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Some properties and construction of multiwavelets related to different symmetric centers
Mathematics and Computers in Simulation 70 (2005) 69–89
Some properties and construction of multiwavelets related to different symmetric centers Li-Hong Cui∗,1 School of Science, Beijing University of Chemical Technology, Beijing 100029, PR China Received 31 March 2005; accepted 13 April 2005 Available online 31 May 2005
Abstract In this paper we are interested in discuss the symmetry property and construction of an m-band compactly supported orthonormal multiwavelets related to the filters with different symmetric centers. With the development of the several equivalent conditions on this type of symmetry in terms of filter sequences and polyphase matrices, we derive several necessary constraints on the number of symmetric filters of the system, which is crucial for the construction of multiwavelets associated with given multiscaling functions with different symmetry centers. Then, we show how to construct multiwavelets with desired symmetric property by matrix extensions. Finally, to illustrate our proposed general scheme, we give two examples in this paper. © 2005 IMACS. Published by Elsevier B.V. All rights reserved. MSC: 42C40; 65T60 Keywords: Different symmetric centers; Orthogonality; Multiscaling function; Matrix extension; Multiwavelet
1. Introduction In the theory and applications of wavelets, certain properties are always desirable. Symmetry, for example, is a crucial property in image processing. For symmetric filter, symmetric extensions transform of the finite length signals can be carried out, which improve the rate-distortion performance in image ∗ 1
Corresponding author. Tel.: +86 1064430220; fax: +86 1064430220. Engages in the research on wavelet analysis and signal processing. E-mail address: [email protected] (L.-H. Cui).
0378-4754/$30.00 © 2005 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.matcom.2005.04.001
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compression [1,2]. Recently, multiwavelets and their application have been much attention [3–13] due to the fact that it can simultaneously possess such as compactly supported, orthogonality and symmetry property, which is not possible for any real-valued 2-band scalar wavelets [14]. The GHM multiwavelet which was constructed by Geronimo et al. [5] using fractal interpolation is the first one to combine symmetry, orthogonality and short support, at the same time. Since then, many have worked on the design of symmetric/antisymmetric orthonormal multiwavelets [8,11,12,15,16]. Most deals with 2-band multiwavelet systems, a few authors have studied m-band multiwavelets, for example, the parameterization of m-channel orthogonal multifilter banks was developed in [17]. However, we note that these research mainly concerned multiscaling functions with all components having the same symmetric center. On the other hand, for a multifilter bank, since the input are vector signals, it is also required that the corresponding multiwavelets be balanced [18,19]. It was shown in [20] that the components of multiscaling functions must have the different symmetric center in order to construct symmetric and balanced multiwavelets. The purpose of this paper is investigate m-band orthonormal multiwavelet systems associated with such type symmetric property, where m > 1 is an arbitrary integer. To this end, we are needed to analysis symmetric property of orthogonal filters for such types of orthonormal multiscaling functions and multiwavelets. We called the corresponding filter to be a filter with different symmetric centers. Matrix extension is one of the important approach for the constructing multiwavelets associated with a given multiscaling functions. The paraunitary matrix extension problems corresponding to the construction of orthonormal multiwavelets have been well studied [10]. Recently, Jiang studied symmetric paraunitary matrix extensions associated with 2-band compact supported orthonormal multiwavelets and shown that such matrix extension is always solvable [16]. In our preceding paper [23], we dealt mainly with the construction of m-band orthonormal multiwavelets corresponding to multiscaling functions that all the components having the same symmetric center. In this paper, we will concerned with orthonormal multiwavelets related to different symmetric center filter. This work is very interesting and more complicated. Several important results are presented but having some new feature. We systematically investigate the some properties of m-band multiwavelets system related to different symmetric center filter. In fact, we present the characterization of multiscaling functions and multiwavelets with the type symmetry in terms of matrix filter and polyphase matrix. Exploiting this property, we derived several necessary constraints on the number of symmetric/antisymmetric filters, which help to design multiwavelets with the desired symmetric property. It follows that we discuss the problem of matrix extension associated with multiscaling functions with different symmetric center for even m and some odd m. This lead to an practical algorithm for the construction multiwavelets with some symmetric property by matrix extension. This can be applied in the construction of multiwavelets if the multiscaling functions and low-pass filter are known. Finally, some examples are given by applying the proposed scheme. Throughout this paper: let A∗ , AT and Tr(A) denote conjugate transpose, transpose and the trace of A, respectively. In denotes the n × n identity matrix, 0s,j denote s × j zero matrix, for convenience, we omit the subscript s × j when it does not cause any confusion. Let O(n) denote the set of all n × n real orthogonal matrices. For two matrices B = (bij ) and C = (cij ), let B ⊗ C = (bij C) denote the Kronecker product of B, C. For n × n antidiagonal matrix Jn , we assumption: J0 ⊗ A = 1. δk is the Dirac sequence such that δ0 = 1 and k ∈ Z\{0} for all δk = 0. · is the floor function. In this paper, we consider causal matrix sequence with real finiteimpulse response (FIR), such sequences can be identified with Laurent polynomial defined by H(z) = Lk=0 h(k)z−k , where 0 and L are the smallest and largest indices that h(k) is nonzero, respectively.
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2. Preliminaries of orthonormal multiwavelets In this section, we introduce some notation and summarize needed results that we will be used later in this paper. An m-band orthonormal multiwavelet systems are composed of compactly support orthonormal multiscaling functions = (φ1 , . . . , φr )T and orthonormal multiwavelets = (ψ1 , . . . , ψr )T , = 1, . . . , m − 1 which satisfy the matrix scaling equations, respectively (x) = m
h(k)(mx − k),
(1)
k∈Z
(x) = m
g (k)(mx − k),
= 1, . . . , m − 1,
(2)
k∈Z
or equivalently
ω ˆ ω ˆ ω , = 1, . . . , m − 1, ˆ (ω) = G ω , (3) m m m m and H and G denote the corresponding where h(k), g (k) is r × r is real-value matrix sequence matrix frequency response which defined by H(ω) = k h(k) e−ikω and G (ω) = k g (k) e−ikω , = 1, . . . , m − 1, respectively. H and G are also called matrix low-pass filter and high-pass filter in the application of signal processing. The set {H, G }1≤
m−1
H
k=0 m−1
H
k=0 m−1 k=0
ω + 2kπ ω + 2kπ G∗ m m
G
ω + 2kπ ω + 2kπ H∗ m m
= Ir ,
(4)
= 0r ,
(5)
ω + 2kπ ω + 2kπ G∗n m m
= δ−n Ir ,
0 < , n < m.
