Time-Limited Functions Represented as Periodic

We had a long discussion about time-limited functions, spectral
representation, etc.

I came away convinced in principle that treating a time-limited function as
a periodic function was OK because it appears that no information is lost.
Now I've been thinking about it and want to get more clarity / detail.

I'm going to do this as much as possible without using sampled data.  So, if
continuous Fourier Transforms seem odd, that's the basis for discussion
nonetheless.  I'm *not* talking about DFT /IDFT pairs here.  I'm talking
about CFT / ICFT (C=continous) even if a domain has a discrete
representation and can be reduced to a discrete sum.

We start with an arbitrary continuous function of time f(t) that is time
limited.
f(t)=0 for t<0 and t>T.
The Fourier transform of f(t), F(w) is continuous and of infinite extent.
f(t) is expressed as an infinite sum of sines and cosines (the inverse
transform expression).
F(w) can be expressed as an infinite sum of sines and cosines or
equivalently as a sum of sincs that are spaced by T and have zero crossings
spaced by T - the latter form is a direct result of f(t) being time limited.

Now, what if we decide that it's more convenient to consider f(t) to be a
periodic waveform g(t) with period T?
We can generate g(t) by convolving f(t) with a dirac sequence with spacing
T.  This has the effect of multiplying F(w) by a dirac sequence with spacing
1/T - yielding a discrete spectrum.  In effect, this is sampling in the
frequency domain.  The sampling described barely meets the Nyquist criterion
for sampling because the sampling period is T - which matches the
time-limitation of f(t).

Doing this introduces a question: is this adequate sampling or not?  I think
the answer is: "usually yes". But if f(t) is a pair of diracs at t=0 and
t=T, then not. If f(t) is zero at the end points assures that this is not
the case. .. just like lowpass filtering a function before sampling in time.

So far so good.  Now we have a general case where it's OK to create g(t)
without apparent loss of information and G(w) is discrete / sampled at
frequency intervals of 1/T.

If this is truly OK, then why bother with expressions of F(w) as sums of
continuous sincs?

How do we know where the edges of f(t) are?  Is that information lost in
creating g(t)?
I guess not if we adopt a convention (and perhaps there is one without
focusing on it) by saying: "all periodic waveforms can be represented as
time-limited waveforms that start at t=0 and end at t=T"

Why is this useful?  Because in PAM, there really are essentially
time-limited pulses.   And, rather than simply repeating in time, each
temporal epoch contains a pulse with a different amplitude.  So, treating
superposition of time-shifted, amplitude variable, pulses is a common thing
to do and it's handy to treat each pulse separately.  The spectra aren't
discrete in this case because the composite waveform isn't periodic -
although the constituent parts are time-limited.

I like Rune's comment about the discrete spectrum (being directly related to
the resolution of the temporal window) also being related to an uncertainly
principle.  It's like Nyquist saying: well, the time span won't allow us to
resolve in frequency any better than this so it must be OK to lump the
spectral energy into samples.  That's equivalent to the (normal time)
sampling theorem saying: well, we cant resolve in time any better than the
bandwidth will allow so it's OK to sample in time.

So, we can use this duality as a "test" of the question above about where
are the edges of f(t).

If we start with a bandlimited function with bandwidth B and band extent 2B,
if we sample in time at rate 1/(2*B), then we make the spectrum periodic.
How do we know where the edges of the spectrum are?  Answer:  the edges of
the spectrum are located at +/-(2B/2)=+/-B.
Also, because normally f(t) is real, then Re[F(w)] is symmetric about B and
Im[F(w)] is antisymmetric about B.

So if we redefine f(t) to be zero for -T/2<=t<=T/2, and make it periodic,
then the edges are easily located at -T/2 and T/2 if g(t), the periodic
version of f(t) is formed.  And if the convention is changed, it's no doubt
just as tractable but perhaps less convenient.

We also realize that we normally don't want to sample a function in time
that has finite energy at fs/2=B.  So, similarly, we should not want to
sample a function in frequency that has finite energy at -T/2 and T/2.
Unlike F(w) symmetry at -B and +B, g(t) has no such symmetry in general
because an arbitrary f(t) may generate a  discontinuous g(t) at -T/2 and
T/2. Only if f(-T/2) = f(T/2) and f'(-T/2)=f'(T/2) will there not be a
discontinuity.

It's always seemed funny to me that we care a lot about lowpass filtering
and often not at all about shorttime liftering(?) - which would bring f(t)
to zero at the edges before making the periodic assumption wouldn't it?
That's a lot like what Glen Hermansfeldt said.  Similarly we seem to focus a
lot about not aliasing in frequency and often not as much about aliasing in
time - even thought the concepts are the same.

Hmmmmm.  if we turn it around then can we say: for every periodic waveform
with discrete spectrum there is a time-limited waveform with continuous
spectrum with no loss of information?  I guess so .... that's just
reconstruction of the spectrum isn't it?

Fred

"Fred Marshall" <fmarshallx@remove_the_x.acm.org> wrote in message
news:JBjPa.2522$Jk5.1647120@feed2.centurytel.net...
> We had a long discussion about time-limited functions, spectral
> representation, etc.
>
> I came away convinced in principle that treating a time-limited function
as
> a periodic function was OK because it appears that no information is lost.
> Now I've been thinking about it and want to get more clarity / detail.
>
> I'm going to do this as much as possible without using sampled data.  So,
if
> continuous Fourier Transforms seem odd, that's the basis for discussion
> nonetheless.  I'm *not* talking about DFT /IDFT pairs here.  I'm talking
> about CFT / ICFT (C=continous) even if a domain has a discrete
> representation and can be reduced to a discrete sum.
>
> We start with an arbitrary continuous function of time f(t) that is time
> limited.
> f(t)=0 for t<0 and t>T.
> The Fourier transform of f(t), F(w) is continuous and of infinite extent.
> f(t) is expressed as an infinite sum of sines and cosines (the inverse
> transform expression).
> F(w) can be expressed as an infinite sum of sines and cosines or
> equivalently as a sum of sincs that are spaced by T and have zero
crossings
> spaced by T - the latter form is a direct result of f(t) being time
limited.
>
> Now, what if we decide that it's more convenient to consider f(t) to be a
> periodic waveform g(t) with period T?
> We can generate g(t) by convolving f(t) with a dirac sequence with spacing
> T.  This has the effect of multiplying F(w) by a dirac sequence with
spacing
> 1/T - yielding a discrete spectrum.  In effect, this is sampling in the
> frequency domain.  The sampling described barely meets the Nyquist
criterion
> for sampling because the sampling period is T - which matches the
> time-limitation of f(t).
>
> Doing this introduces a question: is this adequate sampling or not?  I
think
> the answer is: "usually yes". But if f(t) is a pair of diracs at t=0 and
> t=T, then not. If f(t) is zero at the end points assures that this is not
> the case. .. just like lowpass filtering a function before sampling in
time.

