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IEEE TRANS. SIGNAL PROC. 9
where I 0 (ω) is defined by (15), and that the 3 × 3 matrix 1 √
3
[I 0 (ω + 2πM −T η 0 ), I 0 (ω + 2πM −T η 1 ), I 0 (ω +
2πM −T η 2 )] is unitary for all ω ∈ IR 2 , we know that (19) holds if and only if V (ω) is unitary for all ω ∈ IR 2 ,
namely, V (ω) satisfies
V (ω)V (ω) ∗ = I 3 , ω ∈ IR 2 . (21)
If {p, q (1) , q (2) } is given by (18), then its polyphase matrix V (ω) is
V (ω) = B n D(ω)B n−1 D(ω) · · · B 1 D(ω)B 0 .
Since each D(ω) is unitary, we know that if constant matrices B k , 0 ≤ k ≤ n, are orthogonal, then V (ω) satisfies
(21).
Next, we consider the orthogonality of a matrix B of the from (17). To this regard, let U denote the unitary
matrix:
⎡
⎤
U =
⎢
⎣
1 0 0
0
1 √2
1 √2
0
1 √2 − 1 √
2
Then
⎡ √ ⎤
b 11
UBU ∗ √ 2b12 0
= ⎣ 2b21 b 22 + b 23 0 ⎦ .
0 0 b 22 − b 23
[ √ ]
b
Thus B is orthogonal if and only if 11 2b12
√2b21 is orthogonal and b 22 + b 23 = ±1, which implies that
b 22 + b 23
b ij can be written as
⎥
⎦ .
b 11 = s 0 cos θ, b 12 = 1 √
2
sin θ, b 21 = s 0
√
2
sin θ,
b 22 + b 23 = − cos θ, b 22 − b 23 = s 1 ,
where s 0 = ±1, s 1 = ±1, θ ∈ IR. Thus an orthogonal matrix B of the form (17) has one parameter. If we choose
s 0 = 1, s 1 = 1 and write cos θ = 1−t2
1+t
, sin θ = 2t
2 1+t
, then we have
2
√
b 11 = 1−t2
1+t
, b 2 12 = b 21 = 2t
1+t
, b 2 22 = t2
1+t
, b 2 23 = 1
1+t
. (23)
2
We have therefore the following theorem.
Theorem 1: Suppose {p, q (1) , q (2) } is given by (18). If each B k , 0 ≤ k ≤ n is of the form (17) and its entries b ij
are given by (22) for some θ k , then {p, q (1) , q (2) } is an orthogonal FIR filter bank with 2-fold rotational symmetry.
With such a family of orthogonal FIR filter banks, by selecting the free parameters, one can design the filters
with desirable properties. Here we consider the filters based on the Sobolev smoothness of the associated scaling
functions φ. We say φ to be in the Sobolev space W s for some s > 0 if φ satisfies ∫ IR 2(1 + |ω|2 ) s | ˆφ(ω)| 2 dω < ∞.
To assure that φ ∈ W s , the associated FIR lowpass filter p(ω) has sum rules of certain order. p(ω) is said to have
sum rule order m (with dilation matrix M) provided that p(0, 0) = 1 and
(22)
D α 1
1 Dα 2
2 p(2πM −T η j ) = 0, 1 ≤ j ≤ 2, (24)
for all (α 1 , α 2 ) ∈ Z 2 + with α 1 + α 2 < m, where η j , 1 ≤ j ≤ 2, are defined by (8), D 1 and D 2 denote the
partial derivatives with the first and second variables of p(ω) respectively. Under some condition, sum rule order is
equivalent to the approximation order of φ, see [48]. The Sobolev smoothness of φ can be given by the eigenvalues
of the so-called transition operator matrix T p associated with the lowpass filter p, see [49], [50]. The reader refers
to [26] for the algorithms and Matlab routines to find the Sobolev smoothness order.
We find that if the orthogonal filter bank {p, q (1) , q (2) } is given by (18) with n = 0 or n = 1, then the lowpass
filter p(ω) cannot achieve sum rule order 2, and hence, the smoothness order of φ is very low. In the following
two examples, we consider the filter banks with n = 2 and n = 3.
Example 1: Let {p, q (1) , q (2) } be the orthogonal filter bank with 2-fold rotational symmetry given by (18) for
n = 2: B 2 D 2 (M T ω)B 1 D 1 (M T ω)B 0 I 0 (ω), where D 1 , D 2 are defined in (16), B 0 , B 1 and B 2 are orthogonal