(6)
The sequence H which satisfies (4) is called a conjugate quadrature filter (CQF). If the multifilter banks {H, G }1≤
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functions. Paraunitary matrix extension is one of the important approaches for constructing multiwavelets associated with a given multiscaling functions. Let H(z) = k h(k)z−k is a given low-pass filter which generated m-band compactly supported orthonormal multiscaling functions , then H satisfies (4). To obtain m-band compactly supported orthonormal multiwavelets corresponding to , the key problem is seek the high-pass G , = 1, . . . , m − 1 filter satisfied with (5) and (6), which can be reduced to the problem of paraunitary matrix extensions. The polyphase matrix E(z) of multifilter banks {H, G }, = 1, . . . , m − 1 is the block matrix E(z) = [E,n (z)]0≤,n≤m−1 ,
(7)
where the polyphase symbols E,n are defined by 1 E,n (z) = √ g (mk + n)z−k , m k
(8)
where h = g0 . It is not difficult to verify that (4)–(6) are equivalent to the following form: E(z)ET (z−1 ) = Irm .
(9)
From the above theory, we know that H satisfies (4) if and only if matrix [E0,0 (z), E0,1 (z), . . . , E0,m−1 (z)] = P(z) is paraunitary. An s × n(s ≤ n) matrix filter P(z) is called paraunitary if P(z)P T (z−1 ) = Is ,
z = 0,
that is P(eiω ) is a matrix of orthogonal rows for all ω ∈ R. The basic idea of orthogonal extension of paraunitary matrix is that: for a given r × mr matrix P(z), finding (m − 1)r × mr matrix such that [E0 (z), E1 (z), . . . , Em−1 (z)] = Q(z), where Ej (z), j = 0, 1, . . . , m − 1 is (m − 1)r × r Laurent polynomial matrix and such that matrix defined in (7), i.e.,
P(z) E(z) = Q(z)
is paraunitary. From the literature [10], we know that for a given paraunitary matrix P(z), one always can find its paraunitary extensions matrix Q(z) such that (9) holds.
3. Symmetry property of multiwavelets with different symmetric center In this section, we will analyse some symmetric property on multiscaling functions and multiwavelets with their components not having the same symmetric center. Suppose a scalar function f (x) is symmetric about point a, then f (x) = f (2a − x). Analogously, f (x) is antisymmetric about point a if and only if f (x) = −f (2a − x). For multiscaling functions (x) = (φ1 , . . . , φr )T and multiwavelets (x) = (ψ1 , . . . , ψr )T , we want all their components are symmetric or antisymmetric, i.e., φj (x) = sj φj (2aj − x),
ψj (x) = tj ψj (2bj − x),
(10)
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where sj , tj ∈ {−1, 1}, aj , bj ∈ R. That is sj , tj determine whether a function is symmetric or antisymmetric and aj , bj are the centers of symmetry. We say that will have the different symmetric center if there exist ai = aj . The following lemma present the characterization of multiscaling functions and multiwavelets with this symmetry in terms of matrix filter in the Fourier domain [17]. Lemma 1. Assume that H is an FIR matrix filter and = (φ1 , . . . , φr )T is an m-band compactly ˆ supported multiscaling functions with φ(0) = 0. If H satisfies SDa (mω)H(−ω)SDa (−ω) = H(ω)
(11)
for some a = (a1 , . . . , ar ) ∈ Rr , then φj is symmetric/antisymmetric about
φj
aj − x = sj φj (x), m−1
aj , 2(m−1)
i.e.,
1 ≤ j ≤ r,
(12)
where S is the diagonal matrix having coefficients sj on the diagonal, and Da (ω) = diag(e−ia1 ω/(m−1) , e−ia2 ω/(m−1) , . . . , e−iar ω/(m−1) )
(13)
or a vector a = (a1 , . . . , ar ) ∈ Rr . Conversely, if φ is L2 -stable and maj = ai mod(m − 1), 1 ≤ i, j ≤ r, then (12) implies (11). Lemma 2. Assumed H is an FIR matrix filter satisfying (11) for some a = (a1 , . . . , ar ) ∈ Rr , and ˆ is an m-band compactly supported multiscaling functions with φ(0) = 0. Let = (ψ1 , . . . , ψr )T be a vector-valued function defined by (x) = m
g(k)(mx − k)
k
for some FIR matrix filter G. If G satisfies TDb (mω)G(−ω)TDa (−ω) = G(ω)
(14)
for some b = (b1 , . . . , br ) ∈ Rr , then
ψj
bj − x = tj ψj (x), m−1
1 ≤ j ≤ r,
(15)
where T and Db (ω) are defined analogously with S and Da (ω). Conversely, if is L2 -stable and mbj = ai mod(m − 1), 1 ≤ i, j ≤ r, then (15) implies (14).
(α+1)m−1 (α+1)m−1 In this paper, we always assuming that H(ω) = k=0 h(k) e−ikω , G (ω) = k=0 g (k) e−ikω , 1 ≤ < m is an orthogonal causal FIR multifilter bank generating multiscaling functions and multiwavelets , , = 1, . . . , m − 1. Suppose h(k), g (k) = 0 if k ∈ / [0, (α + 1)m − 1] for some α ∈ α Z+ \{0}, then , , 1 ≤ < m, are supported in 0, 1 + α + m−1 [22]. In the following, we will discuss symmetric orthogonal multifilter banks with corresponding multiscaling functions and multiwavelets with their components not having the same symmetric center. To this end, we first need to find a characterization of this type symmetric in terms of filter sequences. By Lemmas 1 and 2, we may
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consider H(z) =
m(α+1)−1 k=0
h(k)z−k , G (z) =
m(α+1)−1 k=0
g (k)z−k , = 1, . . . , m − 1 satisfy (4)–(6) and
z−((α+1)m−1) diag(S0 zm , s0 )H(z−1 ) diag(S0 , s0 z) = H(z), z−((α+1)m−1) diag(S1 zm , S2 )G (z−1 ) diag(S0 , s0 z) = G (z),
1 ≤ < m,
(16)
where s0 = ±1, S0 , S1 , S2 , 1 ≤ < m are diagonal matrices with diagonal entries 1 or −1. Then we have the following theorem.