Periodic with period T means f(0)=f(T), and that the period of interest is
[0,T), including f(0) but not f(T).

If you want [0,T] it might be that Fourier doesn't help much.  If you make
the period tau, and limit tau --> T+ it might still be that it doesn't work.

There was a discussion before about the Nyquist limit, if it is inclusive or
not.  My answer was that in most cases T Fn was not integer and should be
rounded up.  If it is, then it should be considered strictly less than Fn.

For functions with finite (no dirac functions) discontinuities it is well
known that the Fourier/inverse Fourier gives the average of the two sides of
the discontinuity.   Though such functions are not band limited, and neither
are dirac functions.

> So far so good.  Now we have a general case where it's OK to create g(t)
> without apparent loss of information and G(w) is discrete / sampled at
> frequency intervals of 1/T.
>
> If this is truly OK, then why bother with expressions of F(w) as sums of
> continuous sincs?

For non time limited, or T >> 1/Fs, the sincs are a nice way to look at the
result.  If you take a dirac function and band limit it with a perfect low
pass filter, you will get a sinc().   Also, the limit of a sinc() getting
narrower with unit area will be a dirac function.

sinc() is the Fourier transform of rect().

sinc() also comes out in optics problems when imaging through a finite
aperature, or finite diameter lens, though in radial form.

(snip)

> It's always seemed funny to me that we care a lot about lowpass filtering
> and often not at all about shorttime liftering(?) - which would bring f(t)
> to zero at the edges before making the periodic assumption wouldn't it?
> That's a lot like what Glen Hermansfeldt said.  Similarly we seem to focus
a
> lot about not aliasing in frequency and often not as much about aliasing
in
> time - even thought the concepts are the same.

If you bring f(0) and  f(T) both to a non-zero value you can just add a
constant to the whole transform.  Otherwise, you can just add one sample
point on either end.  Increase T by 1/Fs or 2/Fs.   If T >> 1/Fs that
doesn't change much.

People who record CDs like to have a silent period at the beginning and end,
probably thousands of samples long.

> Hmmmmm.  if we turn it around then can we say: for every periodic waveform
> with discrete spectrum there is a time-limited waveform with continuous
> spectrum with no loss of information?  I guess so .... that's just
> reconstruction of the spectrum isn't it?

-- glen

Fred Marshall wrote:
> We had a long discussion about time-limited functions, spectral
> representation, etc.
> 
> I came away convinced in principle that treating a time-limited function as
> a periodic function was OK because it appears that no information is lost.
> Now I've been thinking about it and want to get more clarity / detail.

Lets say that your time limited function is a "chunk" of a much longer 
duration waveform. Lets us further say that that the much longer 
waveform was chopped up in a bunch of contiguous chunks.  Given the 
linearity and time invariance of the Fourier Transform, I can construct 
  the Fourier transform of the much longer waveform from the transforms 
of the chunks.  This is why the overlap add technique works.  The chunks 
don't have even have to be the same duration.

Considering each time limited chunk as a periodic extension is not 
compatible with the linearity I like in the continuous Fourier transform.

I'll use an argument that Jerry used a while back.  Lets say I have two 
chucks of data of equal length N.  Lets say we peridiodicly extend each 
. Each will have a DFT with N frequencies.  Lets say we concatenate the 
original time waveforms so now there are 2N frequencies.  Are we to 
assume the concatenation of time series is a nonlinear operation because 
we "created" N frequencies?

