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   -> Volume 2, Issue 7

Response to WD Vol 2.6 #8
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Anil A. Bharath, Imperial College London

PostPosted: Fri Nov 29, 2002 3:29 pm    
Subject: Response to WD Vol 2.6 #8
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Response to WD Vol 2.6 #8

Response to Richard Favero's question, Item 8 of Wavelet Digest vol. 2, Issue 6


In order to see how sub-octave resolution might be obtained, have
a look at the excellent paper by Shensa (IEEE Transactions on Signal Proc-
essing, vol. 40, no. 10, October, 1992). In addition, I have found the
paper by Olivier Rioul and Pierre Duhamel (IEEE Transactions on Information
Theory, vol. 38, no. 2, March 1992) to be very useful in implementing sub-
octave decompositions using a combination of the Mallat and "a trous"
algorithms, with families of wavelets. The formulations refer to
sub-octave frequency domain partitioning as a "voice" of the decomposition.

As for the matter of analytic forms for continuous wavelets, I can provide
a few pointers if you e-mail me directly; a common choice which is not
*strictly* admissible as a wavelet (but which is easy to work with,
and which you mentioned in your question) is the Morlet Wavelet

psi(t) = exp^(-t*t/2).e^(j omega_0 t)

where omega_0 must be made sufficiently large that FT{psi(t)}(0)
is very small. I have found this choice quite useful for analyzing
asymptotic signals.

Anil A. Bharath,
Centre for Biological and Medical Systems,
Level 1,
Mech. Eng. Building
Imperial College
London SW7 2BT

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