Theorem 1. Let {H, G }1≤
(17)
where
0 S0 0 Jm−1 ⊗ 1 0 . JS0 = S0 0 Proof. Let P(z) = [E0,0 (z), E0,1 (z), . . . , E0,m−1 (z)], Q (z) = [E,0 (z), E,1 (z), . . . , E,m−1 (z)],
= 1, . . . , m − 1.
To complete the proof, it suffices to prove that z−α diag(S0 z, s0 )P(z−1 ) diag(JS0 , s0 z) = P(z)
(18)
and z−α diag(S1 z, S2 )Q (z−1 ) diag(JS0 , s0 z) = Q (z),
= 1, . . . , m = 1.
(19)
Denote
Eij (z) =
Eij11 (z) Eij12 (z) Eij21 (z) Eij22 (z)
,
i, j = 0, 1, . . . , m − 1,
where Eij11 (z) and Eij12 (z) are the (r − 1) × (r − 1) and (r − 1) × 1 matrix, Eij21 (z) and Eij22 (z) are the 1 × (r − 1) and 1 × 1 matrix.
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We substituting the following
Ir −1 11 12 11 z Ir E00 E00 · · · E0(m−1) m H(z) = P(z ) .. = 21 22 21 . E00 E00 · · · E0(m−1) z−(m−1) Ir
Ir−1 0 −1 z Ir−1 12 E0(m−1) 0 22 . E0(m−1) . . z−(m−1) Ir−1 0
into (16) leads to
z−mα+1 S0
m−1
j 11 −m )S0 j=0 z E0j (z m−1 j 21 −m −mα−(m−1) z s0 j=0 z E0j (z )S0
=
m−1 −j 11 m z E0j (z ) j=0 m−1 −j 21 m j=0 z E0j (z )
z−mα+2 S0
m−1
j 12 −m )s0 j=0 z E0j (z m−1 j 22 −m −mα−(m−2) z s0 j=0 z E0j (z )s0
m−1
−j 12 m j=0 z E0j (z ) m−1 −j 22 m . j=0 z E0j (z )
Then we can obtain 11 −m 11 z−mα+j+1 S0 E0j (z )S0 = z−(m−j−1) E0(m−j−1) (zm ), j = 1, 2, . . . , m − 1; 12 −m 12 z−mα+j+2 S0 E0j (z )s0 = z−(m−j−2) E0(m−j−2) (zm ), j = 1, 2, . . . , m − 2, 12 12 z−mα+m+1 S0 E0(m−1) (z−m )s0 = z−(m−1) E0(m−1) (zm ); 21 −m 21 z−mα−(m−j−1) s0 E0j (z )S0 = z−(m−j−1) E0(m−j−1) (zm ), j = 1, 2, . . . , m − 1; 22 −m 22 (z )s0 = z−(m−j−2) E0(m−j−2) (zm ), j = 1, 2, . . . , m − 2, z−mα−(m−j−2) s0 E0j 22 22 (z−m )s0 = z−(m−1) E0(m−1) (zm ). z−mα+1 s0 E0(m−1)
That is 11 −m 11 z−mα+m S0 E0j (z )S0 = E0(m−j−1) (zm ), j = 1, 2, . . . , m − 1; 12 −m 12 (z )s0 = E0(m−j−2) (zm ), j = 1, 2, . . . , m − 2, z−mα+m S0 E0j 12 12 z−mα+2m S0 E0(m−1) (z−m )s0 = E0(m−1) (zm ); 21 −m 21 z−mα s0 E0j (z )S0 = E0(m−j−1) (zm ), j = 1, 2, . . . , m − 1; 22 −m 22 (z )s0 = E0(m−j−2) (zm ), j = 1, 2, . . . , m − 2, z−mα s0 E0j 22 22 (z−m )s0 = E0(m−1) (zm ). z−mα+m s0 E0(m−1)
0 1 0 z−1
0 z−(m−1)
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It follows that z−mα diag(S0 zm , s0 )P(z−m ) diag(JS0 , s0 zm ) = P(zm ), which implies (18) holds, where
0 S0 0 Jm−1 ⊗ 1 0 . JS0 = S0 0 By using the similar arguments, we can obtain (19). Combining (18) and (19) together, one can obtain (16). Denote J(z) = diag(JS0 , s0 z), then it is not difficult to obtain the following lemma. Lemma 3.
Tr J(z) =
1 + s0 z if m is even, Tr(S0 ) + s0 z if m is odd.
(20)
Theorem 2. Suppose multifilter bank {H, G }1≤
(21)
Taking the trace of both sides of (21), using the fact that Tr(AB) = Tr(BA) and Lemma 3, one can obtain Tr(S(1)) = Tr(E(1) diag(JS0 , s0 )E−1 (1)) = Tr(E−1 (1)E(1) diag(JS0 , s0 ))
= Tr(diag(JS0 , s0 )) = which implies (i), (ii) and (iii) hold.
1 + s0 if m is even, Tr(S0 ) + s0 if m is odd,
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Theorem 3. Suppose multifilter bank {H, G }1≤
Tr(s0 + S21 + · · · + S2m−1 ) = 1.
(22)
Tr(s0 + S21 + · · · + S2m−1 ) = s0 .
(23)
Tr(s0 + S21 + · · · + S2m−1 ) = Tr(S0 ).
(24)
Tr(s0 + S21 + · · · + S2m−1 ) = s0 .
(25)
(2) When m is even and α is odd, Tr(S0 + S11 + · · · + S1m−1 ) = 1, (3) When m is odd and α is even, Tr(S0 + S11 + · · · + S1m−1 ) = s0 , (4) When m is odd and α is odd, Tr(S0 + S11 + · · · + S1m−1 ) = Tr(S0 ),
Proof. From the proof of Theorem 3, we can obtain
Tr(S(1)) =
1 + s0 if m is even, Tr(S0 ) + s0 if m is odd,
Thus we have
Tr(S(−1)) = (−1)
α
1 − s0 if m is even, Tr(S0 ) − s0 if m is odd.
2 2s0
if m is even and α is even, if m is even and α is odd, Tr(S(1)) + Tr(S(−1)) = 2Tr(S0 ) if m is odd and α is even, 2s if m is odd and α is odd. 0 2s0 2
if m is even and α is even, if m is even and α is odd, Tr(S(1)) − Tr(S(−1)) = 2s0 if m is odd and α is even, 2Tr(S ) if m is odd and α is odd. 0 This lead to (22)–(25).
