Appendix A: Types of Fourier Transforms

dbd <dbd@ieee.org> wrote:
> On Jan 10, 4:32&#4294967295;pm, robert bristow-johnson <r...@audioimagination.com>
> wrote:
 
>> one old issue that gets me into fights here is that i disagree with
>> the mathematical factuality of this statement: "Finite x[n] length N,
>> zero elsewhere".

>> it's oft repeated, you see it in print, and it's wrong.
 
> This appears so often because it is the only information that the DFT
> is given. There are many assumptions that can be made about the region
> outside the "length N", but the consequences of  those assumptions are
> properties of the assumptions and not properties of the DFT.
 
> The output of the DFT is not a function of the values outside that
> "length N" The set of all functions I can calculate the DFT on N
> samples of is not limited to periodic functions.

The DFT basis functions, and so the DFT itself, have periodic
boundary conditions.  This becomes noticable if you have data
with a large discontinuity between X[0] and X[N-1], (relative
to pairs of sequential data points.)

Note, in contrast, that DST and DCT don't have periodic 
boundary conditions.  The DST has the boundary condition
that the function (and basis functions) go to zero at the boundary,
and DCT has the derivative go to zero at the boundary.

Both DST and DCT allow odd multiples of a half cycle of the
basis functions over the transform length such that, even
as sums of sines and cosines, are not periodic with length N.
(They are periodic with length 2N, though.)

> It would certainly be convenient and elegant if all signals of which I
> might come across a set of N samples had the characteristic that
> contiguous blocks of the signal had identical samples. Some people
> seem to have been so seduced by the convenience and elegance of a
> universe limited to such signals that they decide that they should
> live only in a world where that is true and they try to convince those
> around them to live there. But there have been many stochastic or non-
> stationary or even discrete frequency signals for which periodicity
> does not hold despite the fact that I have sampled  and performed the
> DFT on N of those samples.

In those cases, you should ask if other transforms would be more
appropriate.  There is a reason for the popularity of DCT in
signal processing.
 
-- glen

On Jan 11, 3:50&#4294967295;pm, glen herrmannsfeldt <g...@ugcs.caltech.edu> wrote:
> dbd <d...@ieee.org> wrote:
> > On Jan 10, 4:32&#4294967295;pm, robert bristow-johnson <r...@audioimagination.com>
> > wrote:
> >> one old issue that gets me into fights here is that i disagree with
> >> the mathematical factuality of this statement: "Finite x[n] length N,
> >> zero elsewhere".
> >> it's oft repeated, you see it in print, and it's wrong.
> > This appears so often because it is the only information that the DFT
> > is given. There are many assumptions that can be made about the region
> > outside the "length N", but the consequences of &#4294967295;those assumptions are
> > properties of the assumptions and not properties of the DFT.
> > The output of the DFT is not a function of the values outside that
> > "length N" The set of all functions I can calculate the DFT on N
> > samples of is not limited to periodic functions.
>
> The DFT basis functions, and so the DFT itself, have periodic
> boundary conditions. &#4294967295;This becomes noticable if you have data
> with a large discontinuity between X[0] and X[N-1], (relative
> to pairs of sequential data points.)
>

The application of a set of basis functions to a set of data samples
does not require "boundary conditions" on the product. These are
assumptions you are free to make for an application. They are not part
of the DFT. You point out that data can violate boundary conditions.
That just means that your assumptions are not correct for the data.
Your assumptions are incorrect for a lot of people's applications and
data.

Dale B. Dalrymple

Dale B. Dalrymple

>On Jan 10, 4:32=A0pm, robert bristow-johnson <r...@audioimagination.com>
>wrote:
>> ...
>
>> one old issue that gets me into fights here is that i disagree with
>> the mathematical factuality of this statement: "Finite x[n] length N,
>> zero elsewhere".
>>
>> it's oft repeated, you see it in print, and it's wrong.
>>
>> r b-j
>
>This appears so often because it is the only information that the DFT
>is given. There are many assumptions that can be made about the region
>outside the "length N", but the consequences of  those assumptions are
>properties of the assumptions and not properties of the DFT.
>
>The output of the DFT is not a function of the values outside that
>"length N" The set of all functions I can calculate the DFT on N
>samples of is not limited to periodic functions.

