Information Technology Reference
In-Depth Information
m
0
C1
X
X
C1
X
f.x/D
S
m
0
;n
m
0
;n
C
T
m;n
m;n
.x/
(11.16)
nD1
mD1
nD1
It can be seen from this equation that the original continuous signal is expressed as
a combination of an approximation of itself, at arbitrary scale index m
0
added to
a succession of signal details form scales m
0
down to negative infinity. The signal
detail at scale m is defined as
C1
X
d
m
.x/ D
T
m;n
m;n
.x/
(11.17)
nD1
and hence one can write Eq. (
11.16
)
m
0
X
f.x/D f
m
0
.t/ C
d
m
.x/
(11.18)
mD1
From this equation it can be shown that
f
m1
.x/ D f
m
.x/ C d
m
.x/
(11.19)
which shows that if one adds the signal detail at an arbitrary scale (index m)to
the approximation at that scale he gets the signal approximation at an increased
resolution (at a smaller scale index m 1). This is the so-called multi-resolution
representation.
11.2.4
Examples of Orthonormal Wavelets
The scaling equation (or dilation equation) describes the scaling function .x/ in
terms of contracted and shifted versions of itself as follows [
2
,
119
]:
.x/ D
X
k
c
k
.2x k/
(11.20)
where .2x k/ is a contracted version of .t/ shifted along the time axis by an
integer step k and factored by an associated scaling coefficient c
k
. The coefficient
of the scaling equation should satisfy the condition
X
c
k
D 2
(11.21)
k
2 if k
0
D 0
0 otherwise
X
c
k
c
kC2k
0
D
(11.22)
k