In the following, we assume that s0 = 1 and suppose S0 = diag(Is , −Ir−s−1 ),
(26)
for a nonnegative integer s ≤ r − 1. Then we have the following assumption because of the above discussion. (i) When m is even, we may suppose S1 = −Ia ,
S2 = diag(Ib , −Ib )
if 2s ≥ r,
S1 = I−a ,
S2 = diag(Ic , −Ic )
if 2s < r, (27)
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where 2s−r 2s−r + 1 if = 2k and r is odd, or is odd and r is even, 2s−r m−1 m−1 m−1 a = 2s−r 2s−r if 2s−r = −2k and r is even, or is odd and r is odd, m−1 m−1 m−1
b=
r−a 2
,c=
r+a 2
, = 1, . . . , m − 1, k is a nonnegative integer.
(ii) When m is odd, both α and r are even, we may suppose S1 = −Ia ,
S2 = diag(Ir/2 , −Ir/2−a ) if 2s ≥ r,
S1 = I−a ,
S2 = diag(Ir/2+a , −Ir/2 ) if 2s < r,
(28)
where
2s − r , = 1, . . . , m − 1. a= m−1 (ii) When both m and α are odd and r is even, we may suppose S1 = diag(Ir−s , −Ir−s ), S1 = diag(Is , −Is ),
S2 = diag(Is−r/2 , −Is−r/2 ) if 2s ≥ r,
S2 = diag(Ir/2−s , −Ir/2−s ) if 2s < r,
(29)
where s ≤ r, = 1, . . . , m − 1. Here we note that the above assumption is just only fashion. There are have many different form. The following lemma is crucial in the following sections. Lemma 4. Let s × (mk − 1)(s ≥ k − 1) real matrix A satisfies A diag(Ix , −Iy )AT = 0,
(30)
where x + y = mk − 1 and x − y = 1. Then there exists v ∈ O(x) such that
vT = 0. A (Iy , 0)
(31)
Proof. The proof of lemma can be carried out similarly as in [12]. Let n denote the rank of A, then n ≤ x. Let {β1 , β2 , . . . , βn } be an orthogonal basis for the columns of the matrix AT . Write
Y1 , [β1 , . . . , βn ] = Z1
Y1 , Z1 are x × n and y × n matrices.
Then we have Y1T Y1 + Z1T Z1 = In and Y1T Y1 = Z1T Z1 . Thus 1 Y1T Y1 = Z1T Z1 = In . 2
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Therefore
√ √ 2 2 Y , Z1 1 2 2
are x × n and y × n matrices of orthogonal columns, respectively. Let Y2 , Z2 be
the x × (x − n) and y × (y − n) matrices such that one has
79
√ 2 [Y1 , Y2 ] 2
∈ O(x),
√ 2 [Z1 , Z2 ] 2
∈ O(x) ∈ O(y). Then
Y1T −Z1T β = 0, Y2T −[Z2 , 0]T j
1 ≤ j ≤ n.
Thus
Y1T −Z1T AT = 0. Y2T −[Z2 , 0]T
Thus (31) holds true with v = −2 diag([Z1 , Z2 ], I)[Y1 , Y2 ]T ∈ O(x).
From the proof of Lemma 4, we know that orthogonal matrices v are constructed by the Gram–Schmidt process of the rows of A.
4. Symmetric paraunitary matrix extension related to different centers In this section, we will show how to construct the corresponding multiwavelets with some desired symmetry for a given compactly supported orthonormal multiscaling functions with different symmetric m(α+1)−1 center. In concrete, given H(z) = k=0 h(k)z−k satisfying z−((α+1)m−1) diag(S0 zm , 1)H(z−1 ) diag(S0 , z) = H(z),
S0 = diag(Is , −Ir−s−1 )
(32)
for a nonnegative integer s ≤ r − 1. In this case, by the theory in Section 3, we know that if H generates mα−1 an orthonormal multiscaling functions = (φ1 , . . . , φr )T , then φ1 , . . . , φs are symmetric about 2(m−1) mα−1 while φs+1 , . . . , φr−s−1 are antisymmetric about 2(m−1) , and φr is symmetric about m(α+1)−2 . 2(m−1) We want construct G (z) =
m(α+1)−1 k=0
gk z−k , = 1, . . . , m − 1 satisfy
z−((α+1)m−1) diag(S1 zm , S2 )G (z−1 ) diag(JS0 , z) = G (z),
0 < < m,
or equivalently z−((α+1)m−1) diag(S11 zm , S 1 , . . . , S1m−1 zm , S2m−1 )G(z−1 ) diag(JS0 , z) = G(z), j
(33)
where S , = 1, . . . , m − 1, j = 1, 2 defined by one of the equations (27)–(29). For example, in the case of (27), the corresponding multiwavelets have the following symmetric properties: (1) if 2s ≥ r, then mα−1 ψ1 , . . . , ψa are antisymmetric about 2(m−1) , ψa+1 , . . . , ψ(r+a)/2 and ψ(r+a)/2+1 , . . . , ψr are symmetric and mα−1 antisymmetric about m(α+1)−2 , respectively; (2) if 2s < r, then ψ1 , . . . , ψ−a are symmetric about 2(m−1) , 2(m−1) m(α+1)−2 ψ1−a , . . . , ψr−a/2 and ψ(r−a)/2+1 , . . . , φr are symmetric and antisymmetric about 2(m−1) , respectively. Here, all a and b are consistent with those in (27).
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From the above, we know that, the key problem to construct a corresponding m-band multiwavelet which have the symmetric property can be reduced to the symmetric paraunitary extension of a given (α+1)m−1 matrix. That is: suppose H(z) = k=0 h(k)z−k is a matrix CQF which satisfies (4) and symmetric conditions (16), then its polyphase matrix P(z) = [E0,0 (z), E0,1 (z), . . . , E0,m−1 (z)] satisfies z−α diag(S0 z, 1)P(z−1 ) diag(JS0 , z) = P(z),
(34)
where
0 S0 0 Jm−1 ⊗ 1 0 . JS0 = S0 0 We want to construct Q(z) = [E0 (z), E1 (z), . . . , Em−1 (z)] such that Q(z) is the paraunitary extension matrix of P(z) and satisfies z−α diag(S11 z, S21 , . . . , S1m−1 z, S2m−1 )Q(z−1 ) diag(JS0 , z) = Q(z),
(35)
where Sj , = 1, . . . , m − 1, j = 1, 2 are defined by one of the equations (27)–(29). Since
Tr JS0 =
1 if m is even, Tr(S0 ) = 2s − r + 1 if m is odd.