That would imply that the validity of the answer from the DFT is not
dependant on the values outside the length N, which is clearly not the
case. The answer is predicated on the assumption of a periodic signal, and
windowing function exist for the express reason of mitigating the effects
of that not always being true.

>It would certainly be convenient and elegant if all signals of which I
>might come across a set of N samples had the characteristic that
>contiguous blocks of the signal had identical samples. Some people
>seem to have been so seduced by the convenience and elegance of a
>universe limited to such signals that they decide that they should
>live only in a world where that is true and they try to convince those
>around them to live there. But there have been many stochastic or non-
>stationary or even discrete frequency signals for which periodicity
>does not hold despite the fact that I have sampled  and performed the
>DFT on N of those samples.
>
>Dale B. Dalrymple

Steve

On Jan 11, 10:23�am, robert bristow-johnson
<r...@audioimagination.com> wrote:

> periodic on one domain implies discrete (with dirac impulses) in the
> other domain and vise-versa.
>
> the converse is also true: &#4294967295;non-periodic on one domain implies
> continuous in the other domain.

Not true.  exp(j w0 n) is periodic with period N iff  w0/2pi = k/N (k
is an integer); otherwise it is not.  But, independent of whether or
not exp(j w0 n) is periodic, its DTFT is 2pi delta(w - w0), which is a
line spectrum.  Hence, periodic in one domain necessarily means "line"
in the other domain, but "line" in one domain doesn't alway imply
periodicity, as the above example shows.

--vv

On Jan 11, 4:49&#4294967295;pm, "steveu" <steveu@n_o_s_p_a_m.coppice.org> wrote:
> >On Jan 10, 4:32=A0pm, robert bristow-johnson <r...@audioimagination.com>
> >wrote:
> >> ...
>
> >> one old issue that gets me into fights here is that i disagree with
> >> the mathematical factuality of this statement: "Finite x[n] length N,
> >> zero elsewhere".
>
> >> it's oft repeated, you see it in print, and it's wrong.
>
> >> r b-j
>
> >This appears so often because it is the only information that the DFT
> >is given. There are many assumptions that can be made about the region
> >outside the "length N", but the consequences of &#4294967295;those assumptions are
> >properties of the assumptions and not properties of the DFT.
>
> >The output of the DFT is not a function of the values outside that
> >"length N" The set of all functions I can calculate the DFT on N
> >samples of is not limited to periodic functions.
>
> That would imply that the validity of the answer from the DFT is not
> dependant on the values outside the length N, which is clearly not the
> case. The answer is predicated on the assumption of a periodic signal, and
> windowing function exist for the express reason of mitigating the effects
> of that not always being true.

If the known data length and the DFT length is finite, the window
function is
automatically rectangular.  The basis functions don't even need to be
defined outside of the DFT aperture.


IMHO. YMMV.
--
rhn A.T nicholson d.0.t C-o-M
 http://www.nicholson.com/rhn/dsp.html

On Jan 12, 1:40&#4294967295;am, VV <vanam...@netzero.net> wrote:
> On Jan 11, 10:23&#4294967295;am, robert bristow-johnson
>
> <r...@audioimagination.com> wrote:
> > periodic on one domain implies discrete (with dirac impulses) in the
> > other domain and vise-versa.
>
> > the converse is also true: &#4294967295;non-periodic on one domain implies
> > continuous in the other domain.
>
> Not true. &#4294967295;exp(j w0 n) is periodic with period N iff &#4294967295;w0/2pi = k/N (k
> is an integer); otherwise it is not. &#4294967295;But, independent of whether or
> not exp(j w0 n) is periodic, its DTFT is 2pi delta(w - w0), which is a
> line spectrum. &#4294967295;Hence, periodic in one domain necessarily means "line"
> in the other domain, but "line" in one domain doesn't always imply
> periodicity, as the above example shows.

yeah, i know.  i was trying to account for that when answering Rune
with the "...it depends on how anal-retentive (or OCD) the engineer
is" paragraph.

