Digital Signal Processing Reference
In-Depth Information
c
j+1
h
h
h
c
j
2
2
2
g
g
g
2
2
2
w
j+1
2
2
2
g
g
g
h
h
h
c
j
2
2
2
2
2
Figure 2.5. Fast pyramidal algorithm associated with the biorthogonal
wavelet transform. (top) Fast analysis transform with a cascade of
filtering with
h
and
g
followed by factor 2 subsampling. (bottom) Fast
inverse transform by progressively inserting zeros and filtering with
dual filters
h
and
g
.
high-frequency channel
w
j
+
1
is left, and the process is iterated with the low-
frequency part
c
j
+
1
. This is displayed in the upper part of Fig. 2.5. In the recon-
struction or synthesis side, the coefficients are up-sampled by inserting a 0 between
each sample and then convolved with the dual filters
h
and
g
; the resulting coeffi-
cients are summed, and the result is multiplied by 2. The procedure is iterated up to
the smallest scale, as depicted in the lower part of Fig. 2.5.
Compared to the CWT, we have far fewer scales because we consider only
dyadic scales
, that is, scales
a
j
that are a power of 2 of the initial scale
a
0
(
a
j
=
2
j
a
0
).
Therefore, for a discrete signal
X
[
n
]with
N
samples, one would typically use
J
1 corresponds to the finest
scale (high frequencies). The algorithm is the following.
=
log
2
N
scales; the indexing here is such that
j
=
Algorithm 2
1-D DWT Algorithm
Task:
Compute DWT of discrete finite-length signal
X
.
Parameters:
Filters
h
h
.
,
Initialization:
c
0
=
X
,
J
=
log
2
N
.
for
j
=
0
to
J
−
1
do
h
Compute
c
j
+
1
=
c
j
, down-sample by a factor 2.
Compute
w
j
+
1
=
g
c
j
, down-sample by a factor 2.
Output:
W ={
w
1
,...,w
J
,
c
J
}
,theDWTof
X
.