Therefore, there exist R ∈ O(mr − 1) defined by A 0 −A √ 0 2 0 if m is even, √ 2 A 0 A R= B 0 −B 2 √ 0 2Ir−1 0 if m is odd, B 0 B
(36)
where A= Jm 2
−1
⊗
1 S0
S0
,
B = J m−1 ⊗ 2
1
S0
such that RJS0 RT = diag(−A, 1, A) holds, where mr
x=
2 (m−1)r 2
if m is even, + s if m is odd,
mr
y=
−1
2 (m−1)r 2
if m is even,
+ r − 1 − s if m is odd.
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From Theorems 2 and 3, we know that multiwavelet systems have two more symmetric filters than antisymmetric filters for all even m and these odd m only when 2s = r. In the following, we consider the symmetric extension for all even m and these odd m only when 2s = r. In this case, mr
x=
2 mr 2
mr
if m is even, if m is odd,
y=
2 mr 2
− 1 if m is even, − 1 if m is odd.
Here, we note that x − y = 1. Let M0 be such a 2r × 2r permutation matrix that M0T diag(−A, 1, A, z)M0 = diag(z, D),
(37)
where D = diag(Ix , −Iy ) ∈ O(2r − 1), x − y = 1. Constructing B(z) = P(z) diag(RT , 1)M0 , then B(z) possessing the following features: (i) B(z) is casual and paraunitary. (ii) P(z) satisfies (18) if and only if B(z) satisfies z−α diag(S0 z, 1)B(z−1 ) diag(z, D) = B(z) holds, where D = diag(Ix , −Iy ). (iii) For α ≥ 1, B(z) has the form of
(38)
0 a0 0 0 0 0 −1 B(z) = 0 0 b0 + 0 0 0 z , c0 d1 d2 0 d1 −d2
(39)
where a0 , b0 are s × x and (r − s − 1) × y matrices, c0 ∈ R and d1 , d2 are 1 × x and 1 × y row vectors. (iv) For α ≥ 2, B(z) has the form of
a0 b0 a1 b1 −1 S0 a0 S0 b1 D −(α−2) B(z) = + z + ··· + z c1 d2 D c0 d0 c1 d1
0 S0 b0 D −(α−1) 00 + z + z−α , c0 d1 D 0 d0 D
(40)
where cj ∈ R, aj and dj are the (r − 1) × 1 and 1 × (mr − 1) vectors, bj is the (r − 1) × (mr − 1) matrix. Now we are prepared to discuss the symmetric extension of B(z). We want to construct a causal filter C(z) is a paraunitary extension of C(z) and satisfies z−α diag(S11 z, S21 , . . . , S1m−1 z, S2m−1 )C(z−1 ) diag(z, D) = C(z). First let us consider the case α = 1. From (39) and the paraunitaryness of B, we have a0 a0T = Is ,
b0 b0T = Ir−s−1 ,
a0 d1T = 0,
b0 d2T = 0,
(41)
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82
d1 d1T = d2 d2T ,
c02 + 4d1 d2T = 1.
Thus we know a0 , b0 are s × x and (r − s − 1) × y matrices of orthonormal rows. Let θ be such a real number that cos θ = c0 ,
sin θ =
1 − c02 .
Then d1 , d2 can be written as d1 =
1 sin θu0 , 2
d2 =
1 sin θv0 , 2
(42)
where u0 and v0 are 1 × x and 1 × y row vectors such that
a0 u0
and
b0 v0
are (s + 1) × x and (r − s) × y matrices of orthonormal to rows. If sin θ = 0, then d1 = 0, d2 = 0 and 2d1 2d2 unit vectors u0 , v0 orthonormal to rows of a0 , b0 , respectively, will do. If sin θ = 0, u0 = sin , v0 = sin . θ θ Consider the case 2s ≥ r. Choose (x − s − 1) × x, (2s − r) × y and (x − s − 1) × y matrices u˜ , v˜ 1 , v˜ such that [a0T , uT0 , u˜ T ] ∈ O(x),
[b0T , vT0 , v˜ T , v˜ T1 ] ∈ O(y),
where u0 , v0 are the vectors satisfying (42). Then C(z) defined by
0 0 2˜v1 0 0 0 −2 sin θ cos θu0 cos θv0 0 cos θu0 − cos θv0 1 1 0 u˜ v˜ u˜ −˜v C(z) = + 0 z−1 2 2 0 −u0 0 u0 v0 v0 0 u˜ v˜ 0 −˜u v˜ is a symmetric paraunitary extension of B(z) with C(z) satisfying (41) for α = 1. For the case 2s < r, choose (r − 2s) × x, (y − r + s) × x and (y − r + s) × y matrices u˜ 1 , u˜ and v˜ such that [a0T , uT0 , u˜ T , u˜ T1 ] ∈ O(x),
[b0T , vT0 , v˜ T ] ∈ O(y),
L.-H. Cui / Mathematics and Computers in Simulation 70 (2005) 69–89
83
where u0 , v0 are the vectors satisfying (42). Then C(z) defined by
0 2u˜ 1 0 0 0 0 −2 sin θ cos θu0 cos θv0 0 cos θu0 − cos θv0 1 1 0 u˜ v˜ u˜ −˜v C(z) = + 0 z−1 2 2 0 −u0 0 u v v 0 0 0 0 u˜ v˜ 0 −˜u v˜ is a symmetric paraunitary extension of B(z) with C(z) satisfying (41) for α = 1. Theorem 4. Suppose B(z) is a casual paraunitary matrix satisfies (32) for α = 1. Then C(z) constructed above is a symmetric paraunitary extension of B(z) satisfying (41) α = 1. In the following, we discuss the case α ≥ 2. We first define
0 0 1 2 0 W(z) = 2 0 w1 0 −w1
0 0 2w2 0 1 0 0 + −Iy 2 0 w1 0 w1 Iy
0 0 −1 z Iy Iy
(43)
for
w1 ∈ O(x), w= w2 where w1 , w2 are y × x and 1 × x matrices, respectively. Then by a direct calculation, one has the following lemma. Lemma 5. Let W(z) be the matrix defined by (43) with some w ∈ O(x). Then (i) W(z)W(z−1 )T = Imr , (ii) z−1 diag(z−1 , D)W(z−1 ) diag(z, D) = W(z). From (40) and the paraunitaryness of B(z), we have
a0 b0 c0 d0
00 0 d0 D
T
= 0,
which leads to
b0 D[b0T , d0T ] = 0. d0
a0 b0 c0 d0
0 S0 b0 D c0 d1 D
T
a1 b1 + c1 d1
00 0 d0 D
T
= 0,
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84
By Lemma 3, we can construct w ∈ O(x) satisfying
b0 d0
wT = 0. (Iy , 0)
(44)
Denote
w1 w= , w2 where w1 , w2 are y × x and 1 × x matrices, respectively. From (44), we have
wT2 wT1 d0 D 0 −Iy
= 0.
