| Author |
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lucy
Guest
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 Posted:
Sat Nov 12, 2005 9:15 am Post subject:
how do you prove a signal that is time-limited cannot be ban |
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Hi all,
I have read about this statement many times... but I did not find a
precise way of proving it...
Do you have a mathematical way of proving it?
Thanks a lot
-L |
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Bevan Weiss
Guest
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| Posted:
Sat Nov 12, 2005 9:16 am Post subject:
Re: how do you prove a signal that is time-limited cannot be |
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lucy wrote:
| Quote: | Hi all,
I have read about this statement many times... but I did not find a
precise way of proving it...
Do you have a mathematical way of proving it?
|
Yes
No worries...
If you want more of a response, you should try explaining where you're
struggling with this problem. It looks an aweful lot like homework etc,
so without a better explanation of what it's for and where you're
particular problem is with it, many people, myself included will be
hesitant to help. |
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Jerry Avins
Guest
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| Posted:
Sat Nov 12, 2005 5:16 pm Post subject:
Re: how do you prove a signal that is time-limited cannot be |
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lucy wrote:
| Quote: | Hi all,
I have read about this statement many times... but I did not find a
precise way of proving it...
Do you have a mathematical way of proving it?
Thanks a lot
|
Lucy,
Among other things, please don't pose a question only in the subject
line. I usually get to the next message by tapping the space bar, not
selecting from a list, so I don't always notice the subject. Try to
follow the guidelines in http://users.rcn.com/jyavins/procfaq.htm.
Jerry
--
Engineering is the art of making what you want from things you can get.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ |
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zzbunker@netscape.net
Guest
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| Posted:
Sat Nov 12, 2005 5:16 pm Post subject:
Re: how do you prove a signal that is time-limited cannot be |
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lucy wrote:
| Quote: | Hi all,
I have read about this statement many times... but I did not find a
precise way of proving it...
|
Well, you don't really, since bandwidth is acrually just
crap physicists and Dan Rather made up to get
jobs with the CIA.
It's only really true with one-sided distributions.
Where it can only be proved if the Big Bang is true.
With two-sided distributions you need to show that
the side-lobes decohere, not that they're band-limited.
| Quote: | Do you have a mathematical way of proving it?
Thanks a lot
-L |
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Jerry Avins
Guest
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| Posted:
Sat Nov 12, 2005 5:16 pm Post subject:
Re: how do you prove a signal that is time-limited cannot be |
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Jerry Avins wrote:
Pay particular attention to the cartoon at the bottom.
Jerry
--
Engineering is the art of making what you want from things you can get.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ |
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Rick Lyons
Guest
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| Posted:
Sat Nov 12, 2005 5:16 pm Post subject:
Re: how do you prove a signal that is time-limited cannot be |
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On Sat, 12 Nov 2005 21:47:22 +1300, Bevan Weiss
<kaizen__@NOSPAM.hotmail.com> wrote:
| Quote: | lucy wrote:
Hi all,
I have read about this statement many times... but I did not find a
precise way of proving it...
Do you have a mathematical way of proving it?
Yes
Thanks a lot
No worries...
If you want more of a response, you should try explaining where you're
struggling with this problem. It looks an aweful lot like homework etc,
so without a better explanation of what it's for and where you're
particular problem is with it, many people, myself included will be
hesitant to help.
|
Hi Bevan,
I agree. This is Lucy's 2nd post today.
Both posts sure look like homework to me.
Maybe Lucy could tell us how he/she attempted
to solve this problem on his/her own. That way
someone here will surely help Lucy out.
[-Rick-] |
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Dr Tim
Guest
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| Posted:
Sun Nov 13, 2005 1:16 am Post subject:
Re: how do you prove a signal that is time-limited cannot be |
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May be try it the other way around:
Show that a band-limited signal has positive energy in every finite
interval.
PS: I think the other posters are a bit harsh.
No harm in giving you a hint, right? |
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robert bristow-johnson
Guest
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| Posted:
Sun Nov 13, 2005 8:06 am Post subject:
Re: how do you prove a signal that is time-limited cannot be |
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in article 1131834614.274273.143040@g47g2000cwa.googlegroups.com, Dr Tim at
timrobinson@paradise.net.nz wrote on 11/12/2005 17:30:
| Quote: | May be try it the other way around:
Show that a band-limited signal has positive energy in every finite
interval.
PS: I think the other posters are a bit harsh.