"...we never, in practical reality, ever have to deal with the
situation you bring up where, say, a million digits for pi ain't
enough."

both k and N must be integers, but i can find a k,N pair that will
approximate whatever w0/(2pi) that you toss me to whatever finite
precision that you require.  maybe N has to be a zillion, to get it
close enough.  then, to the same degree of accuracy, x[n+N]=x[n] .
there would be lots of dirac deltas with zero coefficients and just
one, the kth dirac impulse, that would not be zero.

r b-j

On Jan 11, 12:23�am, robert bristow-johnson
<r...@audioimagination.com> wrote:
> On Jan 10, 10:30&#4294967295;pm, Rune Allnor <all...@tele.ntnu.no> wrote:
>
> > On Jan 10, 9:00&#4294967295;pm, Tim Wescott <t...@seemywebsite.com> wrote:
>
> > > Just to inscribe this on the wall in cyberspace:
> > > &#4294967295;From this you get the four possible combinations:
>
> > > continuous-time infinite --> continuous-frequency infinite.
> > > continuous-time cyclic --> discrete-frequency infinite.
> > > discrete-time infinite --> continuous-frequency cyclic.
> > > discrete-time cyclic --> discrete-frequency, cyclic.
>
> > Almost correct, but crucially wrong: There is no suct thing as
> > a 'cyclic' signal. Those signals are of infinite duration.
> > You confuse the 'cyclic' signal with the 'finite duration'
> > *function*.
>
> sorry, Rune. &#4294967295;but it's you (and many others) that are crucially
> wrong. &#4294967295;and i'm willing to take you on about it. &#4294967295;i've been doing this
> multiple times since my very beginning here at comp.dsp in 1995 or 96.
>
> > Your variants 2 and 4 are the FTs that work
> > on finite-duration domains. For continuous time, that's
> > the Fourier series; for discrete time that's the DFT.
>
> fundamentally, the DFT is nothing other than the DFS. &#4294967295;the DFT maps a
> discrete and periodic function in one domain (we'll call it the "time
> domain") to another discrete and periodic function in the reciprocal
> domain (we'll call it the "frequency domain").
>
> that's what the DFT does.
>
> periodic on one domain implies discrete (with dirac impulses) in the
> other domain and vise-versa.
>
> the converse is also true: &#4294967295;non-periodic on one domain implies
> continuous in the other domain.
>
> just because a function can be fully described by a finite-length
> segment of that function, does not mean that the function is itself
> finite in duration.
>
> r b-j

Robert,

This reminds of a story about three guys looking out a window. The 1st
remarks there is a cow outside eating grass. The 2nd remarks there is
a brown cow outside eating grass. Finally the mathematician says there
is a cow outside eating grass and the side that I can see is brown.

If you approach DFTs from a linear algebra approach where your finite
length of data gets expanded into a linear combination of finite
length basis vectors. You can see nothing is stated about what the
function to be analyzed or the basis functions do outside of the
interval of interest. I don't know the color of the other side of the
cow.

Clay

On 1/10/2011 11:38 PM, Rune Allnor wrote:
>
> And that's where the problem occurs: The condition for the
> FT of a function x(t) to exist (CT infinite domain) is that
>
> integral |x(t)|^2 dt<  infinite
>
> With x(t) = sin(t) that breaks down. Note that this has nothing
> to do with the sine being periodic, it has to do with it having
> infinite energy. (Try the same excercise with y(t) = sin(t)+sin(pi*t)
> to see why.)
>
> To get out of that embarrasment, engineers (*not* mathematicians)
> came up with the ad hoc solution to express the periodic sine as
> a sequence of periods, repeated ad infinitum, and compute the
> Fourier series of one period.
>
> It has no mathematical meaning, as the rather essential property
> of linearity of the FT breaks down (again, use the y(t) above to
> see why).
>
> Again, this is totally trivial.
>
> Rune

I guess mathematicians over time have thus had a lot of fun using stuff 
that engineers came up with....  I fail to acknowledge a fundamental 
difference between the two sets of folks where stuff like this is 
concerned.  Obviously each have their areas of expertise that go beyond.

Dale repeats his point about the DFT being (in my terms) an abstract 
thingy that is simply a mapping of N points - having nothing whatsoever 
to do with any imagined or real samples which may exist outside the 
sample regions.  This is a surely a valid perspective of the FT as a 
*mapping* but I've not reconciled how it fits in my own perspectives. 
It seems that r b-j takes the opposite view which matches better with my 
working framework.  And, to be clear, I think that framework is less 
abstract and related to real-world signals - which is handy at the least.

Let's see then, here is a "constructive" approach to the topic:
(I will likely use the "engineers" convention of "believing in" Diracs 
either explicitly or implied).
BE AWARE: the functions mentioned are only discrete when so defined!!