(45)
˜ by Let Wα (z) be the matrix defined by (43) with w = wα . Defined B ˜ B(z) = B(z)Wα (z−1 ). ˜ Then (44) and (45) imply that P˜ is casual. Since Wα (z) is paraunitary and satisfies (ii) of Lemma 5, B(z) is also paraunitary and satisfies (38) with α = 1. In this way, we construct wα−1 , . . . , w2 ∈ O(r) similarly such that B(z) can be written as ˜ α (z) = · · · = B1 (z)W2 (z) · · · Wα (z), B(z) = BW where Wj is defined by (43) with w = wj and B1 (z) is an r × mr matrix satisfying (38) with α = 1. By Theorem 4, we can construct a casual filter C1 (z) such that C1 (z) is a symmetric paraunitary extension of B1 (z). Let C(z) = C1 (z)W2 (z) · · · Wα (z). Then C(z) is a symmetric extension of B(z) satisfying (41). Define Q(z) = C(z)M0T diag(R, 1). Then Q(z) is a symmetric paraunitary extension of P(z) with Q(z) satisfying (35). Theorem 5. Suppose P(z) = [E0,0 (z), E0,1 (z), . . . , E0,m−1 (z)] is an r × mr paraunitary matrix satisfying (18). Then [E0 (z), E1 (z), . . . , Em−1 (z)] = Q(z)
(46)
obtained by the above algorithm is a symmetric paraunitary extension of P(z) satisfying (35). Furthermore the filter length of Q(z) is not greater than α.
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Let G(z) be the matrix defined by (8). Then G(z) is casual, satisfies (33) and len(G) ≤ m(α + 1) − 1. Corollary 1. Assumed that the casual FIR H generated an orthonormal multiscaling = (φ1 , . . . , φr )T mα−1 with φ1 , . . . , φs and φs+1 , . . . , φr−s−1 symmetric and antisymmetric about 2(m−1) and φr is symmetric about m(α+1)−2 . Let G be the casual matrix filter obtained by the above algorithm. Then = (ψ1 , . . . , ψr )T 2(m−1) defined by (3) possess the following symmetric properties: mα−1 (i) For even m, ψ1 , . . . , ψa are antisymmetric about 2(m−1) , ψa+1 , . . . , ψ(r+a)/2 and ψ(r+a)/2+1 , . . . , ψr are symmetric and antisymmetric about m(α+1)−2 , respectively, for the case 2s ≥ r; and ψ1 , . . . , ψ−a 2(m−1) mα−1 are symmetric about 2(m−1) , ψ1−a , . . . , ψ(r−a)/2 and ψ(r−a)/2+1 , . . . , ψr are symmetric and antisymmetric about m(α+1)−2 , respectively, for the case 2s < r. 2(m−1) (ii) For odd m and r = 2s, ψ1 , . . . , ψr/2 and ψr/2+1 , . . . , ψr are symmetric and antisymmetric about m(α+1)−2 , respectively. 2(m−1)
5. Examples In this section, we will provide some examples by applying the above theory. Example 1. The following H(z) = √ 4 1 1 3 √ 3 2 h(0) = , 2 13 − 42 − 61
1 0 h(2) = 1 2 3+
√ 2 4
0 − 61
,
3 k=0
h(k)z−k is a matrix CQF, where
1
1 3 h(1) = 1 2 3+
1 0 h(3) = 1 2 3−
√ 2 4
√ 2 4
0 0
0 − 16
,
.
We will construct the corresponding symmetric multiwavelets with high-pass filter g(k), k = 0, 1, 2, 3 by the above algorithm. It is not hard to verify that H(z) =
3 k=0
h(k)z−k satisfying symmetric condition provided by (32), that
is z−3 diag(z2 , 1)H(z−1 ) diag(1, z) = H(z), in this case S0 = (1), and
1
1 JS0 = . 1
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Thus, we have
1 0 −1 √ √ 2 0 2 0 , R= 2 10 1
00 0I M0 = 2 10
1 0 . 0
Let E0,0 , E0,1 be the filter defined by (8). Then E0,0
√ = 2(h0 + h2 z−1 ) =
E0,1
√ 2 1 − 2 3
√ = 2(h1 + h3 z−1 ) =
√ 2 4
√ 2 1 + 2 3
+ √ 2 4
1 3 1 3
√ 2 z−1 4
+
1 3 1 3
√
−
2 4
4 3
− 16 (1 + z−1 ) 0
z−1 1
,
.
Thus √ 4 2 0 3 B(z) = [E0,0 , E0,1 ] diag(RT , 1)M0 = 2 1 − 16
√
2 √3 2 3
√ 0 0 0 0 2 √ + z−1 . 2 0 − 16 32 21 − 21
0
Furthermore, by the scheme provided in Theorem 4, we obtain symmetric extension C(z) of B(z):
C(z) =
√ √ − 2 − 62 (1 + z−1 )
1 2 0
− 13 (1 − z−1 )
2 (1 + z−1 ) 3 √ 2 2 (1 − z−1 ) 3
−
√
2 (1 2
− z−1 )
−(1 + z−1 )
.
Again by Q(z) = [E10 (z), E11 (z)] = C(z)M0T diag(R, 1), we obtain
√
√
√
√
√ − 2 + 23 + 22 + 23 z−1 − 13 (1 + z−1 ) 22 + 23 + − 22 + 23 z−1 −2 2 2 √ √ √ √ √ Q(z) = . 4 −1 + 2 3 2 + −1 − 2 3 2 z−1 − 32 (1 − z−1 ) 1 + 2 3 2 + 1 − 2 3 2 z−1 0 So we have √ 3 I2 2 2 2 = g(k)z−k , [E10 (z ), E11 (z )] −1 G(z) = z I2 2 k=0
where
g(0) =
√ 4−3 2 24 √ −3+2 2 24
1 − 12 √
− 122
,
g(1) =
√ 4+3 2 24√ 3+2 2 24
− 21 0
,
L.-H. Cui / Mathematics and Computers in Simulation 70 (2005) 69–89
√ 4+3 2 −1 24 √ √ 12 2 −3−2 2 24 12
g(2) =
g(3) =
,
√ 4−3 2 24√ 3−2 2 24
0 0
87
.
