Here we derive sufficient conditions on the coefficients used
in the scaling equation and wavelet scaling equation that
ensure, for every
k∈Z
k
, that the sets
φ
k
,
n
t
n∈Z
φ
k
,
n
t
n
and
ψ
k
,
n
t
n∈Z
ψ
k
,
n
t
n
have the orthonormality properties described in The Scaling Equation and
The Wavelet Scaling
Equation.

For
φ
k
,
n
t
n∈Z
φ
k
,
n
t
n
to be orthonormal at all
kk, we certainly need
orthonormality when
k=1
k
1
. This is equivalent to

δm=〈
φ
1
,
0
t,
φ
1
,
m
t〉=〈∑nhnφt−n,∑ℓhℓφt−ℓ−2m〉=∑nhn∑ℓhℓ〈(φt−n,φt−ℓ−2m)〉
δ
m
φ
1
,
0
t
φ
1
,
m
t
n
h
n
φ
t
n
ℓ
h
ℓ
φ
t
ℓ
2
m
n
h
n
ℓ
h
ℓ
φ
t
n
φ
t
ℓ
2
m

(1)
where

δn−ℓ+2m=〈φt−n,φt−ℓ−2m〉
δ
n
ℓ
2
m
φ
t
n
φ
t
ℓ
2
m
δm=∑n=−∞∞hnhn−2m
δ
m
n
h
n
h
n
2
m

(2)
There is an interesting frequency-domain interpretation of the
previous condition. If we define

pm=hm*h−m=∑nhnhn−m
p
m
h
m
h
m
n
h
n
h
n
m

(3)
then we see that our condition is equivalent to

p2m=δm
p
2
m
δ
m
. In the

zz-domain,
this yields the pair of conditions

Pz=HzHz-1
P
z
H
z
H
z
-1

(4)
1=1/2∑p=01Pz1/2ei2π2p=1/2Pz1/2+1/2P−z1/2
1
12
p
0
1
P
z
12
2
2
p
12
P
z
12
12
P
z
12
Putting these together,

2=Hz1/2Hz−1/2+H−z1/2H−z−1/2
2
H
z
12
H
z
12
H
z
12
H
z
12

(5)
⇔
2=HzHz-1+H−zH−z-1
⇔
2
H
z
H
z
-1
H
z
H
z
-1
⇔
2=|Heiω|2+|Hei(π−ω)|2
⇔
2
H
ω
2
H
ω
2
where the last property invokes the fact that

hn∈R
h
n
and that real-valued impulse responses yield
conjugate-symmetric DTFTs. Thus we find that

hn
h
n
are the impulse response coefficients of a
power-symmetric filter. Recall that this property was also
shared by the analysis filters in an orthogonal
perfect-reconstruction FIR filterbank.

Given orthonormality at level
k=0
k
0
, we have now derived a condition on
hn
h
n
which is necessary and sufficient for orthonormality
at level
k=1
k
1
. Yet the same condition is necessary and sufficient
for orthonormality at level
k=2
k
2
:

δm=〈
φ
2
,
0
t,
φ
2
,
m
t〉=〈∑nhn
φ
1
,
n
t,∑ℓhℓ
φ
1
,
ℓ
+
2
m
t〉=∑nhn∑ℓhℓ〈(
φ
1
,
n
t,
φ
1
,
ℓ
+
2
m
t)〉=∑n=−∞∞hnhn−2m
δ
m
φ
2
,
0
t
φ
2
,
m
t
n
h
n
φ
1
,
n
t
ℓ
h
ℓ
φ
1
,
ℓ
+
2
m
t
n
h
n
ℓ
h
ℓ
φ
1
,
n
t
φ
1
,
ℓ
+
2
m
t
n
h
n
h
n
2
m

(6)
where

δn−ℓ+2m=〈
φ
1
,
n
t,
φ
1
,
ℓ
+
2
m
t〉
δ
n
ℓ
2
m
φ
1
,
n
t
φ
1
,
ℓ
+
2
m
t
. Using induction, we conclude that the previous
condition will be necessary and sufficient for orthonormality
of

φ
k
,
n
t
n∈Z
φ
k
,
n
t
n
for all

k∈Z
k
.