If one takes an infinite continuous function and samples it then I guess 
we should say that it should start out "bandlimited" just to be safe 
(although I'm not sure that latter caution is necessary here - see 
Footnote #1).

Now, we compute the Fourier Transform of this function
f(t) > F(w) ... continuous/infinite

And, we compute the Fourier Transform of the sampled version:
f(nT) > F'(w) with F'(w) continuous/infinite
And, we recognize that the Fourier Transform in this case can be 
simplified into a discrete (infinite) Fourier Transform .. that is the 
integral becomes a sum over the discrete samples in time.  But that's 
only a trivial simplification so far so we still have:
f(nT) > F'(w) with F'(w) continuous/infinite

Now I will assert that our continuous/infinite F'(w) is periodic with 
period 1/T. [If this assertion is warranted, I could use some help with 
that right now].

We can now compute the Inverse Fourier Transform of F'(w).  And, because 
F'(w) is periodic, we recognize that the Fourier Transform can be 
simplified from an infinite integral to a finite integral over one 
period which becomes a finite (not discrete) Fourier Transform which we 
recognize as the computation of the Fourier Series coefficients which 
should be the same as f(nT).  But just to be careful, let's call this 
f'(nT), OK?  [see Footnote #1]

At this point we have infinite, discrete f'(nT) and continuous, periodic 
F'(w).  And, so far, I think this is in a context that we can all 
understand.

But wait!  Having continuous F'(w) is really inconvenient isn't it? 
And, having infinite f'(nT) is also really inconvenient isn't it?
What we'd really like is for f'(nT) to be finite.
And, what we'd really like is for F'(w) to be discrete so we can 
represent it with numbers instead of some mathematical functional 
expression.  Where to start?

If we time-limit f'(nT) we also convolve F'(w) with a sinc.  So that 
introduces spectral spreading or a type of aliasing.

If we sample the infinite F'(w), we make f'(nT) periodic.

Of course, in the end we want to do both but I wonder if folks don't 
often think of this as being one or the other - or just don't think 
about it at all?

OK.  So let's start out by sampling F'(w).  In order to avoid aliasing, 
we would like to pick the frequency sample interval W and in order to 
avoid temporal aliasing or overlap at all, we need the extent of f'(nT) 
to be less than 1/2W.

So, it appears that making F'(w) discrete and making f'(nT) time-limited 
really amount to the same thing.  We have to accept some spectral 
spreading if indeed f'(nT) starts out being infinite and we have to 
accept some temporal aliasing if F'(w) is going to be sampled.

What has happened of course is that we all accept these potential 
"problems".
- In fact, we don't encounter temporal aliasing because we *never* start 
with an infinite f(nT) in the real world.  This means that the samples 
f'(nT) are perfect over the interval NT and that F'(w) is a perfect 
mapping that can be inversed (still continuous, peridic here).  We only 
have to deal with the potential for temporal aliasing when doing 
circular convolution.  This aspect aligns with Dale's view.
- And we're all used to dealing with the spectral spreading caused by 
time limiting f'(nT) to the range of n being limited to N.

So, let's NOT start out by sampling F'(w) then.
Let's start by time-limiting f(nT) with the range of n being limited to 
N.  That shouldn't bother anyone too much because it's what we almost 
always do anyway!
Now that f'(nT) is time-limited in the normal fashion, we can consider 
sampling F'(w).  We just need to pick the sample interval W.

Well, we have decided that f'(nT) is going to be time-limited already. 
And, we don't want to cause temporal aliasing or overlap by sampling 
F'(w) too sparsely.  What is the sample interval that will *just* avoid 
such overlap?
If the length of f'(nT) is NT then the temporal period introduced by 
sampling F'(w) must be >= NT.  [And, as we discussed recently, this 
means that the period is of duration => NT and is, if you will, 
"spanned" by at least N+1 samples where the end samples are equal). 
This means that W=>1/NT and we normally choose W=NT.

Thus sampling, we have: F'(kW) = F'(k/NT) where F'(w) was already 
periodic over 1/T so we have a range of k limited to N.
Having sampled F'(w), we now have a periodic version of the assumed 
time-limited f'(nT) which we'll call f''(nT).

Thus, we have taken an acceptable time-limited sample in time and 
converted it to a sampled and periodic "time function" in order to also 
be able to have a sampled "frequency function" which happens to also be 
periodic.

