Appendix A: Types of Fourier Transforms

On Jan 12, 8:36&#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.

I garee with Dale on this.

>&#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's the *only* perspective of the FT, in any of its
shades, shapes or forms.


> If one takes an infinite continuous function and samples it then...

Why do you bring sampling into this? We were discussing
the FT up till this point, not sampling.

> OK. &#4294967295;So let's start out by sampling F'(w). &#4294967295;

Keeping in tune with your 'practical' approach: How do
you 'sample' F(w)? What kinds of ADCs work in frequency
domain?

All you have achieved is to swap a quagmire for quick sand.

Rune

On Jan 12, 8:36&#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.

I garee with Dale on this.

>&#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's the *only* perspective of the FT, in any of its
shades, shapes or forms.


> If one takes an infinite continuous function and samples it then...

Why do you bring sampling into this? We were discussing
the FT up till this point, not sampling.

> OK. &#4294967295;So let's start out by sampling F'(w). &#4294967295;

Keeping in tune with your 'practical' approach: How do
you 'sample' F(w)? What kinds of ADCs work in frequency
domain?

All you have achieved is to swap a quagmire for quick sand.

Rune

On Jan 12, 8:36&#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.

I garee with Dale on this.

>&#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's the *only* perspective of the FT, in any of its
shades, shapes or forms.


> If one takes an infinite continuous function and samples it then...

Why do you bring sampling into this? We were discussing
the FT up till this point, not sampling.

> OK. &#4294967295;So let's start out by sampling F'(w). &#4294967295;

Keeping in tune with your 'practical' approach: How do
you 'sample' F(w)? What kinds of ADCs work in frequency
domain?

All you have achieved is to swap a quagmire for quick sand.

Rune

On Jan 12, 8:36=A0pm, 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< =A0infinite
>
> > With x(t) =3D 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) =3D 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.... =A0I fail to acknowledge a fundamental
> difference between the two sets of folks where stuff like this is
> concerned. =A0Obviously 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.

I garee with Dale on this.

>=A0This 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's the *only* perspective of the FT, in any of its
shades, shapes or forms.


> If one takes an infinite continuous function and samples it then...

Why do you bring sampling into this? We were discussing
the FT up till this point, not sampling.

> OK. =A0So let's start out by sampling F'(w). =A0

Keeping in tune with your 'practical' approach: How do
you 'sample' F(w)? What kinds of ADCs work in frequency
domain?

All you have achieved is to swap a quagmire for quick sand.

Rune

On Jan 12, 8:36=A0pm, 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< =A0infinite
>
> > With x(t) =3D 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) =3D 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.... =A0I fail to acknowledge a fundamental
> difference between the two sets of folks where stuff like this is
> concerned. =A0Obviously 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.

I garee with Dale on this.

>=A0This 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's the *only* perspective of the FT, in any of its
shades, shapes or forms.


> If one takes an infinite continuous function and samples it then...

Why do you bring sampling into this? We were discussing
the FT up till this point, not sampling.

> OK. =A0So let's start out by sampling F'(w). =A0

Keeping in tune with your 'practical' approach: How do
you 'sample' F(w)? What kinds of ADCs work in frequency
domain?

All you have achieved is to swap a quagmire for quick sand.

Rune

On Jan 12, 8:36=A0pm, 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< =A0infinite
>
> > With x(t) =3D 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) =3D 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.... =A0I fail to acknowledge a fundamental
> difference between the two sets of folks where stuff like this is
> concerned. =A0Obviously 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.

I garee with Dale on this.

>=A0This 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's the *only* perspective of the FT, in any of its
shades, shapes or forms.


> If one takes an infinite continuous function and samples it then...

Why do you bring sampling into this? We were discussing
the FT up till this point, not sampling.

> OK. =A0So let's start out by sampling F'(w). =A0

Keeping in tune with your 'practical' approach: How do
you 'sample' F(w)? What kinds of ADCs work in frequency
domain?

All you have achieved is to swap a quagmire for quick sand.

Rune

On Jan 12, 9:57&#4294967295;pm, Clay <c...@claysturner.com> wrote:
> On Jan 12, 2:36&#4294967295;pm, Fred Marshall <fmarshall_xremove_the...@xacm.org>

> 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.

Which is the main problem. There is no reason to think the
various FTs are interrelated. They work in different
mathemathical domains.

> 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.

Exactly.

> 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.

The justification for wanting the DFT is very straightforward:
It is the only variant of the FT that works on a finite set of
discrete data points. The DTFT works on an infinite set of
discrete data points, the other two variants work on continuous
data.

Only finite sets of discrete data points can be handled with
electronic computers.

> But
> one should be careful as to how they differ.

The justification to study all these limits and relations
between the FT variants is to find out what adjustments
one must do to get from the computed result, the DFT, to
the desired result.

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

Nope. That's an intelligent mind coming through.

> 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 - &#4294967295;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.

Nope. It's precise terminology that aids the mind towards
insight.

Rune

On Jan 13, 12:52�am, eric.jacob...@ieee.org (Eric Jacobsen) wrote:

> I don't think it is all that important which point of view a person
> takes, as long as the limitations and nuances are understood.