> 
> I'm going to do this as much as possible without using sampled data.  So, if
> continuous Fourier Transforms seem odd, that's the basis for discussion
> nonetheless.  I'm *not* talking about DFT /IDFT pairs here.  I'm talking
> about CFT / ICFT (C=continuous) even if a domain has a discrete
> representation and can be reduced to a discrete sum.
> 
> We start with an arbitrary continuous function of time f(t) that is time
> limited.
> f(t)=0 for t<0 and t>T.
> The Fourier transform of f(t), F(w) is continuous and of infinite extent.
> f(t) is expressed as an infinite sum of sines and cosines (the inverse
> transform expression).
> F(w) can be expressed as an infinite sum of sines and cosines or
> equivalently as a sum of sincs that are spaced by T and have zero crossings
> spaced by T - the latter form is a direct result of f(t) being time limited.
> 
> Now, what if we decide that it's more convenient to consider f(t) to be a
> periodic waveform g(t) with period T?
> We can generate g(t) by convolving f(t) with a dirac sequence with spacing
> T.  This has the effect of multiplying F(w) by a dirac sequence with spacing
> 1/T - yielding a discrete spectrum.  In effect, this is sampling in the
> frequency domain.  The sampling described barely meets the Nyquist criterion
> for sampling because the sampling period is T - which matches the
> time-limitation of f(t).
> 
> Doing this introduces a question: is this adequate sampling or not?  I think
> the answer is: "usually yes". But if f(t) is a pair of diracs at t=0 and
> t=T, then not. If f(t) is zero at the end points assures that this is not
> the case. .. just like lowpass filtering a function before sampling in time.
> 
> So far so good.  Now we have a general case where it's OK to create g(t)
> without apparent loss of information and G(w) is discrete / sampled at
> frequency intervals of 1/T.
> 
> If this is truly OK, then why bother with expressions of F(w) as sums of
> continuous sincs?
> 
> How do we know where the edges of f(t) are?  Is that information lost in
> creating g(t)?
> I guess not if we adopt a convention (and perhaps there is one without
> focusing on it) by saying: "all periodic waveforms can be represented as
> time-limited waveforms that start at t=0 and end at t=T"
> 
> Why is this useful?  Because in PAM, there really are essentially
> time-limited pulses.   And, rather than simply repeating in time, each
> temporal epoch contains a pulse with a different amplitude.  So, treating
> superposition of time-shifted, amplitude variable, pulses is a common thing
> to do and it's handy to treat each pulse separately.  The spectra aren't
> discrete in this case because the composite waveform isn't periodic -
> although the constituent parts are time-limited.
> 
> I like Rune's comment about the discrete spectrum (being directly related to
> the resolution of the temporal window) also being related to an uncertainly
> principle.  It's like Nyquist saying: well, the time span won't allow us to
> resolve in frequency any better than this so it must be OK to lump the
> spectral energy into samples.  That's equivalent to the (normal time)
> sampling theorem saying: well, we cant resolve in time any better than the
> bandwidth will allow so it's OK to sample in time.
> 
> So, we can use this duality as a "test" of the question above about where
> are the edges of f(t).
> 
> If we start with a bandlimited function with bandwidth B and band extent 2B,
> if we sample in time at rate 1/(2*B), then we make the spectrum periodic.
> How do we know where the edges of the spectrum are?  Answer:  the edges of
> the spectrum are located at +/-(2B/2)=+/-B.
> Also, because normally f(t) is real, then Re[F(w)] is symmetric about B and
> Im[F(w)] is antisymmetric about B.
> 
> So if we redefine f(t) to be zero for -T/2<=t<=T/2, and make it periodic,
> then the edges are easily located at -T/2 and T/2 if g(t), the periodic
> version of f(t) is formed.  And if the convention is changed, it's no doubt
> just as tractable but perhaps less convenient.
> 
> We also realize that we normally don't want to sample a function in time
> that has finite energy at fs/2=B.  So, similarly, we should not want to
> sample a function in frequency that has finite energy at -T/2 and T/2.
> Unlike F(w) symmetry at -B and +B, g(t) has no such symmetry in general
> because an arbitrary f(t) may generate a  discontinuous g(t) at -T/2 and
> T/2. Only if f(-T/2) = f(T/2) and f'(-T/2)=f'(T/2) will there not be a
> discontinuity.
> 
> It's always seemed funny to me that we care a lot about lowpass filtering
> and often not at all about shorttime liftering(?) - which would bring f(t)
> to zero at the edges before making the periodic assumption wouldn't it?
> That's a lot like what Glen Hermansfeldt said.  Similarly we seem to focus a
> lot about not aliasing in frequency and often not as much about aliasing in
> time - even thought the concepts are the same.
> 
> Hmmmmm.  if we turn it around then can we say: for every periodic waveform
> with discrete spectrum there is a time-limited waveform with continuous
> spectrum with no loss of information?  I guess so .... that's just
> reconstruction of the spectrum isn't it?
> 
> Fred
> 
> 
> 
> 
> 
> 
> 
> 
> 
> 
> 
> 
> 
> 
> 
> 
> 
> 
> 

"Glen Herrmannsfeldt" <gah@ugcs.caltech.edu> wrote in message
news:inkPa.29849$H17.8847@sccrnsc02...
>

.....................
>
> Periodic with period T means f(0)=f(T), and that the period of interest is
> [0,T), including f(0) but not f(T).
>

Yep.  About [0,T), I'm often sloppy about that because it's hard to
visualize and visualized math is the best kind - even if the "image" is an
abstraction.  Jerry would probably agree with the approach.  :-)

The convolution with a Dirac sequence to get g(t) should have been clear
enough.  In general there will be a discontinuity.

Better said: f(0)=f(T) and f(0) is not equal to f(T-e) for e>0.
So, g(t) is periodic with a discontinuity.

..........................

>> So far so good.  Now we have a general case where it's OK to create g(t)
>> without apparent loss of information and G(w) is discrete / sampled at
>> frequency intervals of 1/T.
>>
>> If this is truly OK, then why bother with expressions of F(w) as sums of
>> continuous sincs?

>For non time limited, or T >> 1/Fs, the sincs are a nice way to look at the
>result.  If you take a dirac function and band limit it with a perfect low
>pass filter, you will get a sinc().   Also, the limit of a sinc() getting
>narrower with unit area will be a dirac function.

>sinc() is the Fourier transform of rect().

>sinc() also comes out in optics problems when imaging through a finite
>aperature, or finite diameter lens, though in radial form.

Well, right.  But I was saying that F(w) is made up of sincs, not f(t) - by
virtue of the time-limitation.  So, I would have to translate what you said
to:

"For non band limited, or fs >> 1/T (meaning that N is large), the sincs are
a nice way to look at the result.  If you take a dirac function in frequency
and time limit it with a perfect short time lifter, you will get a sinc() in
frequency.   Also, the limit of a sinc() getting narrower with unit area
will be a dirac function."

My philosophical question was: if the discrete spectral representation (for
a time-limited function made periodic) was OK, then why do we bother with
*continuous* frequency sincs for such things??  Isn't it easier to deal with
a discrete spectrum?  Is that a discrepancy or an oversight in practice?
Or, is there an example in practice that I've overlooked.

I'm not asking the trivial question: "couldn't one express the frequency
sinc sequence in a discrete form?"  That's already evident.  I'm asking "why
don't we jump to the discrete form more often, more readily, etc.?"

