Assume that we start with a signal
xt∈
ℒ
2
x
t
ℒ
2
. Denote the best approximation at the
0
th
0
th
level of coarseness by
x0
t
x0
t
. (Recall that
x0
t
x0
t
is the orthogonal projection of
xt
x
t
onto
V
0
V
0
.) Our goal, for the moment, is to decompose
x0
t
x0
t
into scaling coefficients and wavelet coefficients at higher
levels. Since
x0
t∈
V
0
x0
t
V
0
and
V0
=
V1
⊕
W1
V0
V1
W1
, there exist coefficients
c
0
n
c
0
n
,
c
1
n
c
1
n
, and
d
1
n
d
1
n
such that
x
0
t=∑n
c
0
n
φ
0
,
n
t=∑n
c
1
n
φ
1
,
n
t+∑n
d
1
n
ψ
1
,
n
t
x
0
t
n
n
c
0
n
φ
0
,
n
t
n
n
c
1
n
φ
1
,
n
t
n
n
d
1
n
ψ
1
,
n
t
(1)
Using the fact that
{
φ
1
,
n
t|n∈ℤ}
φ
1
,
n
t
n
is an orthonormal basis for
V
1
V
1
, in conjunction with the scaling equation,
c
1
n=<
x
0
t,
φ
1
,
n
t>=<∑m
c
0
m
φ
0
,
m
t,
φ
1
,
n
t>=∑m
c
0
m<
φ
0
,
m
t,
φ
1
,
n
t>=∑m
c
0
m<φt−m,∑ℓhℓφt−ℓ−2n>=∑m
c
0
m∑ℓhℓ<φt−m,φt−ℓ−2n>=∑m
c
0
mhm−2n
c
1
n
x
0
t
φ
1
,
n
t
m
m
c
0
m
φ
0
,
m
t
φ
1
,
n
t
m
m
c
0
m
φ
0
,
m
t
φ
1
,
n
t
m
m
c
0
m
φ
t
m
ℓ
ℓ
h
ℓ
φ
t
ℓ
2
n
m
m
c
0
m
ℓ
ℓ
h
ℓ
φ
t
m
φ
t
ℓ
2
n
m
m
c
0
m
h
m
2
n
(2)
where
δt−ℓ−2n=<φt−m,φt−ℓ−2n>
δ
t
ℓ
2
n
φ
t
m
φ
t
ℓ
2
n
. The previous expression (
Equation 2) indicates that
c
1
n
c
1
n
results from convolving
c
0
m
c
0
m
with a time-reversed version of
hm
h
m
then downsampling by factor two (
Figure 1).
Using the fact that
{
ψ
1
,
n
t|n∈ℤ}
ψ
1
,
n
t
n
is an orthonormal basis for
W
1
W
1
, in conjunction with the wavelet scaling equation,
d
1
n=<
x0
t,
ψ
1
,
n
t>=<∑m
c
0
m
φ
0
,
m
t,
ψ
1
,
n
t>=∑m
c
0
m<
φ
0
,
m
t,
ψ
1
,
n
t>=∑m
c
0
m<φt−m,∑ℓgℓφt−ℓ−2n>=∑m
c
0
m∑ℓgℓ<φt−m,φt−ℓ−2n>=∑m
c
0
mgm−2n
d
1
n
x0
t
ψ
1
,
n
t
m
m
c
0
m
φ
0
,
m
t
ψ
1
,
n
t
m
m
c
0
m
φ
0
,
m
t
ψ
1
,
n
t
m
m
c
0
m
φ
t
m
ℓ
ℓ
g
ℓ
φ
t
ℓ
2
n
m
m
c
0
m
ℓ
ℓ
g
ℓ
φ
t
m
φ
t
ℓ
2
n
m
m
c
0
m
g
m
2
n
(3)
where
δt−ℓ−2n=<φt−m,φt−ℓ−2n>
δ
t
ℓ
2
n
φ
t
m
φ
t
ℓ
2
n
.
The previous expression (Equation 3) indicates that
d
1
n
d
1
n
results from convolving
c
0
m
c
0
m
with a time-reversed version of
gm
g
m
then downsampling by factor two (Figure 2).
