z sJ '
d J
ω=
(5)
d J-1
where
sJ = (sJ,1,sJ,2,...,sJ,n/2J)'
dJ = (dJ,1,dJ,2,...,dJ,n/2J)'
dJ-1 = (dJ-1,1,dJ-1,2,...,dJ-1,n/2J-1)'
d1 = (d1,1,d1,2,...,d1,n/2 )'
Each of the sets of coefficients sj, d∣.d1 is called a crystal.
Expression (2) can be rewritten as
f(t) ≈ Sj(t)+Dj(t)+Dj-1(t)+...+D1(t) (6)
where
SJ(t) = ∑sJ,kφJ,k(t) (7a)
k
DJ(t)=∑dj,kψJ,k(t) (7b)
k
are denominated the smooth signal and the detail signals, respectively.
The terms in expression (6) represent a decomposition of the signal into orthogonal
signal components S∣(t), D∣(t), D∣ 1(t), ...,D1(t) at different scales. These terms are
components of the signal at different resolutions. That is why the approximation in (6) is
called a multi-resolution decomposition (MRD).
III Data and Estimation Results
3.1 Description of the Data