Fred

"Fred Marshall" <fmarshallx@remove_the_x.acm.org> wrote in message
news:ljlPa.2524$Jk5.1651670@feed2.centurytel.net...
>

(snip)

> Well, right.  But I was saying that F(w) is made up of sincs, not f(t) -
by
> virtue of the time-limitation.  So, I would have to translate what you
said
> to:

The symmetry of the Fourier transform.

> "For non band limited, or fs >> 1/T (meaning that N is large), the sincs
are
> a nice way to look at the result.  If you take a dirac function in
frequency
> and time limit it with a perfect short time lifter, you will get a sinc()
in
> frequency.   Also, the limit of a sinc() getting narrower with unit area
> will be a dirac function."
>
> My philosophical question was: if the discrete spectral representation
(for
> a time-limited function made periodic) was OK, then why do we bother with
> *continuous* frequency sincs for such things??  Isn't it easier to deal
with
> a discrete spectrum?  Is that a discrepancy or an oversight in practice?
> Or, is there an example in practice that I've overlooked.
>
> I'm not asking the trivial question: "couldn't one express the frequency
> sinc sequence in a discrete form?"  That's already evident.  I'm asking
"why
> don't we jump to the discrete form more often, more readily, etc.?"

If one low pass filters the dirac function, one will get the sinc()s.  So if
one claims to be reconstructing the original, band limited, function one
should express it in sinc() form, instead of dirac form.    Otherwise, it
doesn't make much difference to me.

It could be, though, that people are used to thinking of the world in
continuous terms, instead of discete terms.

Note that quantum mechanics considers the world in discrete terms, but we
tend to average over that and pretend it is continuous.    One of my
favorite quantum mechanics problems: How many different speeds can a
baseball pitcher pitch inside a baseball stadium?  (Pick your favorite
values for the mass of a baseball and the diameter of the stadium.)

So we are actually discretizing a continuous model of a discrete world.

-- glen

"Glen Herrmannsfeldt" <gah@ugcs.caltech.edu> wrote in message
news:TzmPa.30683$H17.9354@sccrnsc02...


> Note that quantum mechanics considers the world in discrete terms, but we
> tend to average over that and pretend it is continuous.    One of my
> favorite quantum mechanics problems: How many different speeds can a
> baseball pitcher pitch inside a baseball stadium?  (Pick your favorite
> values for the mass of a baseball and the diameter of the stadium.)
>
> So we are actually discretizing a continuous model of a discrete world.

Glen,
We don't always do this. When one looks ot the allowed energy states for a
simple case such as hydrogen, we always appeal to the discrete bound states,
but there are also continuum states which are continuous in the energy
domain.

The baseball problem becomes discrete due to the implication that the ball
is bound to stay within the ballpark. If you remove this constraint, then
the energy becomes continuous again. Notice how there is no quantization
with Planc's law for photons. The unbound particles can have any energy you
want. Even when they are produced by bound particles (electrons in atoms) a
simple doppler shift can move the energy level to any arbitrary point.

>
> -- glen
>
>

"Fred Marshall" <fmarshallx@remove_the_x.acm.org> wrote in message
news:JBjPa.2522$Jk5.1647120@feed2.centurytel.net...
> We had a long discussion about time-limited functions, spectral
> representation, etc.
>
> I came away convinced in principle that treating a time-limited function
as
> a periodic function was OK because it appears that no information is lost.
> Now I've been thinking about it and want to get more clarity / detail.

Hello Fred,
Here is a little food for thought.

Imagine we are letting some light diffract through an aperture. The far
field distribution is given by the Fourier transform of the near field
distribution. Now there is a neat property known as Babinet's Principle that
says if the the aperture is complemented, the far field radiation pattern
stays the same other than a phase/polarization shift. It is possible to set
up light distributions such the distribution is not periodic relative to the
width of one of the aperatures. So in this case the effect of zeroing out
portions of the function outside of some boundary doesn't have to assume
periodicity. But basically there is a relationship between an original
function, a windowed version of the function, and the complementary windowed
version of the function. And Babinet's principle is a statement of this. But
I think when you are talking about the FT of a windowed function, you can
represent it in a continuous way so the reconstruction outside of the window
is zero, or you can make the windowed portion periodic, and get a Fourier
series, but you can also do all kinds of extensions outside of the interval
of interest and these will each result in a different FT. What you do just
depends on what you want the function outside of the interval to look like
after reconstruction. Often we don't care, and we can then pick a simple
periodic extension.

Earlier you had asked about why all of the sines/cosines (with continuous
frequency) when analysing a windowed function. This allows the
reconstruction outside of the interval to be nonperiodic (uusally it makes
it all zero). If the reconstruction is periodic, then most of the
sines/cosines cancel out leaving us with a discrete sprectrum. Another way
of looking at it is to take the periodic representation (discrete spectrum)
and convolve it with the spectrum of the window. This can be seen as
smearing the discrete spectral lines about and making the FT a nonline
spectrum.

I hope this helps

Clay

"Clay S. Turner" <physicsNOOOOSPPPPAMMMM@bellsouth.net> wrote in message
news:srzPa.2$Y_1.1@fe04.atl2.webusenet.com...
>
> "Fred Marshall" <fmarshallx@remove_the_x.acm.org> wrote in message
> news:JBjPa.2522$Jk5.1647120@feed2.centurytel.net...
> > We had a long discussion about time-limited functions, spectral
> > representation, etc.
> >
> > I came away convinced in principle that treating a time-limited function
> as
> > a periodic function was OK because it appears that no information is
lost.
> > Now I've been thinking about it and want to get more clarity / detail.
>
> Hello Fred,
> Here is a little food for thought.
>
> Imagine we are letting some light diffract through an aperture. The far
> field distribution is given by the Fourier transform of the near field
> distribution. Now there is a neat property known as Babinet's Principle
that
> says if the the aperture is complemented, the far field radiation pattern
> stays the same other than a phase/polarization shift. It is possible to
set
> up light distributions such the distribution is not periodic relative to
the
> width of one of the aperatures. So in this case the effect of zeroing out
> portions of the function outside of some boundary doesn't have to assume
> periodicity. But basically there is a relationship between an original
> function, a windowed version of the function, and the complementary
windowed
> version of the function. And Babinet's principle is a statement of this.
But
> I think when you are talking about the FT of a windowed function, you can
> represent it in a continuous way so the reconstruction outside of the
window
> is zero, or you can make the windowed portion periodic, and get a Fourier
> series, but you can also do all kinds of extensions outside of the
interval
> of interest and these will each result in a different FT. What you do just
> depends on what you want the function outside of the interval to look like
> after reconstruction. Often we don't care, and we can then pick a simple
> periodic extension.
>
> Earlier you had asked about why all of the sines/cosines (with continuous
> frequency) when analysing a windowed function. This allows the
> reconstruction outside of the interval to be nonperiodic (uusally it makes
> it all zero). If the reconstruction is periodic, then most of the
> sines/cosines cancel out leaving us with a discrete sprectrum. Another way
> of looking at it is to take the periodic representation (discrete
spectrum)
> and convolve it with the spectrum of the window. This can be seen as
> smearing the discrete spectral lines about and making the FT a nonline
> spectrum.
>
> I hope this helps
>
> Clay

