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Bart
science forum beginner
Joined: 07 Jul 2005
Posts: 39
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 Posted: Mon Jun 26, 2006 8:58 am Post subject:
application of Poisson's summation formula
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I'm trying to understand a certain step in a proof which goes
like this:
Denote by I the imaginary unit and suppose K_N(theta) is defined
as follows (call this `identity A'):
K_N(theta) = 1/N * sum_{j=0}^{N-1} w(j/N) exp(2*Pi*I*j*theta)
Then in the proof it is mentioned that by Poisson's summation
formula, this yields the identity (call this `identity B'):
K_N(theta) = lim_{nu->infty} sum_{l=-nu}^{+nu} 1/N * int_{0}^{N} w(x/N)*exp(2*Pi*I*theta*x-2*Pi*I*l*x) dx
Using Google, I have found that Poisson's summation formula
defines the relation between the sum of a time based serie and
the sum of it's transformed serie, but I cannot figure out the
small steps to go from `identity A' to `identity B'.
Could anyone give me some advice here so I can understand more
how to get from identity A to identity B?
Thanks,
Bart
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Robert B. Israel
science forum Guru
Joined: 24 Mar 2005
Posts: 2151
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| Posted: Tue Jun 27, 2006 12:03 am Post subject:
Re: application of Poisson's summation formula
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In article <1151312290.289702@seven.kulnet.kuleuven.ac.be>,
Bart <Reply.In@This.Group> wrote:
| Quote: | I'm trying to understand a certain step in a proof which goes
like this:
Denote by I the imaginary unit and suppose K_N(theta) is defined
as follows (call this `identity A'):
K_N(theta) = 1/N * sum_{j=0}^{N-1} w(j/N) exp(2*Pi*I*j*theta)
Then in the proof it is mentioned that by Poisson's summation
formula, this yields the identity (call this `identity B'):
K_N(theta) = lim_{nu->infty} sum_{l=-nu}^{+nu} 1/N * int_{0}^{N}
w(x/N)*exp(2*Pi*I*theta*x-2*Pi*I*l*x) dx
Using Google, I have found that Poisson's summation formula
defines the relation between the sum of a time based serie and
the sum of it's transformed serie, but I cannot figure out the
small steps to go from `identity A' to `identity B'.
Could anyone give me some advice here so I can understand more
how to get from identity A to identity B?
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One form of Poisson's summation formula says, for suitably "nice"
functions f (e.g. it is sufficient for f to be integrable and of
bounded variation with
f(x) = lim_{d -> 0} (f(x+d) + f(x-d))/2
for all integers x),
sum_{j=-infty}^infty f(j) = sum_{l=-infty}^infty F(2 pi l)
where F is the Fourier transform of f:
F(p) = int_{-infty}^infty f(x) exp(-i p x) dx
Your left side can be written as sum_{j=-infty}^infty f(j)
if f(j) = 1/N w(j/N) exp(2 pi i j theta) and w(x) = 0 outside
the interval 0 <= x < 1. Then
F(2 pi l) = int_{0}^N 1/N w(x/N) exp(2 pi i x (theta - l)) dx
(the integral is only from 0 to N because w(x/N) is 0 outside that
interval). Since the sum from -infty to infty is the limit
of the sum from -nu to nu as nu -> infty, this gives you your
formula B.
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada |
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Bart
science forum beginner
Joined: 07 Jul 2005
Posts: 39
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| Posted: Wed Jun 28, 2006 9:30 am Post subject:
Re: application of Poisson's summation formula
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On 2006-06-27, Robert Israel <israel@math.ubc.ca> wrote:
| Quote: |
One form of Poisson's summation formula says, for suitably "nice"
functions f (e.g. it is sufficient for f to be integrable and of
bounded variation with
f(x) = lim_{d -> 0} (f(x+d) + f(x-d))/2
for all integers x),
sum_{j=-infty}^infty f(j) = sum_{l=-infty}^infty F(2 pi l)
where F is the Fourier transform of f:
F(p) = int_{-infty}^infty f(x) exp(-i p x) dx
Your left side can be written as sum_{j=-infty}^infty f(j)
if f(j) = 1/N w(j/N) exp(2 pi i j theta) and w(x) = 0 outside
the interval 0 <= x < 1. Then
F(2 pi l) = int_{0}^N 1/N w(x/N) exp(2 pi i x (theta - l)) dx
(the integral is only from 0 to N because w(x/N) is 0 outside that
interval). Since the sum from -infty to infty is the limit
of the sum from -nu to nu as nu -> infty, this gives you your
formula B.
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Thanks for your clear explanation. I guess I was a bit confused
by the different forms in which the Fourier transform can be
written down (see e.g.
http://mathworld.wolfram.com/FourierTransform.html)
However, I guess I was also confused by the different forms in
which Poisson's sum formula can be stated. I can't remember on
which website i've found it, but it seems like i had written down
also the following forms of Poisson's sum formula:
FORM 1: with a factor 2*Pi
------------------------
sum_{n=-infty}^{infty} F(n) = 2*Pi*sum_{m=-infty}^{infty}f(2*Pi*m)
where
F(n) = int_{-infty}^{+infty}exp(i*n*t)f(t)dt is the Fourier transform
FORM 2: without a factor
------------------------
I had also written down something like `for the interval [0,1]':
sum_{n=-infty}^{+infty} F(2*Pi*n) = sum_{m=-infty}^{infty}f(m)
which is the form as you stated it.
FORM 3: with factor sqrt(2*Pi)
-------------------------------
The last form i have found is with a factor sqrt(2*Pi):
If the Fourier transform is defined
F(omega) = 1/(2*Pi) int_{k=-infty}^{infty} f(k)exp(-i*k*omega)
Then Poisson's summation formula tells that
sum_{n in Z} f(n) = sqrt(2*Pi) * sum(k in Z) F(2*Pi*k)
Sorry I can't remember/find the website i got these formula's
from, but if anybody has a good reference that could explain me where the
difference between all these forms of Poisson's sum formula come
from, that would be something interesting to read...
Thanks again,
Bart
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