Given coefficients
hn
h
n
that satisfy the regularity conditions, we can
iteratively calculate samples of
φt
φ
t
on a fine grid of points
t
t
using the cascade algorithm. Once we
have obtained
φt
φ
t
, the wavelet scaling equation can be used to construct
ψt
ψ
t
.
In this discussion we assume that
Hz
H
z
is causal with impulse response length
NN. Recall, from our discussion of the regularity conditions, that this implies
φt
φ
t
will have compact support on the interval
0N-1
0
N
1
. The cascade algorithm is described below.
- Consider the scaling function at integer times
t=m∈0…N-1
t
m
0
…
N
1
:
φm=2∑n=0N-1hnφ2m-n
φ
m
2
n
0
N
1
h
n
φ
2
m
n
Knowing that
φt=0
φ
t
0
for
t∉0N-1
t
0
N
1
, the previous equation can be written using an
NNxNN
matrix. In the case where
N=4
N
4
, we have
φ0φ1φ2φ3=2h0000h2h1h000h3h2h1000h3φ0φ1φ2φ3
φ
0
φ
1
φ
2
φ
3
2
h
0
0
0
0
h
2
h
1
h
0
0
0
h
3
h
2
h
1
0
0
0
h
3
φ
0
φ
1
φ
2
φ
3
(1)
where
H=h0000h2h1h000h3h2h1000h3
where
H
h
0
0
0
0
h
2
h
1
h
0
0
0
h
3
h
2
h
1
0
0
0
h
3
The matrix HH is
structured as a row-decimated convolution
matrix. From the matrix equation above (Equation 1), we see that
φ0φ1φ2φ3T
φ
0
φ
1
φ
2
φ
3
must be (some scaled version of) the eigenvector
of HH
corresponding to eigenvalue
2-1
2
-1
. In general, the nonzero values of
{φn|n∈ℤ}
φ
n
n
, i.e.,
φ0φ1…φN-1T
φ
0
φ
1
…
φ
N
1
, can be calculated by appropriately scaling the eigenvector
of the
NNxNN
row-decimated convolution matrix HH corresponding to the
eigenvalue
2-1
2
-1
. It can be shown that this eigenvector must be
scaled so that
∑n=0N-1φn=1
n
0
N
1
φ
n
1
.
- Given
{φn|n∈ℤ}
φ
n
n
, we can use the scaling equation to determine
{φn2|n∈ℤ}
φ
n
2
n
:
φm2=2∑n=0N-1hnφm2-n
φ
m
2
2
n
0
N
1
h
n
φ
m
2
n
(2)
This produces the
2N-1
2
N
1
non-zero samples
φ0φ1/2φ1φ3/2…φN-1
φ
0
φ
12
φ
1
φ
32
…
φ
N
1
.
-
Given
{φn2|n∈ℤ}
φ
n
2
n
, the scaling equation can be used to find
{φn4|n∈ℤ}
φ
n
4
n
:
φm4=2∑n=0N-1hnφm2-n=2∑p evenhp2φm-p2=2∑p
h
↑
2
p
φ
1
2
m-p
φ
m
4
2
n
0
N
1
h
n
φ
m
2
n
2
p
p even
h
p
2
φ
m
p
2
2
p
p
h
↑
2
p
φ
1
2
m
p
(3)
where
h
↑
2
p
h
↑
2
p
denotes the impulse response of
Hz2
H
z
2
, i.e., a 2-upsampled version of
hn
h
n
, and where
φ
1
2
m=φm2
φ
1
2
m
φ
m
2
. Note that
{φn4|n∈ℤ}
φ
n
4
n
is the result of convolving
h
↑
2
n
h
↑
2
n
with
φ
1
2
n
φ
1
2
n
.
-
Given
{φn4|n∈ℤ}
φ
n
4
n
, another convolution yields
{φn8|n∈ℤ}
φ
n
8
n
:
φm8=2∑n=0N-1hnφm4-n=2∑p
h
↑
4
p
φ
1
4
m-p
φ
m
8
2
n
0
N
1
h
n
φ
m
4
n
2
p
p
h
↑
4
p
φ
1
4
m
p
(4)
where
h
↑
4
n
h
↑
4
n
is a 4-upsampled version of
hn
h
n
and where
φ
1
4
m=φm4
φ
1
4
m
φ
m
4
.
-
At the
ℓ
th
ℓ
th
stage,
φn2ℓ
φ
n
2
ℓ
is calculated by convolving the result of the
ℓ
−
1
th
ℓ
−
1
th
stage with a
2ℓ-1
2
ℓ
1
-upsampled version of
hn
h
n
:
φ
1
2
ℓ
m=2∑p
h
↑
2
ℓ
−
1
p
φ
1
2
ℓ
−
1
m-p
φ
1
2
ℓ
m
2
p
p
h
↑
2
ℓ
−
1
p
φ
1
2
ℓ
−
1
m
p
(5)
For
ℓ≈10
ℓ
10
, this gives a very good approximation of
φt
φ
t
. At this point, you could verify the key properties of
φt
φ
t
, such as orthonormality and the satisfaction of the
scaling equation.
In Figure 1 we show steps 1 through 5
of the cascade algorithm, as well as step 10, using Daubechies'
db2 coefficients (for which
N=4
N
4
).