Clay,

Yes - thanks.  Babinet's principle is simply about aperture illumination
phase isn't it?  Also, it sounds like we're talking about looking at an
aperture space factor (beam pattern) in either polar coordinates (sin theta)
or in angular coordinates (theta) where the former is circular like the z
transform and the latter includes functional description "outside the
visible region".  Well, maybe that's an example of what you're talking
about - I get the idea.

I don't know that I was asking "why all the sines cosines".  I knew that
much.
I was taking off from Glen's observation that the periodic version could be
used instead of the continuous with no loss of information.  I initially
thought that's the same as saying we're only going to look in "the visible
region" of a space factor.  But, I had to think about it and the question I
had was: why don't we do this more often then?

As an aside: I hadn't thought about it before but we *can* design filters
that are "supergained"?  I see the term coming up in image processing now -
and it was used much longer ago in continuous array design.  Generally this
meant that the edges of the array would be "hot" (large amplitude) and the
rest of the array weighting would be small in comparison.  It put
"sidelobes" outside the visible region.  I think the answer is yes - after
all, it wasn't generally what you wanted to do!

Continuing the array analogy, are we saying this:
"If I have a continuous array of finite aperture, then I can view the
aperture as infinite / periodic and then I can view the broadside space
factor as discrete"?  Well, why not?  As you point out, the real space
factor would then have to be reconstructed with a sinc related to
re-limiting the aperture to being non-periodic."?

Now, of course we're very used to having arrays of discrete elements.  We're
not so used to looking at the space factor / beam pattern as a discrete
function of infinitely narrow beams! (Which I must hasten to add that folks
*do* create narrow beams with long discrete, perhaps "periodic" arrays).

What this all boils down to is flipping the Fourier Transform domains around
and making "normal" observations in the opposite domain.  Doing this isn't
so "normal" and causes us to think - even though there's duality in the FT.
So, in some sense I've launched a trivial discussion.  On the other hand,
there are elements that it seems to me the DSP community doesn't think about
very often.  For example: we very often talk about Nyquist and the adequacy
of a sample rate in the time domain.  How often do folks worry about the
adequacy of a sample rate in the frequency domain?  The answer, I believe,
is that *sometimes* we do worry about temporal aliasing - such as assuring
M+N-1 array lengths for circular convolution.  But putting this in terms of
the Nyquist criterion in frequency is an unusual perspective - er... I think
it's unusual.

The most common example I can think of is this:
Someone decides to take a time signal, generate discrete samples (presumably
meeting the Nyquist criterion (if not perfectly at least adequately), take
an epoch or chunk of these samples and compute the Discrete Fourier
Transform.
The length of the chunk determines the *spectral* sample rate.
Is this sample rate adequate to avoid temporal aliasing - would Mr. Myquist
approve?  Why?
Why do we often not care?

I think this is the answer:
If our objective is to recreate an entire continuous temporal function then
the answer is *no* because, presumably, f(t) continues in some unique way
beyond the epoch of length T.  The Nyquist sampling crieterion (applied in
frequency) isn't met.
So, it's convenient to view f(n) or f(t) over epoch length T as the *only*
representation of interest.
In that case, it's OK to view f(n) as either time-limited or periodic
Then we can get fancier and do saves & overlaps, etc..... with due caution
of course!

Fred

"Fred Marshall" <fmarshallx@remove_the_x.acm.org> wrote in message news:<JBjPa.2522$Jk5.1647120@feed2.centurytel.net>...
> We had a long discussion about time-limited functions, spectral
> representation, etc.

Sure. Now, I've read this post a couple of times and I think there is 
something hidden inside here, that I may learn something from. Being 
who I am, I'll try to bring whatever this is out by playing one of my 
favourite games -- "The Devil's Advocate". You are warned...
 
> I came away convinced in principle that treating a time-limited function as
> a periodic function was OK because it appears that no information is lost.
> Now I've been thinking about it and want to get more clarity / detail.
> 
> I'm going to do this as much as possible without using sampled data.  So, if
> continuous Fourier Transforms seem odd, that's the basis for discussion
> nonetheless.  I'm *not* talking about DFT /IDFT pairs here.  I'm talking
> about CFT / ICFT (C=continous) even if a domain has a discrete
> representation and can be reduced to a discrete sum.
> 
> We start with an arbitrary continuous function of time f(t) that is time
> limited.
> f(t)=0 for t<0 and t>T.

OK...

> The Fourier transform of f(t), F(w) is continuous and of infinite extent.
> f(t) is expressed as an infinite sum of sines and cosines (the inverse
> transform expression).

These appear to be contradictory statements according to the "usual" 
definitions of the FT. The spectrum is *either* continuous (f(t) is an
integral over w) *or* the signal is a sum of sine/cosines.

