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> Volume 4, Issue 7
Question: Window Fourier Transform

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Haikel Hichri (hah134@cs.usask.ca) Guest

Posted: Tue Dec 03, 2002 1:22 pm Subject: Question: Window Fourier Transform




Question: Window Fourier Transform
Hello everyone
I am currently studying wavelets. I have a few questions
that I couldn't answer from the books and papers I am reading.
the question is about Windowed Fourier Transform :
I am reading in a lot of places (for example a fairly good book titled
" a freindly guide to wavelets " by Gerald Kaiser) about the
Heisenberg inequality and I don't understand why is it a problem.
now, the regular fourier analysis as i understand so far, has a
problem with discontiniouties which makes the fourier expansion to
include high frequency harmonics to aproxmate the function to a
certain degree. and these high frequency harmonics extend over the
whole time period, right.. ok, now they say that the WFT solves this
problem by looking at the function through windows, using a function g
suported on interval I. that way only the coeficients near the
discontineouty are affected. or in other words only high frequency
harmonics supported on the interval I are needed.
then they go on and prove the "Heisenberg inequality" which says
in the book mentioned above that dt.df>c where c is a constant and df
and dt are defined by the standard deviation of g divided by its
energy and df is the standard deviation of Fourier transform(g)
devided by the same energy again. now they say this a problem because
two pulses closer than dt can't be descriminated and also tow waves
with frequencies closer than df can't be descriminated. but why? how
on earth did they get this result? everybody talks about it as if it
is intiutive well it is not for me!
this makes me even more confused :
why this resolution problem exists for WFT but for the FT? I have no
problem with the mathematical proof as usual but I don't understand
the meaning of it, the intuition behind it.
I hope this is clear enough.
i hope I will get an answer from you guys
thanks,
Hichri Haikel (H)(306) 3734842
Department of Computer Science
University of Saskachewan, Saskatoon CANADA.
[Answer] 





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