Putting these two operations together, we arrive at what looks
like the analysis portion of an FIR filterbank (Figure 3):
We can repeat this process at the next higher level. Since
V
1
=
W
2
⊕
V
2
V
1
W
2
V
2
, there exist coefficients
c
2
n
c
2
n
and
d
2
n
d
2
n
such that
x
1
t=∑n
c
1
n
φ
1
,
n
t=∑n
d
2
n
ψ
2
,
n
t+∑n
c
2
n
φ
2
,
n
t
x
1
t
n
n
c
1
n
φ
1
,
n
t
n
n
d
2
n
ψ
2
,
n
t
n
n
c
2
n
φ
2
,
n
t
(4)
Using the same steps as before we find that
c
2
n=∑m
c
1
mhm−2n
c
2
n
m
m
c
1
m
h
m
2
n
(5)
d
2
n=∑m
c
1
mgm−2n
d
2
n
m
m
c
1
m
g
m
2
n
(6)
which gives a cascaded analysis filterbank (
Figure 4):
If we use
V
0
=
W
1
⊕
W
2
⊕
W
3
⊕⋯⊕
W
k
⊕
V
k
V
0
W
1
W
2
W
3
⋯
W
k
V
k
to repeat this process up to the
k
th
k
th
level, we get the iterated analysis filterbank (Figure 5).
As we might expect, signal reconstruction can be accomplished
using cascaded two-channel synthesis filterbanks. Using the
same assumptions as before, we have:
c
0
m=<
x
0
t,
φ
0
,
m
t>=<∑n
c
1
n
φ
1
,
n
t+∑n
d
1
n
ψ
1
,
n
t,
φ
0
,
m
t>=∑n
c
1
n<
φ
1
,
n
t,
φ
0
,
m
t>+∑n
d
1
n<
ψ
1
,
n
t,
φ
0
,
m
t>=∑n
c
1
nhm−2n+∑n
d
1
ngm−2n
c
0
m
x
0
t
φ
0
,
m
t
n
n
c
1
n
φ
1
,
n
t
n
n
d
1
n
ψ
1
,
n
t
φ
0
,
m
t
n
n
c
1
n
φ
1
,
n
t
φ
0
,
m
t
n
n
d
1
n
ψ
1
,
n
t
φ
0
,
m
t
n
n
c
1
n
h
m
2
n
n
n
d
1
n
g
m
2
n
(7)
where
hm−2n=<
φ
1
,
n
t,
φ
0
,
m
t>
where
h
m
2
n
φ
1
,
n
t
φ
0
,
m
t
and
gm−2n=<
ψ
1
,
n
t,
φ
0
,
m
t>
and
g
m
2
n
ψ
1
,
n
t
φ
0
,
m
t
which can be implemented using the block diagram in
Figure 6.
The same procedure can be used to derive
c
1
m=∑n
c
2
nhm−2n+∑n
d
2
ngm−2n
c
1
m
n
n
c
2
n
h
m
2
n
n
n
d
2
n
g
m
2
n
(8)
from which we get the diagram in
Figure 7.
To reconstruct from the
k
th
k
th
level, we can use the iterated synthesis filterbank (Figure 8).
The table makes a
direct comparison between wavelets and the two-channel
orthogonal PR-FIR filterbanks.
Table 1
|
|
Discrete Wavelet Transform
|
2-Channel Orthogonal PR-FIR Filterbank
|
|
Analysis-LPF
|
Hz-1
H
z
-1
|
H
0
z
H
0
z
|
|
Power Symmetry
|
HzHz-1+H-zH-z-1=2
H
z
H
z
-1
H
z
H
z
-1
2
|
H
0
z
H
0
z-1+
H
0
-z
H
0
-z-1=1
H
0
z
H
0
z
-1
H
0
z
H
0
z
-1
1
|
|
Analysis HPF
|
Gz-1
G
z
-1
|
H
1
z
H
1
z
|
| Spectral Reverse |
∀P,P is odd:Gz=±z-PH-z-1
P
P is odd
G
z
±
z
P
H
z
-1
|
∀N,N is even:
H
1
z=±z-(N−1)
H
0
-z-1
N
N is even
H
1
z
±
z
N
1
H
0
z
-1
|
|
Synthesis LPF
|
Hz
H
z
|
G
0
z=2z-(N−1)
H
0
z-1
G
0
z
2
z
N
1
H
0
z
-1
|
|
Synthesis HPF
|
Gz
G
z
|
G
1
z=2z-(N−1)
H
1
z-1
G
1
z
2
z
N
1
H
1
z
-1
|
From the table, we see that the discrete wavelet transform that
we have been developing is identical to two-channel orthogonal
PR-FIR filterbanks in all but a couple details.
-
Orthogonal PR-FIR filterbanks employ synthesis filters with
twice the gain of the analysis filters, whereas in the DWT
the gains are equal.
-
Orthogonal PR-FIR filterbanks employ causal filters of length
NN, whereas the DWT filters are
not constrained to be causal.
For convenience, however, the wavelet filters
Hz
H
z
and
Gz
G
z
are usually chosen to be causal. For both to have
even impulse response length
NN, we
require that
P=N−1
P
N
1
.
