta functions can only consist of
> samples of signals that are sums of components at the frequencies of
> the basis functions of the DFT? Aperiodic components and components
> periodic at other frequencies than the DFT's basis functions don't
> meet the "periodic boundary conditions".
>
> Dale B. Dalrymple
Dale,
Yes, I understand and agree to a point. Except for "when?".
Let us say that some signal with aperiodic components (relative to our
intended DFT) is sampled for a long time.
Now let us select some N or temporal window and do a DFT on those
samples. The result is a length N discrete transform (complex usually).
Now we have a transform pair. At this stage there are no aperiodic
components .. even though the original aperiodic components may have
affected the original samples. In effect what we have is a *new*
periodic sequence which deviates from the original *underlying* periodic
components. Actually this occurred when we selected N. When we select
the sample rate and N, we are one way or another asserting its
periodicity. Well, I should add I guess "if we are going to compute a
DFT". I suppose there are other applications of those N samples that
wouldn't suggest such a thing.
Perhaps another way to put it is:
- once you've sampled there's no going back .. perfect reconstruction
being the only counter example that I can think of.
- once you've windowed i.e. selected N samples .. except for perhaps
things like concatenation of sequences with their neighboring samples ..
there's no going back.
- once you've DFT'd, you've put things in a context where everything is
periodic and there's no going back.
I'm not trolling for arguments or counter examples. Maybe these
assertions are just a "mind set". Without "proof" I think it's a useful
framework.
So, anyway, that's how I deal with the question of aperiodic components
.. which are only evident before N is selected. And, N, conceptually at
least, becomes a period of something that was originally not periodic
when we consider that we allow a discrete version of the FT of those
samples.
Obviously we can compute the FT of the N samples and get a continuous
transform. Then the temporal periodicity wouldn't come up. But that's
not what we do. So, I call "what we do" a context. Then there are more
rigorous mathematical treatments to say the same thing but I rather
think that this somewhat philosophical treatment is worthwhile.
And, of course, we all know that we can select "good" values of the
sample interval T and number of samples N and "bad" values of the same
such that some strong periodic component is grabbed with an integer plus
1/2 of a period is in the window. That's pretty "aperiodic" and the
underlying (i.e. original) boundary conditions are ugly and the
resulting DFT is ugly too. But calling the DFT "ugly" is a perspective
while saying "it is what it is", I think, is more to the point.
Agreed? I can't comment on "many readers". Perhaps so.
Fred
Reply by dbd●December 9, 20112011-12-09
On Dec 8, 8:47�am, Fred Marshall <fmarshallxremove_th...@acm.org>
wrote:
> On 12/7/2011 7:59 PM, glen herrmannsfeldt wrote:
>
> > You can consider the DFT as the FT of delta functions at the data
> > points, and periodic boundary conditions.
>
> > -- glen
>
> Glen,
>
> What an elegant way to put it! �I don't think in all the discussions
> that I've seen it expressed this way.
>
> I've always just thought: this is how it is .. period.
>
> What does this imply about "and if you don't" [consider the DFT as the
> FT of delta functions....etc.]?
>
> Fred
Fred
The statement is very succinct, anyway. How many readers of comp.dsp
do you think correctly convert "periodic boundary conditions" to the
required assumption that the delta functions can only consist of
samples of signals that are sums of components at the frequencies of
the basis functions of the DFT? Aperiodic components and components
periodic at other frequencies than the DFT's basis functions don't
meet the "periodic boundary conditions".
Dale B. Dalrymple
Reply by robert bristow-johnson●December 8, 20112011-12-08
On 12/8/11 11:47 AM, Fred Marshall wrote:
> On 12/7/2011 7:59 PM, glen herrmannsfeldt wrote:
>> You can consider the DFT as the FT of delta functions at the data
>> points, and periodic boundary conditions.
do the periodic boundary conditions apply in which domain? if only in
the frequency domain, that's the DTFT. if both domains, then it's the DFT.
> What an elegant way to put it! I don't think in all the discussions that
> I've seen it expressed this way.
it sorta sounds more diplomatic than how i generally insist what the DFT
is (as an invertible mapping of one discrete and periodic sequence to
another discrete and periodic sequence of the same period), but isn't it
equivalent?
> I've always just thought: this is how it is .. period.
really, Fred? you certainly saw it as doing the same sorta thing as the
Fourier transform (or series) in that you're representing one function
as a collection of sinusoids (lest it be named after someone other than
Fourier), no?
> What does this imply about "and if you don't" [consider the DFT as the
> FT of delta functions....etc.]?
i think we consider the DFT to be the uniform *sampling* of the FT of a
finite set of uniformly-spaced delta functions. what does that sampling
of the FT imply back in the domain of those delta functions on the other
side of the FT?
i'm skating close to the periodic (or ongoing) political or theological
or philosophical dispute we have here about the DFT. <<but me still
thinks our side is Right: The DFT is one and the same as the Discrete
Fourier Series because God made it so and if you don't believe that then
yer going to hell fer sher.>> but i guess we shouldn't be burning to
the stake the folks who don't see it so. <<tolerance is hard, it's
funner to just burn the heretics.>>
--
r b-j rbj@audioimagination.com
"Imagination is more important than knowledge."