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IEEE TRANS. SIGNAL PROC. 8
We hope to construct filter banks {p, q (1) , q (2) } given by the product of appropriate block matrices. If we can write
a symmetric FIR filter bank [p(ω), q (1) (ω), q (2) (ω)] T as a product B(M T ω)[p s (ω), q s (1) (ω), q s
(2) (ω)] T , where M
is M 1 defined in (2), B(ω) is a 3 × 3 matrix whose entries are trigonometric polynomials, and {p s , q s (1) , q s
(2) } is
another FIR filter bank with 2-fold rotational symmetry, then (12) implies that B(ω) satisfies
where M 0 is the matrix defined by (13).
Denote
B(−M T ω) = M 0 B(M T ω)M −1
0 , (14)
I 0 (ω) = [1, e −iω1 , e iω1 ] T . (15)
Clearly, I 0 (ω) satisfies (12). Therefore, 1-tap filter bank {1, e −iω 1 , eiω 1
} has 2-fold rotational symmetry, and it
could be used as the initial symmetric filter bank.
Denote
D 1 (ω) = diag(1, e −iω2 , e iω2 ),
D 1 (ω) = diag(1, e −i(ω 1+ω ) 2
, e i(ω 1+ω ) 2
),
(16)
D 3 (ω) = diag(1, e −iω 1 , eiω 1 ).
Then one can easily verify that D 1 (ω), D 2 (ω), D 3 (ω), D 1 (−ω), D 2 (−ω), D 3 (−ω) satisfy (14), and thus they
could be used to build the block matrices. Next we use B(ω) = BD(ω) as the block matrix, where B is a 3 × 3
(real) constant matrix, and D(ω) is D j (ω) or D j (−ω) for some j, 1 ≤ j ≤ 3. Based on the above discussion, we
know that B(ω) = BD(ω) satisfies (14) if and only if B satisfies M 0 BM0 −1 = B, which is equivalent to that B
has the form:
⎡
⎤
b 11 b 12 b 12
B = ⎣ b 21 b 22 b 23
⎦ . (17)
b 21 b 23 b 22
Thus we conclude that if {p, q (1) , q (2) } is given by
[p(ω), q (1) (ω), q (2) (ω)] T = 1 √
3
B n D(M T ω)B n−1 D(M T ω) · · · B 1 D(M T ω)B 0 I 0 (ω) (18)
for some n ∈ Z + , where I 0 (ω) is defined by (15), B k , 0 ≤ k ≤ n are constant matrices of the form (17), and each
D(ω) is D j (ω) or D j (−ω) for some j, 1 ≤ j ≤ 3, then {p, q (1) , q (2) } is an FIR filter bank with 2-fold rotational
symmetry.
Next, we show that the block structure in (18) yields 2-fold symmetric orthogonal FIR filter banks.
For an FIR filter bank {p, q (1) , q (2) }, denote q (0) (ω) = p(ω). Let U(ω) be a 3 × 3 matrix defined by U(ω) =
[
q (l) (ω + η j ) ] 0≤l,j≤2 ,where η 0, η 1 , η 2 are given in (8). Then {p, q (1) , q (2) } is orthogonal if U(ω) is unitary for
all ω ∈ IR 2 , that is it satisfies
U(ω)U(ω) ∗ = I 3 , ω ∈ IR 2 . (19)
Next, we write q (l) (ω), 0 ≤ l ≤ 2 as
q (l) (ω) = √ 1 (
)
q (l)
0 (M T ω) + q (l)
1 (M T ω)e −iω 1
+ q (l)
2 (M T ω)e iω 1
,
3
where q (l)
k
(ω) are trigonometric polynomials. Let V (ω) denote the polyphase matrix (with dilation matrix M) of
{p(ω), q (1) (ω), q (2) (ω)}:
⎡
p 0 (ω) p 1 (ω) p 2 (ω)
⎢
V (ω) = ⎣ q (1)
0 (ω) q(1) 1 (ω) q(1) 2 (ω) ⎥
⎦ . (20)
From the facts that
q (2)
0
(ω) q(2) 1
(ω) q(2)
2 (ω) ⎤
[p(ω), q (1) (ω), q (2) (ω)] T =
1 √
3
V (M T ω)I 0 (ω),