Here we give a quick description of what is
probably the most popular family of filter coefficients
hn
h
n
and
gn
g
n
— those proposed by Daubechies.
Recall the iterated synthesis filterbank. Applying the Noble
identities, we can move the up-samplers before the filters, as
illustrated in Figure 1.
The properties of the ii-stage
cascaded lowpass filter
H
(
i
)
z=∏k=0i−1Hz2k ,
i≥1
i
1
H
(
i
)
z
k
0
i
1
H
z
2
k
(1)
in the limit
i→∞
i
give an important characterization of the wavelet
system. But how do we know that
limit i→
∞
H
(
i
)
ejω
i
H
(
i
)
ω
converges to a response in
ℒ
2
ℒ
2
? In fact, there are some rather strict conditions on
Hejω
H
ω
that must be satisfied for this convergence to
occur. Without such convergence, we might have a finite-stage
perfect reconstruction filterbank, but we will
not have a countable wavelet basis for
ℒ
2
ℒ
2
. Below we present some "regularity conditions" on
Hejω
H
ω
that ensure convergence of the iterated synthesis
lowpass filter.
The convergence of the
lowpass filter implies convergence of all other filters in the
bank.
Let us denote the impulse response of
H
(
i
)
z
H
(
i
)
z
by
h
(
i
)
n
h
(
i
)
n
. Writing
H
(
i
)
z=Hz2i−1
H
(
i
−
1
)
z
H
(
i
)
z
H
z
2
i
1
H
(
i
−
1
)
z
in the time domain, we have
h
(
i
)
n=∑kkhk
h
(
i
−
1
)
n−2i−1k
h
(
i
)
n
k
k
h
k
h
(
i
−
1
)
n
2
i
1
k
Now define the function
φ
(
i
)
t=2i2∑nn
h
(
i
)
n
ℐ
[
n
/
2
i
,
n
+
1
/
2
i
)
t
φ
(
i
)
t
2
i
2
n
n
h
(
i
)
n
ℐ
[
n
/
2
i
,
n
+
1
/
2
i
)
t
where
ℐ
[
a
,
b
)
t
ℐ
[
a
,
b
)
t
denotes the indicator function over the interval
a
b
a
b
:
ℐ
[
a
,
b
)
t={1 if t∈
a
b
0 if t∉
a
b
ℐ
[
a
,
b
)
t
1
t
a
b
0
t
a
b
The definition of
φ
(
i
)
t
φ
(
i
)
t
implies
h
(
i
)
n=2−i2
φ
(
i
)
t ,
t∈
n2i
n+12i
t
t
n
2
i
n
1
2
i
h
(
i
)
n
2
i
2
φ
(
i
)
t
(2)
h
(
i
−
1
)
n−2i−1k=2−i−12
φ
(
i
−
1
)
2t−k ,
t∈
n2i
n+12i
t
t
n
2
i
n
1
2
i
h
(
i
−
1
)
n
2
i
1
k
2
i
1
2
φ
(
i
−
1
)
2
t
k
(3)
and plugging the two previous expressions into the equation
for
h
(
i
)
n
h
(
i
)
n
yields
φ
(
i
)
t=2∑kkhk
φ
(
i
−
1
)
2t−k
.
φ
(
i
)
t
2
k
k
h
k
φ
(
i
−
1
)
2
t
k
.
(4)
Thus, if
φ
(
i
)
t
φ
(
i
)
t
converges pointwise to a continuous function, then it
must satisfy the scaling equation, so that
limit i→
∞
φ
(
i
)
t=φt
i
φ
(
i
)
t
φ
t
.
Daubechies showed that, for
pointwise convergence of
φ
(
i
)
t
φ
(
i
)
t
to a continuous function in
ℒ
2
ℒ
2
, it is sufficient that
Hejω
H
ω
can be factored as
Hejω=21+ejω2PRejω ,
P≥1
P
P
1
H
ω
2
1
ω
2
P
R
ω
(5)
for
Rejω
R
ω
such that
sup
ω
|Rejω|<2P−1
sup
ω
R
ω
2
P
1
(6)
Here
PP denotes the number of
zeros that
Hejω
H
ω
has at
ω=π
ω
. Such conditions are called
regularity
conditions because they ensure the regularity, or
smoothness of
φt
φ
t
. In fact, if we make the previous condition
stronger:
sup
ω
|Rejω|<2P−1−ℓ ,
ℓ≥1
ℓ
ℓ
1
sup
ω
R
ω
2
P
1
ℓ
(7)
then
limit i→
∞
φ
(
i
)
t=φt
i
φ
(
i
)
t
φ
t
for
φt
φ
t
that is
ℓℓ-times
continuously differentiable.
There is an interesting and important by-product of the
preceding analysis. If
hn
h
n
is a causal length-NN
filter, it can be shown that
h
(
i
)
n
h
(
i
)
n
is causal with length
N
i
=2i(N−1)+1
N
i
2
i
1
N
1
1
. By construction, then,
φ
(
i
)
t
φ
(
i
)
t
will be zero outside the interval
0
2i(N−1)+12i
0
2
i
1
N
1
1
2
i
. Assuming that the regularity conditions are
satisfied so that
limit i→
∞
φ
(
i
)
t=φt
i
φ
(
i
)
t
φ
t
, it follows that
φt
φ
t
must be zero outside the interval
0
N−1
0
N
1
. In this case we say that
φt
φ
t
has compact support. Finally, the
wavelet scaling equation implies that, when
φt
φ
t
is compactly supported on
0
N−1
0
N
1
and
gn
g
n
is length NN,
ψt
ψ
t
will also be compactly supported on the interval
0
N−1
0
N
1
.
Daubechies constructed a family of
Hz
H
z
with impulse response lengths
N∈46810…
N
4
6
8
10
…
which satisfy the regularity conditions. Moreover,
her filters have the maximum possible number of zeros at
ω=π
ω
, and thus are maximally regular (i.e., they yield
the smoothest possible
φt
φ
t
for a given support interval). It turns out that
these filters are the maximally flat filters
derived by Herrmann long before
filterbanks and wavelets were in vogue. In Figure 2 and Figure 3 we show
φt
φ
t
,
ΦΩ
Φ
Ω
,
ψt
ψ
t
, and
ΨΩ
Ψ
Ω
for various members of the Daubechies' wavelet
system.
See Vetterli and Kovacivić for a
more complete discussion of these matters.
-
I. Daubechies. (1988, Nov). Orthonormal bases of compactly supported wavelets. Commun. on Pure and Applied Math, 41, 909-996.
-
O. Herrmann. (1971). On the approximation problem in nonrecursive digital filter design. IEEE Trans. on Circuit Theory, 18, 411-413.
-
M. Vetterli and J. Kovacivić. (1995). Wavelets and Subband Coding. Englewood Cliffs, NJ: Prentice Hall.