As above, one could decide using Dale's framework that there is a finite 
sequence in time ... which is something that we're all very used to 
anyway ... and it has a corresponding finite sequence in frequency.  I 
see no big problem with that but it's not the way I like to *think* 
about it.  And, I've been prone to saying that something *is* periodic 
when perhaps I should say that *I prefer to think of it* as periodic. 
There's certainly a connection to the literature, physical systems, etc.

I see very little difference looking at a finite sequence on a line and 
mapping that sequence over the finite length into a circle.  One can 
choose to traverse a function plotted on the circle just once (which is 
equivalent to it being finite on an infinite line) or consider it to be 
representative of a periodic function and traverse the circle 
continuously.  There is precedent for this:
In antenna and array design we can plot the beam pattern as a periodic 
function of the look angle in a polar plot OR we can plot the beam 
pattern as a function of an infinite-ranged look angle.  The historic 
van der Maas function for antenna patterns was done on the latter and 
has infinite extent.  In this case the antenna is continuous of finite 
length and the beam pattern is continuous and infinite.  Then we get 
into terminology like "the visible region" etc. etc.


Fred


Footnote #1:
If f(t) is *not* strictly bandlimited to B < 1/2T:
then the computation of continuous/infinite F'(w) will involve some 
overlap / "aliasing".
Thus, the Inverse Fourier Transform of F'(w) will not match the original 
f(nT) so we call it f'(nT).

I think this must be what Rune was referring to re: linearity.....

Anyway, given f'(nT) we can compute its Fourier Transform to get F'(w).
So, now we have a consistent transform pair.
f'(nT) and F'(w)
but f'(nT) here is no longer necessarily a perfect replica of anything 
that may have existed at the "beginning".

Footnote #2:
When we're dealing with real-world signals there's no such thing as 
strictly bandlimited nor infinite extent including infinite periodic. 
But there is such thing as strictly time-limited which analytically 
means infinite bandwidth.  We have to live with the discrepancy here and 
do so by accepting effective time spans and effective bandwidths.