> F(w) can be expressed as an infinite sum of sines and cosines or
> equivalently as a sum of sincs that are spaced by T and have zero crossings
> spaced by T - the latter form is a direct result of f(t) being time limited.

I don't see this equivalence. Could you elaborate, please?
 
> Now, what if we decide that it's more convenient to consider f(t) to be a
> periodic waveform g(t) with period T?
> We can generate g(t) by convolving f(t) with a dirac sequence with spacing
> T.  This has the effect of multiplying F(w) by a dirac sequence with spacing
> 1/T - yielding a discrete spectrum.  In effect, this is sampling in the
> frequency domain.  The sampling described barely meets the Nyquist criterion
> for sampling because the sampling period is T - which matches the
> time-limitation of f(t).

I see what you mean, but I have a hard time following your line of thought. 
You stated before that you consider continuous functions - why introduce 
sampling? I see it as somewhat unmotivated...

> Doing this introduces a question: is this adequate sampling or not?  I think
> the answer is: "usually yes". But if f(t) is a pair of diracs at t=0 and
> t=T, then not. If f(t) is zero at the end points assures that this is not
> the case. .. just like lowpass filtering a function before sampling in time.

I'm lost.
 
> So far so good.  Now we have a general case where it's OK to create g(t)
> without apparent loss of information and G(w) is discrete / sampled at
> frequency intervals of 1/T.
> 
> If this is truly OK, then why bother with expressions of F(w) as sums of
> continuous sincs?
> 
> How do we know where the edges of f(t) are?  Is that information lost in
> creating g(t)?
> I guess not if we adopt a convention (and perhaps there is one without
> focusing on it) by saying: "all periodic waveforms can be represented as
> time-limited waveforms that start at t=0 and end at t=T"

This creates a formal problem. If you do that, you loose periodicity
by making the interval closed. What you need for your ilne of argument 
to work, is a half-open interval and some convention to define g(T)=g(0).
Otherwise, your g(t) will not be periodic with period T. Which leaves 
one end point in the "modified" f(t) undefined. I don't like that.

> Why is this useful?  Because in PAM, there really are essentially
> time-limited pulses.   And, rather than simply repeating in time, each
> temporal epoch contains a pulse with a different amplitude.  So, treating
> superposition of time-shifted, amplitude variable, pulses is a common thing
> to do and it's handy to treat each pulse separately.  The spectra aren't
> discrete in this case because the composite waveform isn't periodic -
> although the constituent parts are time-limited.

What's important is not the temporal behaviour of the signal, even inside 
the observation window, it's the observation length T. Finite or infinite, 
that's all that matters. 

> I like Rune's comment about the discrete spectrum (being directly related to
> the resolution of the temporal window) also being related to an uncertainly
> principle.  It's like Nyquist saying: well, the time span won't allow us to
> resolve in frequency any better than this so it must be OK to lump the
> spectral energy into samples. 

I wouldn't say Nyquist, rather Heissenberg. Apart from that, I think 
pursuing the effects of finite T is the way to go, and leave any assumed
periodic bahaviour of f(t) or g(t) on the side. Somebody said in a recent 
thread that finite-T signals, discrete or contiuous, contained finite 
information and thus needed only a "reduced", i.e. discrete, spectrum. 
Signals of infinite duration, on the other hand, contain infinite 
information and need to be represented by contiuous spectra. Although my
phrasing of this particular line of thought leaves a bit to be desired, 
this mental model does somehow appeal to me.

> That's equivalent to the (normal time)
> sampling theorem saying: well, we cant resolve in time any better than the
> bandwidth will allow so it's OK to sample in time.

Again, why bring in temporal sampling? You *did* specify f(t) continuous?

> So, we can use this duality as a "test" of the question above about where
> are the edges of f(t).
> 
> If we start with a bandlimited function with bandwidth B and band extent 2B,
> if we sample in time at rate 1/(2*B), then we make the spectrum periodic.
> How do we know where the edges of the spectrum are?  Answer:  the edges of
> the spectrum are located at +/-(2B/2)=+/-B.
> Also, because normally f(t) is real, then Re[F(w)] is symmetric about B and
> Im[F(w)] is antisymmetric about B.
> 
> So if we redefine f(t) to be zero for -T/2<=t<=T/2, and make it periodic,
> then the edges are easily located at -T/2 and T/2 if g(t), the periodic
> version of f(t) is formed.  And if the convention is changed, it's no doubt
> just as tractable but perhaps less convenient.
> 
> We also realize that we normally don't want to sample a function in time
> that has finite energy at fs/2=B.  So, similarly, we should not want to
> sample a function in frequency that has finite energy at -T/2 and T/2.
> Unlike F(w) symmetry at -B and +B, g(t) has no such symmetry in general
> because an arbitrary f(t) may generate a  discontinuous g(t) at -T/2 and
> T/2. Only if f(-T/2) = f(T/2) and f'(-T/2)=f'(T/2) will there not be a
> discontinuity.

Doesn't this impose severe restrictions on f(t)? Sure, there is such a 
thing as Gibbs' phenomenon, but does it appear on the end points of the
continuous f(t)? As far as I know, Gibbs' phenomenon only appears with 
dicontiuities *internal* to the interval [0,T]? You need to explain this 
anomality in the dicontinuities.
 
> It's always seemed funny to me that we care a lot about lowpass filtering
> and often not at all about shorttime liftering(?) - which would bring f(t)
> to zero at the edges before making the periodic assumption wouldn't it?
> That's a lot like what Glen Hermansfeldt said.  Similarly we seem to focus a
> lot about not aliasing in frequency and often not as much about aliasing in
> time - even thought the concepts are the same.

Yes? I think it's common knowledge to choose frame length long enough 
to avoid the effects of circular convolution.

> Hmmmmm.  if we turn it around then can we say: for every periodic waveform
> with discrete spectrum there is a time-limited waveform with continuous
> spectrum with no loss of information?  I guess so .... that's just
> reconstruction of the spectrum isn't it?

Glen keeps coming back to the example with the CD that's played 
through analog speakers...