The first and the second components of the corresponding = (ψ1 , ψ2 )T is symmetric and antisymmetric about 1, respectively. Example 2. An important multiwavelet system was constructed by J. Geronimo, D. Hardin and P. Massopust (GHM). The GHM multiscaling function (x) = (φ1 (x), φ2 (x))T satisfies the following equation [9]: (x) = 2
3
h(k)(2x − k),
k=0
where
√ 1 12 16 2 √ , h(0) = 40 − 2 −6
1 0√ 0 h(2) = , 40 9 2 −6
1 12√ 0 h(1) = , 40 9 2 20
1 0√ 0 h(3) = . 40 − 2 0
We will construct the corresponding symmetric multiwavelets with high-pass filter g(k), k = 0, 1, 2, 3 by the above algorithm. It is not hard to verify that H(z) = 3k=0 h(k)z−k satisfying symmetric condition provided by (32), that is z−3 diag(z2 , 1)H(z−1 ) diag(1, z) = H(z), in this case S0 = (1), and
1
1 JS0 = . 1 Thus, we have
1 0 −1 √ √ 2 0 2 0 , R= 2 10 1
00 M0 = 0 I2 10
1 0 . 0
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Let E0,0 , E0,1 be the filter defined by (8). Then
E0,0
√ = 2(h0 + h2 z−1 ) =
E0,1
√ = 2(h0 + h2 z−1 ) =
√ 3 2 10
√
4 5
1 − 20 (1 − 9z−1 ) − 3202 (1 + z−1 )
√ 3 2 10 1 (9 20
0
− z−1 )
√ 2 2
,
.
Thus
B(z) = [E0,0 , E0,1 ] diag(RT , 1)M0 =
4 3 5√ √5 √ 2 − 3202 52 2
0
0 −
√ 2 4
+
0
0
√
0 − 3202
0 0
√ √ 2 2 5 4
z−1 .
Furthermore, by the scheme provided in Theorem 4, we obtain symmetric extension C(z) of B(z):
C(z) =
√ √ − 2 − 3102
1 2 0
− 35
√ √ 2 2 2 5 2 4 1 5
+
√
0 − 3102
1 2 0
3 5
√ 2 2 5 − 45
−
√ 2 2
1
z−1 .
Again by Q(z) = [E10 (z), E11 (z)] = C(z)M0T diag(R, 1), we obtain
Q(z) =
1 (9 − z−1 ) √20 2 (1 + 9z−1 ) 20
√
− 3202 (1 + z−1 ) 3 − 10 (1 − z−1 )
1 (1 + 9z−1 ) 20 √ 2 (9 + z−1 ) 20
−
√ 2 2
0
.
So we have √ 3 I2 2 2 2 G(z) = [E10 (z ), E11 (z )] −1 = g(k)z−k , z I2 2 k=0
where √
9 2 −6
1 √ , g(0) = 40 18 −6 2 √ 1 − 2 −6 √ , g(2) = 6 2 40 2
√ 1 − 2 −20 g(1) = , 40 −2 0 √ 1 9 20 g(3) = . 40 −18 0
The first and the second components of the corresponding = (ψ1 , ψ2 )T is symmetric and antisymmetric about 1, respectively.
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Acknowledgement The work is supported by Study Foundation for Young Teacher of Beijing University of Chemical Technology QN.0414.
References [1] G. Strang, T.Q. Nguyen, Wavelets and Filter Banks, Wellesley-Cambridge Press, 1996. [2] T. Xia, Q.T. Jiang, Optimal multifilter banks: design, related symmetric extension transform and application to image compression, IEEE Trans. Signal Process. 47 (1999) 1878–1889. [3] T.N.T. Goodman, S.L. Lee, W.S. Tang, Wavelets in wandering subspaces, Trans. Am. Math. Soc. 338 (1993) 639–654. [4] T.N.T. Goodman, S.L. Lee, Wavelets of multiplicity r, Trans. Am. Math. Soc. 342 (1994) 307–324. [5] J. Geronimo, D.P. Hardin, P. Massopust, Fractal functions and wavelet expansions based on several scaling functions, J. Approx. Theory 78 (1994) 373–401. [6] G. Strang, V. Strela, Short wavelets and matrix dilation equations, IEEE Trans. Signal Process. 43 (1995) 108–115. [7] G. Donovan, J. Geronimo, D.P. Hardin, Intertwining multiresolution analysis and the construction of piecewise polynomial wavelets, SIAM J. Math. Anal. 27 (1996) 1791–1815. [8] C.K. Chui, J. Lian, A study on orthonormal multiwavelets, Appl. Numer. Math. 20 (1996) 273–298. [9] G.C. Donovan, J. Geronimo, D.P. Hardin, Construction of orthogonal wavelets using fractal interpolation functions, SIAM J. Math. Anal. 27 (1996) 1158–1192. [10] W. Lawton, S.L. Lee, Z.W. Shen, An algorithm for matrix extension and wavelet construction, Math. Comput. 65 (1996) 723–737. [11] P. Rieder, Parameterization of symmetric multiwavelets, in: Proceedings of the ICASSP, Munich, Germany, 1997, pp. 21–24. [12] Q.T. Jiang, On the design of multifilter banks and orthogonal multiwavelet bases, IEEE Trans. Signal Process. 46 (1998) 3292–3303. [13] B. Han, Approximation properties and construction of Hermite interpolants and biorthogonal multiwavelets, J. Approx. Theory 110 (2001) 18–53. [14] I. Daubechies, Ten Lecture on Wavelet, SIAM, Philadelphia, PA, 1992. [15] L. Shen, H.H. Tan, J.Y. Tham, Symmetric–antisymmetric orthonormal multiwavelets and related scalar wavelets, Appl. Comput. Harm. Anal. 8 (2000) 258–279. [16] Q.T. Jiang, Symmetric paraunitary matrix extension and parameterization of symmetric orthogonal multifilter banks, SIAM J. Matrix Anal. Appl. 22 (2001) 138–166. [17] Q.T. Jiang, Parameterization of M-channel orthogonal multifilter banks, Adv. Comput. Math. 12 (2000) 189–211. [18] J. Leburn, M. Vetterli, Balanced multiwavelets theory and design, IEEE Trans. Signal Process. 46 (1998) 1119–1124. [19] I.W. Selesnick, Balanced multiwavelet bases based on symmetric FIR filters, IEEE Trans. Signal Process. 48 (2000) 184– 191. [20] Q.T. Jiang, Parameterization of symmetric orthogonal multifilter banks with different filter lengths, Linear Algebra Appl. 311 (2000) 79–96. [21] Z.W. Shen, Refinable function vectors, SIAM J. Math. Anal. 29 (1998) 235–250. [22] Q.T. Jiang, Multivariate matrix refinable functions with arbitrary matrix dilation, Trans. Am. Math. Soc. 351 (1999) 2407– 2438. [23] L. Cui, Z. Cheng, An algorithm for constructing symmetric orthogonal multiwavelets by matrix symmetric extension, Appl. Math. Comput. 149 (2004) 227–243.