On Jan 12, 2:36&#4294967295;pm, Fred Marshall <fmarshall_xremove_the...@xacm.org>
wrote:
> On 1/10/2011 11:38 PM, Rune Allnor wrote:
>
>
>
>
>
>
>
> > And that's where the problem occurs: The condition for the
> > FT of a function x(t) to exist (CT infinite domain) is that
>
> > integral |x(t)|^2 dt< &#4294967295;infinite
>
> > With x(t) = sin(t) that breaks down. Note that this has nothing
> > to do with the sine being periodic, it has to do with it having
> > infinite energy. (Try the same excercise with y(t) = sin(t)+sin(pi*t)
> > to see why.)
>
> > To get out of that embarrasment, engineers (*not* mathematicians)
> > came up with the ad hoc solution to express the periodic sine as
> > a sequence of periods, repeated ad infinitum, and compute the
> > Fourier series of one period.
>
> > It has no mathematical meaning, as the rather essential property
> > of linearity of the FT breaks down (again, use the y(t) above to
> > see why).
>
> > Again, this is totally trivial.
>
> > Rune
>
> I guess mathematicians over time have thus had a lot of fun using stuff
> that engineers came up with.... &#4294967295;I fail to acknowledge a fundamental
> difference between the two sets of folks where stuff like this is
> concerned. &#4294967295;Obviously each have their areas of expertise that go beyond.
>
> Dale repeats his point about the DFT being (in my terms) an abstract
> thingy that is simply a mapping of N points - having nothing whatsoever
> to do with any imagined or real samples which may exist outside the
> sample regions. &#4294967295;This is a surely a valid perspective of the FT as a
> *mapping* but I've not reconciled how it fits in my own perspectives.
> It seems that r b-j takes the opposite view which matches better with my
> working framework. &#4294967295;And, to be clear, I think that framework is less
> abstract and related to real-world signals - which is handy at the least.
>
> Let's see then, here is a "constructive" approach to the topic:
> (I will likely use the "engineers" convention of "believing in" Diracs
> either explicitly or implied).
> BE AWARE: the functions mentioned are only discrete when so defined!!
>
> If one takes an infinite continuous function and samples it then I guess
> we should say that it should start out "bandlimited" just to be safe
> (although I'm not sure that latter caution is necessary here - see
> Footnote #1).
>
> Now, we compute the Fourier Transform of this function
> f(t) > F(w) ... continuous/infinite
>
> And, we compute the Fourier Transform of the sampled version:
> f(nT) > F'(w) with F'(w) continuous/infinite
> And, we recognize that the Fourier Transform in this case can be
> simplified into a discrete (infinite) Fourier Transform .. that is the
> integral becomes a sum over the discrete samples in time. &#4294967295;But that's
> only a trivial simplification so far so we still have:
> f(nT) > F'(w) with F'(w) continuous/infinite
>
> Now I will assert that our continuous/infinite F'(w) is periodic with
> period 1/T. [If this assertion is warranted, I could use some help with
> that right now].
>
> We can now compute the Inverse Fourier Transform of F'(w). &#4294967295;And, because
> F'(w) is periodic, we recognize that the Fourier Transform can be
> simplified from an infinite integral to a finite integral over one
> period which becomes a finite (not discrete) Fourier Transform which we
> recognize as the computation of the Fourier Series coefficients which
> should be the same as f(nT). &#4294967295;But just to be careful, let's call this
> f'(nT), OK? &#4294967295;[see Footnote #1]
>
> At this point we have infinite, discrete f'(nT) and continuous, periodic
> F'(w). &#4294967295;And, so far, I think this is in a context that we can all
> understand.
>
> But wait! &#4294967295;Having continuous F'(w) is really inconvenient isn't it?
> And, having infinite f'(nT) is also really inconvenient isn't it?
> What we'd really like is for f'(nT) to be finite.
> And, what we'd really like is for F'(w) to be discrete so we can
> represent it with numbers instead of some mathematical functional
> expression. &#4294967295;Where to start?
>
> If we time-limit f'(nT) we also convolve F'(w) with a sinc. &#4294967295;So that
> introduces spectral spreading or a type of aliasing.
>
> If we sample the infinite F'(w), we make f'(nT) periodic.
>
> Of course, in the end we want to do both but I wonder if folks don't
> often think of this as being one or the other - or just don't think
> about it at all?
>
> OK. &#4294967295;So let's start out by sampling F'(w). &#4294967295;In order to avoid aliasing,
> we would like to pick the frequency sample interval W and in order to
> avoid temporal aliasing or overlap at all, we need the extent of f'(nT)
> to be less than 1/2W.
>
> So, it appears that making F'(w) discrete and making f'(nT) time-limited
> really amount to the same thing. &#4294967295;We have to accept some spectral
> spreading if indeed f'(nT) starts out being infinite and we have to
> accept some temporal aliasing if F'(w) is going to be sampled.
>
> What has happened of course is that we all accept these potential
> "problems".
> - In fact, we don't encounter temporal aliasing because we *never* start
> with an infinite f(nT) in the real world. &#4294967295;This means that the samples
> f'(nT) are perfect over the interval NT and that F'(w) is a perfect
> mapping that can be inversed (still continuous, peridic here). &#4294967295;We only
> have to deal with the potential for temporal aliasing when doing
> circular convolution. &#4294967295;This aspect aligns with Dale's view.
> - And we're all used to dealing with the spectral spreading caused by
> time limiting f'(nT) to the range of n being limited to N.
>
> So, let's NOT start out by sampling F'(w) then.
> Let's start by time-limiting f(nT) with the range of n being limited to
> N. &#4294967295;That shouldn't bother anyone too much because it's what we almost
> always do anyway!
> Now that f'(nT) is time-limited in the normal fashion, we can consider
> sampling F'(w). &#4294967295;We just need to pick the sample interval W.
>
> Well, we have decided that f'(nT) is going to be time-limited already.
> And, we don't want to cause temporal aliasing or overlap by sampling
> F'(w) too sparsely. &#4294967295;What is the sample interval that will *just* avoid
> such overlap?
> If the length of f'(nT) is NT then the temporal period introduced by
> sampling F'(w) must be >= NT. &#4294967295;[And, as we discussed recently, this
> means that the period is of duration => NT and is, if you will,
> "spanned" by at least N+1 samples where the end samples are equal).
> This means that W=>1/NT and we normally choose W=NT.
>
> Thus sampling, we have: F'(kW) = F'(k/NT) where F'(w) was already
> periodic over 1/T so we have a range of k limited to N.
> Having sampled F'(w), we now have a periodic version of the assumed
> time-limited f'(nT) which we'll call f''(nT).
>
> Thus, we have taken an acceptable time-limited sample in time and
> converted it to a sampled and periodic "time function" in order to also
> be able to have a sampled "frequency function" which happens to also be
> periodic.
>
> As above, one could decide using Dale's framework that there is a finite
> sequence in time ... which is something that we're all very used to
> anyway ... and it has a corresponding finite sequence in frequency. &#4294967295;I
> see no big problem with that but it's not the way I like to *think*
> about it. &#4294967295;And, I've been prone to saying that something *is* periodic
> when perhaps I should say that *I prefer to think of it* as periodic.
> There's certainly a connection to the literature, physical systems, etc.
>
> I see very little difference looking at a finite sequence on a line and
> mapping that sequence over the finite length into a circle. &#4294967295;One can
> choose to traverse a function plotted on the circle just once (which is
> equivalent to it being finite on an infinite line) or consider it to be
> representative of a periodic function and traverse the circle
> continuously. &#4294967295;There is precedent for this:
> In antenna and array design we can plot the beam pattern as a periodic
> function of the look angle in a polar plot OR we can plot the beam
> pattern as a function of an infinite-ranged look angle. &#4294967295;The historic
> van der Maas function for antenna patterns was done on the latter and
> has infinite extent. &#4294967295;In this case the antenna is continuous of finite
> length and the beam pattern is continuous and infinite. &#4294967295;Then we get
> into terminology like "the visible region" etc. etc.
>
> Fred
>
> Footnote #1:
> If f(t) is *not* strictly bandlimited to B < 1/2T:
> then the computation of continuous/infinite F'(w) will involve some
> overlap / "aliasing".
> Thus, the Inverse Fourier Transform of F'(w) will not match the original
> f(nT) so we call it f'(nT).
>
> I think this must be what Rune was referring to re: linearity.....
>
> Anyway, given f'(nT) we can compute its Fourier Transform to get F'(w).
> So, now we have a consistent transform pair.
> f'(nT) and F'(w)
> but f'(nT) here is no longer necessarily a perfect replica of anything
> that may have existed at the "beginning".
>
> Footnote #2:
> When we're dealing with real-world signals there's no such thing as
> strictly bandlimited nor infinite extent including infinite periodic.
> But there is such thing as strictly time-limited which analytically
> means infinite bandwidth. &#4294967295;We have to live with the discrepancy here and
> do so by accepting effective time spans and effective bandwidths.- Hide quoted text -
>
> - Show quoted text -