Now, I think there are a couple of flaws in the basis for your line of 
thoughts. I have pointed out those I have found. I have an opinion what 
the flaw is on some points, on others I don't understand how you think. 

Rune

Rune,

Comments interspersed below:

"Rune Allnor" <allnor@tele.ntnu.no> wrote in message
news:f56893ae.0307111004.7388305a@posting.google.com...
> "Fred Marshall" <fmarshallx@remove_the_x.acm.org> wrote in message
news:<JBjPa.2522$Jk5.1647120@feed2.centurytel.net>...
> > We had a long discussion about time-limited functions, spectral
> > representation, etc.
>
> Sure. Now, I've read this post a couple of times and I think there is
> something hidden inside here, that I may learn something from. Being
> who I am, I'll try to bring whatever this is out by playing one of my
> favourite games -- "The Devil's Advocate". You are warned...
>
> > I came away convinced in principle that treating a time-limited function
as
> > a periodic function was OK because it appears that no information is
lost.
> > Now I've been thinking about it and want to get more clarity / detail.
> >
> > I'm going to do this as much as possible without using sampled data.
So, if
> > continuous Fourier Transforms seem odd, that's the basis for discussion
> > nonetheless.  I'm *not* talking about DFT /IDFT pairs here.  I'm talking
> > about CFT / ICFT (C=continous) even if a domain has a discrete
> > representation and can be reduced to a discrete sum.
> >
> > We start with an arbitrary continuous function of time f(t) that is time
> > limited.
> > f(t)=0 for t<0 and t>T.
>
> OK...
>
> > The Fourier transform of f(t), F(w) is continuous and of infinite
extent.
> > f(t) is expressed as an infinite sum of sines and cosines (the inverse
> > transform expression).
>
> These appear to be contradictory statements according to the "usual"
> definitions of the FT. The spectrum is *either* continuous (f(t) is an
> integral over w) *or* the signal is a sum of sine/cosines.

***I probably should have said infinite *integral* rather than *sum* -
that's what I meant.

>
> > F(w) can be expressed as an infinite sum of sines and cosines or
> > equivalently as a sum of sincs that are spaced by T and have zero
crossings
> > spaced by T - the latter form is a direct result of f(t) being time
limited.
>
> I don't see this equivalence. Could you elaborate, please?

***Again "infinite integral".  If f(t) is time limited then F(w) can be
expressed as an integral of sincs or as an integral of sines and cosines.
It can be shown that the sinc is an infinite sum of sines and cosines - so a
sum of sincs is a linear combination of those same sines and cosines.
That's what I meant....  I hope this clarifies enough.

>
> > Now, what if we decide that it's more convenient to consider f(t) to be
a
> > periodic waveform g(t) with period T?
> > We can generate g(t) by convolving f(t) with a dirac sequence with
spacing
> > T.  This has the effect of multiplying F(w) by a dirac sequence with
spacing
> > 1/T - yielding a discrete spectrum.  In effect, this is sampling in the
> > frequency domain.  The sampling described barely meets the Nyquist
criterion
> > for sampling because the sampling period is T - which matches the
> > time-limitation of f(t).
>
> I see what you mean, but I have a hard time following your line of
thought.
> You stated before that you consider continuous functions - why introduce
> sampling? I see it as somewhat unmotivated...

*** I said I was going to avoid sampling as much as possible.  But the
discrete spectrum was part of the question I was asking in the first place.
Periodic time <-> Discrete spectrum.

>
> > Doing this introduces a question: is this adequate sampling or not?  I
think
> > the answer is: "usually yes". But if f(t) is a pair of diracs at t=0 and
> > t=T, then not. If f(t) is zero at the end points assures that this is
not
> > the case. .. just like lowpass filtering a function before sampling in
time.
>
> I'm lost.

***If we take a time-limited function of length T and make it periodic with
period T then we are making the spectrum discrete.  This corresponds with
sampling in the frequency domain.  Is the sample rate in frequency adequate?
That's the question.

>
> > So far so good.  Now we have a general case where it's OK to create g(t)
> > without apparent loss of information and G(w) is discrete / sampled at
> > frequency intervals of 1/T.
> >
> > If this is truly OK, then why bother with expressions of F(w) as sums of
> > continuous sincs?
> >
> > How do we know where the edges of f(t) are?  Is that information lost in
> > creating g(t)?
> > I guess not if we adopt a convention (and perhaps there is one without
> > focusing on it) by saying: "all periodic waveforms can be represented as
> > time-limited waveforms that start at t=0 and end at t=T"
>
> This creates a formal problem. If you do that, you loose periodicity
> by making the interval closed. What you need for your ilne of argument
> to work, is a half-open interval and some convention to define g(T)=g(0).
> Otherwise, your g(t) will not be periodic with period T. Which leaves
> one end point in the "modified" f(t) undefined. I don't like that.

Well, I guess the interval for the period has to be [0,T).  Whatever works.
What I'm trying to convey is a function g(t) that makes time-limited f(t)
into a periodic function with a periodic unit step discontinuity if
necessary.

>
> > Why is this useful?  Because in PAM, there really are essentially
> > time-limited pulses.   And, rather than simply repeating in time, each
> > temporal epoch contains a pulse with a different amplitude.  So,
treating
> > superposition of time-shifted, amplitude variable, pulses is a common
thing
> > to do and it's handy to treat each pulse separately.  The spectra aren't
> > discrete in this case because the composite waveform isn't periodic -
> > although the constituent parts are time-limited.
>
> What's important is not the temporal behaviour of the signal, even inside
> the observation window, it's the observation length T. Finite or infinite,
> that's all that matters.

***I'm saying there are examples when treating time-limited functions is
useful - that's all.  As compared to changing them to periodic infinite
extent functions.