Wrong. If one gets the basics right first time around, one can
either get things done more quickly or easily, or one can achieve
more than one otherwise would.

THis stuff is simple. No need to spend lots of time concocting
explanations involving dependencies on numbers that never enter
the computations.

>� If one
> can understand and appreciate both points of view, and others as well,
> then I think it maximizes an individual's potential understanding, and
> also makes it easier to talk to folks of any specific point of view.

No. That kind of thing only spawns chaos and confusion.

Rune

dbd <dbd@ieee.org> wrote:
(snip)

> I appreciate the convenience and comfort of having a digital method
> that can produce a Fourier Transform. For a special signal space, the
> linear combinations of the independent basis vectors, that abstract
> mapping DFT thingie Dale talks about does just that. We should teach
> that, but not leave out explaining the signal space restrictions.
 
> My motivation does not come from a mathematical background. The
> companies I have worked for (and many others) have sold digital
> spectrum analyzers to customers who don't limit themselves to that
> special sparse signal space. They want DFTs on stochastic data and non-
> stationary data as well as periodic data and they know that their
> periodic data have components of non-commensurate periods. Not a one
> of them wants to see a 2D plot of perfectly periodic data. This is
> real world, every day, in the field, instrumentation not mathematical
> abstraction. I don't care that some people aren't comfortable going
> honestly there.

If you understand the transform, then it can be used with non-periodic
data, but it should be done knowing the transform boundary conditions.

Some time ago there was a discussion on the FT of a Gaussian being
a Gaussian, and the desire to compute one using the DFT.  It isn't
hard, most important being to center the Gaussian on zero.  If the
transform is long enough, the boundaries are far from the important
part, and it usually works well enough.

> Teaching that "the DFT calculates the Fourier Transform" and "anything
> I calculate a DFT of is periodic" is harmful. Whether these have been
> intentional in all cases or not, they are what is often getting
> across. 

The beginning of band theory in solid-state physics classes
is discussed with periodic boundary conditions.  There are no
blocks of silicon that have periodic boundary conditions (in one,
two, or three dimensions), but the calculation works well if you
want to know the wave function far from the surface.  A not-so-big
piece of silicon has 1e21 atoms, or, in a cube, 1e7 on a side.
Periodic boundary conditions are a very good approximation to the
conditions not so near the surface.

On the other hand, when you get to surface physics, especially
of semiconductors, the surface states are very important, and
periodic boundary conditions don't help at all.

> We get these amazing questions from DSP beginners (and not so
> beginners) who have computed the FFT on some sequence and want to know
> why their (widely used) FFT library routine doesn't work. Then there
> are the "my white noise is lumpy" broken FFT questions. And the "FFT
> can't be a set of filters, it's a Fourier Transform". We get those
> questions and responses because they have been miss-taught.

-- glen
 

On Jan 12, 8:57&#4294967295;pm, Clay <c...@claysturner.com> wrote:
> 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 ...
>
> read more &#4294967295;

I like the book: "The Fast Fourier Transform" by E O Brigham (mine is
the 1974 edition which I bought when doing a PhD based on applications
of Fourier Transforms). This discusses the derivation and
interpretation of Discrete Fourier Transforms as well as the FFT).

Brigham's chapter on the Discrete Fourier Transform follows 5 chapters
on Fourier Transforms and Fourier Series, and begins with the
following relevant stattement:

"Normally a discussion of the discrete Fourier Transform is based on
an initial definition of the finite length discrete transform; from
this assumed axiom those properties of the transform implied by this
definition are derived. This approach is unrewarding in that at its
conclusion there is always the question, 'How does the discrete
Fourier transform relate to the continuous Fourier transform?"

This is why Brigham derives the discrete Fourier transform as a
special case of continuous Fourier transform theory. He does so
through both graphical and theoretical developments. I tend to think
of Brigham as a definitive text on the DFT and FFT - its is certainly
very thorough (ISBN 0-13-307496-X, Prentice Hall if you want to look
it up). His derivation certainly leads to taking a view on 'what
happens outside the interval', although he is careful to be very clear
about what assumptions he allows.

I find this difference of viewpoint remains stark: some regard the DFT
as independent of contionuous FT theory, and some take Brigham's
viewpoint in order to relate the discrete and continuous transforms. I
think I like Brigham's approach not only because I am used to it but
also because it realtes well to Shannon's derivationof Sampling Theory
which he proves using the continuous Fourier transform (although I am
familiar with proofs of Sampling Theory that rely only on definitions
of the DFT). When I teach the DFT and FFT, or explain their
application in practical implementations, I do find colleagues who
disagree (sometimes vigorously) with the Brigham-style interpretation:
in those cases I tend to 'win' the argument because the Brigham
approach seems to appeal to those who seek an 'intuitive'
understanding of 'what really happens' so less specialised colleagues
tend to outvote the less intutive interpretation. However, for myself
I am very aware that the more 'pure' and 'mathematical' view is
equally valid, and especially that it is vital always to be very clear
about any assumptions or models that one is applying - sometimes you
have to work hard to realise what you are assuming. On the other hand
one cannot afford to debate endlessly, since no system of logic can be
consistent anyway and we often have to make stuff in a finite time.

Chris
=====================
Chris Bore
BORES Signal Processing
www.bores.com