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   -> Volume 6, Issue 4

Question: most compact time function ...
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Author Message (Alain WITTMANN)

PostPosted: Thu Mar 06, 1997 5:28 pm    
Subject: Question: most compact time function ...
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#21 Question: most compact time function ...

Question: I would like to find a time function that is as "compact" as
possible in the time domain, and that has a defined frequency content
at a fundamental frequency and a finite number of its harmonics. The
frequency content outside these harmonic frequencies is irrelevant. By
compact, I mean that the energy should be gathered around t=0, and the
duration should be as short as possible. I will try to formulate the
problem in a more mathematical way :

Find f(t) such that :

for k=0,...N, F(k.w0)=Ak, with F(w) = TF[f(t)], that minimizes the integral

/ +oo
| 2
C= | f(t) w(t) dt, where w(t) is a cost function that penalizes time content
/ -oo 2
far from t=0. It could be a function like w(t)=e .

Has this problem or a similar one been addressed in the wavelets
litterature ? Please forgive the naive formulation of the problem, I
am new in the wavelets field. Do not hesitate to provides basic

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