No harm in giving you a hint, right?
|
how's this (it ain't a perfect proof)? it's a sorta proof by contradiction:
we know that we can multiply a time-limited function by a rectangular pulse
of sufficient width and it changes not the time-limited function.
multiplying in the time domain is the same as convoluting in the frequency
domain. and we know that the Fourier Transform is an invertible 1 to 1
operator.
the Fourier Transform of a rectangular pulse is a sinc() function in the
frequency domain and that is clearly not bandlimited.
now, suppose that the F.T. of our time-limited function was truly
bandlimited. that is, it is zero forever outside of a finite region in
frequency. if we multiply by the rect() in the time domain, that must be
convolving in the frequency domain and any thin component of the F.T., when
convolved with the sinc() will go on forever in both directions. so it no
longer is bandlimited and we have a contradiction.
the only thing missing is how to show that they cannot somehow add to
precisely zero (in the region outside of the hypothesized bandlimited
region) when the sinc() functions from all of those "thin components" are
added together. someone want to solve that one rigorously?
--
r b-j rbj@audioimagination.com
"Imagination is more important than knowledge." |
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glen herrmannsfeldt
Guest
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| Posted:
Sun Nov 13, 2005 5:16 pm Post subject:
Re: how do you prove a signal that is time-limited cannot be |
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lucy wrote: (in the subject line)
Re: how do you prove a signal that is time-limited cannot be bandlimited?
| Quote: | I have read about this statement many times... but I did not find a
precise way of proving it...
Do you have a mathematical way of proving it?
|
You won't get me on this one. At least when it comes to Fourier
transforms I argue for periodic signals, which can be band limited.
(A favorite trick in solid state physics is using periodic boundary
conditions for the band structure of crystals. Assuming they are
sufficiently large, the answer comes out right.)
If it isn't periodic, what happens at the ends?
-- glen |
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David C. Ullrich
Guest
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| Posted:
Sun Nov 13, 2005 5:16 pm Post subject:
Re: how do you prove a signal that is time-limited cannot be |
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On Sun, 13 Nov 2005 04:25:47 -0800, glen herrmannsfeldt
<gah@ugcs.caltech.edu> wrote:
| Quote: | lucy wrote: (in the subject line)
Re: how do you prove a signal that is time-limited cannot be bandlimited?
I have read about this statement many times... but I did not find a
precise way of proving it...
Do you have a mathematical way of proving it?
You won't get me on this one. At least when it comes to Fourier
transforms I argue for periodic signals, which can be band limited.
|
But which are not likely to be time-limited...
Also not likely to actually come up in the real world (for
example a periodic signal must have begun propagating some
time before the big bang.)
| Quote: | (A favorite trick in solid state physics is using periodic boundary
conditions for the band structure of crystals. Assuming they are
sufficiently large, the answer comes out right.)
If it isn't periodic, what happens at the ends?
-- glen
|
************************
David C. Ullrich |
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David C. Ullrich
Guest
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| Posted:
Sun Nov 13, 2005 5:16 pm Post subject:
Re: how do you prove a signal that is time-limited cannot be |
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On Sat, 12 Nov 2005 21:06:13 -0500, robert bristow-johnson
<rbj@audioimagination.com> wrote:
| Quote: | in article 1131834614.274273.143040@g47g2000cwa.googlegroups.com, Dr Tim at
timrobinson@paradise.net.nz wrote on 11/12/2005 17:30:
May be try it the other way around:
Show that a band-limited signal has positive energy in every finite
interval.
PS: I think the other posters are a bit harsh.
No harm in giving you a hint, right?
how's this (it ain't a perfect proof)? it's a sorta proof by contradiction:
we know that we can multiply a time-limited function by a rectangular pulse
of sufficient width and it changes not the time-limited function.
multiplying in the time domain is the same as convoluting in the frequency
domain. and we know that the Fourier Transform is an invertible 1 to 1
operator.
the Fourier Transform of a rectangular pulse is a sinc() function in the
frequency domain and that is clearly not bandlimited.
now, suppose that the F.T. of our time-limited function was truly
bandlimited. that is, it is zero forever outside of a finite region in
frequency. if we multiply by the rect() in the time domain, that must be
convolving in the frequency domain and any thin component of the F.T., when
convolved with the sinc() will go on forever in both directions. so it no
longer is bandlimited and we have a contradiction.
the only thing missing is how to show that they cannot somehow add to
precisely zero (in the region outside of the hypothesized bandlimited
region) when the sinc() functions from all of those "thin components" are
added together.
|
In other words, the only thing missing from the proof is a
proof that the fourier transform cannot actually be
band-limited.
| Quote: | someone want to solve that one rigorously?
|
An actual mathematical proof is extremely trivial. Alas it
involves mathematics:
If f is time-limited then the fourier transform of f
is actually an entire function in the plane ("entire"
in the sense of complex analysis): If f(t) vanishes for
|t| > A then
F(z) = int_{-A}^A f(t) exp(-itz) dt
makes sense for any complex number z, and it's easy to
show that F is actually differentiable.
But an entire function cannot have compact support,
unless it vanishes identically.