To find conditions on
gn
g
n
ensuring that the set
ψ
k
,
n
t
n∈Z
ψ
k
,
n
t
n
is orthonormal at every
kk, we can repeat the steps above
but with
gn
g
n
replacing
hn
h
n
,
ψ
k
,
n
t
ψ
k
,
n
t
replacing
φ
k
,
n
t
φ
k
,
n
t
, and the wavelet-scaling equation replacing the
scaling equation. This yields

δm=∑n=−∞∞gngn−2m
δ
m
n
g
n
g
n
2
m

(7)
⇔
2=GzGz-1+G−zG−z-1
⇔
2
G
z
G
z
-1
G
z
G
z
-1
Next derive a condition which guarantees that

W
k
⊥
V
k
⊥
W
k
V
k
, as required by our definition

W
k
=
V
k
⊥
W
k
V
k
⊥
, for all

k∈Z
k
. Note that, for any

k∈Z
k
,

W
k
⊥
V
k
⊥
W
k
V
k
is guaranteed by

ψ
k
,
n
t
n∈Z
⊥
φ
k
,
n
t
n∈Z
⊥
ψ
k
,
n
t
n
φ
k
,
n
t
n
which is equivalent to

0=〈
ψ
k
+
1
,
0
t,
φ
k
+
1
,
m
t〉=〈∑ngn
φ
k
,
n
t,∑ℓhℓ
φ
k
,
ℓ
+
2
m
t〉=∑ngn∑ℓhℓ〈(
φ
k
,
n
t,
φ
k
,
ℓ
+
2
m
t)〉=∑ngnhn−2m
0
ψ
k
+
1
,
0
t
φ
k
+
1
,
m
t
n
g
n
φ
k
,
n
t
ℓ
h
ℓ
φ
k
,
ℓ
+
2
m
t
n
g
n
ℓ
h
ℓ
φ
k
,
n
t
φ
k
,
ℓ
+
2
m
t
n
g
n
h
n
2
m

(8)
for all

mm where

δn−ℓ+2m=〈
φ
k
,
n
t,
φ
k
,
ℓ
+
2
m
t〉
δ
n
ℓ
2
m
φ
k
,
n
t
φ
k
,
ℓ
+
2
m
t
. In other words, a 2-downsampled version of

gn*h−n
g
n
h
n
must consist only of zeros. This necessary and
sufficient condition can be restated in the frequency domain
as

0=1/2∑p=01Gz1/2e−(i2π2p)Hz−1/2ei2π2p
0
12
p
0
1
G
z
12
2
2
p
H
z
12
2
2
p

(9)
⇔
0=Gz1/2Hz−1/2+G−z1/2H−z−1/2
⇔
0
G
z
12
H
z
12
G
z
12
H
z
12
⇔
0=GzHz-1+G−zH−z-1
⇔
0
G
z
H
z
-1
G
z
H
z
-1
The choice

∀odd P:Gz=±z−PH−z-1
odd P
G
z
±
z
P
H
z
-1

(10)
satisfies our condition, since

GzHz-1+G−zH−z-1=±z−PH−z-1Hz-1∓z−PHz-1H−z-1=0
G
z
H
z
-1
G
z
H
z
-1
∓
±
z
P
H
z
-1
H
z
-1
z
P
H
z
-1
H
z
-1
0
In the time domain, the condition on

Gz
G
z
and

Hz
H
z
can be expressed

∀odd P:gn=±-1nhP−n
.
odd P
g
n
±
-1
n
h
P
n
.

(11)
Recall that this property was satisfied by the analysis
filters in an orthogonal perfect reconstruction FIR
filterbank.

Note that the two conditions
∀odd P:Gz=±z−PH−z-1
odd P
G
z
±
z
P
H
z
-1
2=HzHz-1+H−zH−z-1
2
H
z
H
z
-1
H
z
H
z
-1
are sufficient to ensure that both
φ
k
,
n
t
n∈Z
φ
k
,
n
t
n
and
ψ
k
,
n
t
n∈Z
ψ
k
,
n
t
n
are orthonormal for all
kk and that
W
k
⊥
V
k
⊥
W
k
V
k
for all kk, since they
satisfy the condition
2=GzGz-1+G−zG−z-1
2
G
z
G
z
-1
G
z
G
z
-1
automatically.