>
> > I like Rune's comment about the discrete spectrum (being directly
related to
> > the resolution of the temporal window) also being related to an
uncertainly
> > principle.  It's like Nyquist saying: well, the time span won't allow us
to
> > resolve in frequency any better than this so it must be OK to lump the
> > spectral energy into samples.
>
> I wouldn't say Nyquist, rather Heissenberg. Apart from that, I think
> pursuing the effects of finite T is the way to go, and leave any assumed
> periodic bahaviour of f(t) or g(t) on the side. Somebody said in a recent
> thread that finite-T signals, discrete or contiuous, contained finite
> information and thus needed only a "reduced", i.e. discrete, spectrum.
> Signals of infinite duration, on the other hand, contain infinite
> information and need to be represented by contiuous spectra. Although my
> phrasing of this particular line of thought leaves a bit to be desired,
> this mental model does somehow appeal to me.

***Right.  Agreed.  Unless the infinite duration happens to be periodic.
>
> > That's equivalent to the (normal time)
> > sampling theorem saying: well, we cant resolve in time any better than
the
> > bandwidth will allow so it's OK to sample in time.
>
> Again, why bring in temporal sampling? You *did* specify f(t) continuous?

***Yes indeed.  I was mentioning the equivalence here is all - because
sampling in time is familiar territory for many.

>
> > So, we can use this duality as a "test" of the question above about
where
> > are the edges of f(t).
> >
> > If we start with a bandlimited function with bandwidth B and band extent
2B,
> > if we sample in time at rate 1/(2*B), then we make the spectrum
periodic.
> > How do we know where the edges of the spectrum are?  Answer:  the edges
of
> > the spectrum are located at +/-(2B/2)=+/-B.
> > Also, because normally f(t) is real, then Re[F(w)] is symmetric about B
and
> > Im[F(w)] is antisymmetric about B.
> >
> > So if we redefine f(t) to be zero for -T/2<=t<=T/2, and make it
periodic,
> > then the edges are easily located at -T/2 and T/2 if g(t), the periodic
> > version of f(t) is formed.  And if the convention is changed, it's no
doubt
> > just as tractable but perhaps less convenient.
> >
> > We also realize that we normally don't want to sample a function in time
> > that has finite energy at fs/2=B.  So, similarly, we should not want to
> > sample a function in frequency that has finite energy at -T/2 and T/2.
> > Unlike F(w) symmetry at -B and +B, g(t) has no such symmetry in general
> > because an arbitrary f(t) may generate a  discontinuous g(t) at -T/2 and
> > T/2. Only if f(-T/2) = f(T/2) and f'(-T/2)=f'(T/2) will there not be a
> > discontinuity.
>
> Doesn't this impose severe restrictions on f(t)? Sure, there is such a
> thing as Gibbs' phenomenon, but does it appear on the end points of the
> continuous f(t)? As far as I know, Gibbs' phenomenon only appears with
> dicontiuities *internal* to the interval [0,T]? You need to explain this
> anomality in the dicontinuities.

***Once we've constructed a periodic waveform that allows step jumps then
the Gibbs' phenomenon is internal to the periodic, infinite waveform g(t).

***Yes, it would appear to impose restriction on f(t).  That was the point I
was trying to make - question I was asking. I would add that this
restriction is no different than the restrictions we put on H(w) before
deciding what temporal sample rate we might use on h(t) - the Nyquist
criterion.

>
> > It's always seemed funny to me that we care a lot about lowpass
filtering
> > and often not at all about shorttime liftering(?) - which would bring
f(t)
> > to zero at the edges before making the periodic assumption wouldn't it?
> > That's a lot like what Glen Hermansfeldt said.  Similarly we seem to
focus a
> > lot about not aliasing in frequency and often not as much about aliasing
in
> > time - even thought the concepts are the same.
>
> Yes? I think it's common knowledge to choose frame length long enough
> to avoid the effects of circular convolution.

***Agreed - that is a clear cut case in practice.  However, how often do you
hear someone ask the question:
Is it OK to consider this temporal sample as periodic (i.e. sample the
spectrum)?  Is the important temporal extent of the signal less than T?
This is just the Nyquist criterion expressed by flipping the domains.  If we
don't consider it, we could have temporal aliasing by virtue of making the
record periodic in time / by virtue of sampling in frequency.

>
> > Hmmmmm.  if we turn it around then can we say: for every periodic
waveform
> > with discrete spectrum there is a time-limited waveform with continuous
> > spectrum with no loss of information?  I guess so .... that's just
> > reconstruction of the spectrum isn't it?
>
> Glen keeps coming back to the example with the CD that's played
> through analog speakers...

***Right.  If we apply his experiment to this statement we get:
If we play the same CD tune over and over again, the spectrum is discrete
(albeit with very tiny spectral steps).  If we play it only once it's the
same as reconstructing the spectrum with sincs for just one temporal epoch,
then we get back the continuous spectrum.  That's dealing with sampling in
frequency.

***Rune - you will appreciate this.  Once upon a time I was building a
ship's acoustic signature simulator.  We would start with an actual
recording of the ship's radiated noise and attempt to put it on a record
that could be played back in an endless loop.  At the time, microprocessors
and memory chips were just coming into wide use and the memories were fairly
small.  Our concentration was on propellor beats and such low frequency
things and we hypothesized that we could put a few such beats into a small
memory and play it repeatedly.
The time sample rate was high and was fixed.  We built a variable length
memory, would fill it with sound samples and, once filled, would switch to
repetitively play back out of the memory.
The result was a 1-second loop that repeated.  It was periodic with period 1
second.  We had succeeded in sampling the spectrum at 1/1sec=1Hz steps.
Guess what we saw on a spectrum analyzer?  Nothing but lines!!!  Oooppppss!
It sounded great but was clearly sampled in frequency / periodic in time.

***I hope this helps.  It's not a very big deal in the math but it does
raise a question about how ready we are to do DFTs with arbitrary T.  It is
*very* much associated with windowing in time to reduce spectral leakage.
If we window, we place that "restriction" you mentioned on f(t) and the
effects of temporal aliasing are diminished.  Spectral leakage is a
consequence of temporal aliasing - I think is fair to say..... but now I'm
arm-waving a bit.

Fred