************************
David C. Ullrich |
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Rob Johnson
Guest
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| Posted:
Sun Nov 13, 2005 11:59 pm Post subject:
Re: Re: how do you prove a signal that is time-limited canno |
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In article <BF9C09C5.C21F%rbj@audioimagination.com>,
robert bristow-johnson <rbj@audioimagination.com> wrote:
| Quote: | in article 1131834614.274273.143040@g47g2000cwa.googlegroups.com, Dr Tim at
timrobinson@paradise.net.nz wrote on 11/12/2005 17:30:
May be try it the other way around:
Show that a band-limited signal has positive energy in every finite
interval.
PS: I think the other posters are a bit harsh.
No harm in giving you a hint, right?
how's this (it ain't a perfect proof)? it's a sorta proof by contradiction:
we know that we can multiply a time-limited function by a rectangular pulse
of sufficient width and it changes not the time-limited function.
multiplying in the time domain is the same as convoluting in the frequency
domain. and we know that the Fourier Transform is an invertible 1 to 1
operator.
the Fourier Transform of a rectangular pulse is a sinc() function in the
frequency domain and that is clearly not bandlimited.
now, suppose that the F.T. of our time-limited function was truly
bandlimited. that is, it is zero forever outside of a finite region in
frequency. if we multiply by the rect() in the time domain, that must be
convolving in the frequency domain and any thin component of the F.T., when
convolved with the sinc() will go on forever in both directions. so it no
longer is bandlimited and we have a contradiction.
the only thing missing is how to show that they cannot somehow add to
precisely zero (in the region outside of the hypothesized bandlimited
region) when the sinc() functions from all of those "thin components" are
added together. someone want to solve that one rigorously?
|
The Fourier Transform of a compactly supported integrable function is
entire. In fact, we can even estimate the size of the terms of the
power series from the support of the function. Suppose that f(x) = 0
for |x| > R:
1 d n (-i2pi)^n |\+oo -i2pi xy n
| -- ( -- ) FT(f)(y) | = | --------- | f(x) e x dx |
n! dy n! \| -oo
(2piR)^n
<= -------- ||f||
n! L^1
The rate of growth of the terms in the power series indicates an
infinite radius of convergence.
Finally, no entire function can have compact support, they can't even
be identically 0 on a non-empty open set.
Rob Johnson <rob@trash.whim.org>
take out the trash before replying |
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Rob Johnson
Guest
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| Posted:
Tue Nov 15, 2005 5:16 pm Post subject:
Re: how do you prove a signal that is time-limited cannot be |
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In article <1132051797.412443.177690@f14g2000cwb.googlegroups.com>,
"hurry" <hurrynarain@gmail.com> wrote:
| Quote: | the only thing missing is how to show that they cannot somehow add to
precisely zero (in the region outside of the hypothesized bandlimited
region) when the sinc() functions from all of those "thin components"
are
added together. someone want to solve that one rigorously?
--------------------------
the same when done on the freq domain accounts for the other case as
welllllll
thou, an uncanny approach still thnk it holds.
|
I will repeat, "no entire function can have compact support". The
Fourier Transform of an integrable function of compact support is
entire. Even sinc functions are entire and so adding them to other
entire functions cannot produce a compactly supported function unless
they cancel everywhere producing a function that is identically 0
everywhere.
Rob Johnson <rob@trash.whim.org>
take out the trash before replying |
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Jerry Avins
Guest
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| Posted:
Tue Nov 15, 2005 5:16 pm Post subject:
Re: how do you prove a signal that is time-limited cannot be |
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Randy Yates wrote:
| Quote: | David C. Ullrich <ullrich@math.okstate.edu> writes:
[...]
An actual mathematical proof is extremely trivial. Alas it
involves mathematics:
I find your response pompous, given that the proof you provide
requires knowledge of functional analysis, which is relatively
advanced even for mathematicians, and given that the OP is probably
an undergraduate electrical engineer.
|
Ahh, give him a pass, Randy. Everyone has an off day now and then. It
usually comes when one is trying to be clever. I think that's the
original meaning of "too clever by half."
Jerry
--
Engineering is the art of making what you want from things you can get.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ |
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Randy Yates
Guest
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| Posted:
Tue Nov 15, 2005 5:16 pm Post subject:
Re: how do you prove a signal that is time-limited cannot be |
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David C. Ullrich <ullrich@math.okstate.edu> writes:
| Quote: | [...]
An actual mathematical proof is extremely trivial. Alas it
involves mathematics:
|
I find your response pompous, given that the proof you provide
requires knowledge of functional analysis, which is relatively
advanced even for mathematicians, and given that the OP is probably
an undergraduate electrical engineer.
--
% Randy Yates % "Midnight, on the water...
%% Fuquay-Varina, NC % I saw... the ocean's daughter."
%%% 919-577-9882 % 'Can't Get It Out Of My Head'
%%%% <yates@ieee.org> % *El Dorado*, Electric Light Orchestra
http://home.earthlink.net/~yatescr |
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