Fred,

Thanks for your summing up the two different philosophies here.

I think one issue is viewing the DFT as some sort of limiting form of
a Fourier Series which in turn can be derived from Fourier Transforms.
A program to calculate a DFT does not perform some other operation
first like finding the Fourier Series or Fourier Transform and then
apply some sort of conversion (simplification or approximation) to
arrive at the Discrete Fourier Transform. It simply performs a finite
number of vector dot products with finite length vectors to find the
DFT directly from the finite length of data. So while one may view a
DFT as a limiting form of another transformation, it is not the other
transformation, therefore not all properties associated with the other
transforms are carried forward. Operationally the DFT performs a
mapping. That is how I look at it. Now as to why one wants the DFT of
a set of data maybe justified by these transform relationships. But
one should be careful as to how they differ.

Maybe this is my mathematician background coming through, but oh well.

It is sort of like when one says two triangles are the same. I say no
they are not the same since they are two triangles. We say they are
congruent -  everything we can measure about them is the same, but
they themselves are not the same. Maybe it is a form of pedantic hair
splitting.

Clay

On Jan 11, 4:49&#4294967295;pm, "steveu" <steveu@n_o_s_p_a_m.coppice.org> wrote:

> dbd:
> >The output of the DFT is not a function of the values outside that
> >"length N" The set of all functions I can calculate the DFT on N
> >samples of is not limited to periodic functions.
>
> That would imply that the validity of the answer from the DFT is not
> dependant on the values outside the length N, which is clearly not the
> case.

Just what equation do you use to calculate the N point DFT then?

> The answer is predicated on the assumption of a periodic signal, and
> windowing function exist for the express reason of mitigating the effects
> of that not always being true.

You are free to chose that assumption. It can be useful at times. It
is damaging at other times, which is why many people don't make that
assumption. They still find the DFT useful, in many ways that your
assumption isn't.

>
> Steve

Dale B. Dalrymple