Some properties and construction of multiwavelets related to different symmetric centers Li-Hong Cui∗,1 School of Science, Beijing University of Chemical Technology, Beijing 100029, PR China Received 31 March 2005; accepted 13 April 2005 Available online 31 May 2005
Abstract In this paper we are interested in discuss the symmetry property and construction of an m-band compactly supported orthonormal multiwavelets related to the filters with different symmetric centers. With the development of the several equivalent conditions on this type of symmetry in terms of filter sequences and polyphase matrices, we derive several necessary constraints on the number of symmetric filters of the system, which is crucial for the construction of multiwavelets associated with given multiscaling functions with different symmetry centers. Then, we show how to construct multiwavelets with desired symmetric property by matrix extensions. Finally, to illustrate our proposed general scheme, we give two examples in this paper. © 2005 IMACS. Published by Elsevier B.V. All rights reserved. MSC: 42C40; 65T60 Keywords: Different symmetric centers; Orthogonality; Multiscaling function; Matrix extension; Multiwavelet
1. Introduction In the theory and applications of wavelets, certain properties are always desirable. Symmetry, for example, is a crucial property in image processing. For symmetric filter, symmetric extensions transform of the finite length signals can be carried out, which improve the rate-distortion performance in image ∗ 1
Corresponding author. Tel.: +86 1064430220; fax: +86 1064430220. Engages in the research on wavelet analysis and signal processing. E-mail address: [email protected] (L.-H. Cui).
0378-4754/$30.00 © 2005 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.matcom.2005.04.001
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compression [1,2]. Recently, multiwavelets and their application have been much attention [3–13] due to the fact that it can simultaneously possess such as compactly supported, orthogonality and symmetry property, which is not possible for any real-valued 2-band scalar wavelets [14]. The GHM multiwavelet which was constructed by Geronimo et al. [5] using fractal interpolation is the first one to combine symmetry, orthogonality and short support, at the same time. Since then, many have worked on the design of symmetric/antisymmetric orthonormal multiwavelets [8,11,12,15,16]. Most deals with 2-band multiwavelet systems, a few authors have studied m-band multiwavelets, for example, the parameterization of m-channel orthogonal multifilter banks was developed in [17]. However, we note that these research mainly concerned multiscaling functions with all components having the same symmetric center. On the other hand, for a multifilter bank, since the input are vector signals, it is also required that the corresponding multiwavelets be balanced [18,19]. It was shown in [20] that the components of multiscaling functions must have the different symmetric center in order to construct symmetric and balanced multiwavelets. The purpose of this paper is investigate m-band orthonormal multiwavelet systems associated with such type symmetric property, where m > 1 is an arbitrary integer. To this end, we are needed to analysis symmetric property of orthogonal filters for such types of orthonormal multiscaling functions and multiwavelets. We called the corresponding filter to be a filter with different symmetric centers. Matrix extension is one of the important approach for the constructing multiwavelets associated with a given multiscaling functions. The paraunitary matrix extension problems corresponding to the construction of orthonormal multiwavelets have been well studied [10]. Recently, Jiang studied symmetric paraunitary matrix extensions associated with 2-band compact supported orthonormal multiwavelets and shown that such matrix extension is always solvable [16]. In our preceding paper [23], we dealt mainly with the construction of m-band orthonormal multiwavelets corresponding to multiscaling functions that all the components having the same symmetric center. In this paper, we will concerned with orthonormal multiwavelets related to different symmetric center filter. This work is very interesting and more complicated. Several important results are presented but having some new feature. We systematically investigate the some properties of m-band multiwavelets system related to different symmetric center filter. In fact, we present the characterization of multiscaling functions and multiwavelets with the type symmetry in terms of matrix filter and polyphase matrix. Exploiting this property, we derived several necessary constraints on the number of symmetric/antisymmetric filters, which help to design multiwavelets with the desired symmetric property. It follows that we discuss the problem of matrix extension associated with multiscaling functions with different symmetric center for even m and some odd m. This lead to an practical algorithm for the construction multiwavelets with some symmetric property by matrix extension. This can be applied in the construction of multiwavelets if the multiscaling functions and low-pass filter are known. Finally, some examples are given by applying the proposed scheme. Throughout this paper: let A∗ , AT and Tr(A) denote conjugate transpose, transpose and the trace of A, respectively. In denotes the n × n identity matrix, 0s,j denote s × j zero matrix, for convenience, we omit the subscript s × j when it does not cause any confusion. Let O(n) denote the set of all n × n real orthogonal matrices. For two matrices B = (bij ) and C = (cij ), let B ⊗ C = (bij C) denote the Kronecker product of B, C. For n × n antidiagonal matrix Jn , we assumption: J0 ⊗ A = 1. δk is the Dirac sequence such that δ0 = 1 and k ∈ Z\{0} for all δk = 0. · is the floor function. In this paper, we consider causal matrix sequence with real finiteimpulse response (FIR), such sequences can be identified with Laurent polynomial defined by H(z) = Lk=0 h(k)z−k , where 0 and L are the smallest and largest indices that h(k) is nonzero, respectively.
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2. Preliminaries of orthonormal multiwavelets In this section, we introduce some notation and summarize needed results that we will be used later in this paper. An m-band orthonormal multiwavelet systems are composed of compactly support orthonormal multiscaling functions = (φ1 , . . . , φr )T and orthonormal multiwavelets = (ψ1 , . . . , ψr )T , = 1, . . . , m − 1 which satisfy the matrix scaling equations, respectively (x) = m
h(k)(mx − k),
(1)
k∈Z
(x) = m
g (k)(mx − k),
= 1, . . . , m − 1,
(2)
k∈Z
or equivalently
ω ˆ ω ˆ ω , = 1, . . . , m − 1, ˆ (ω) = G ω , (3) m m m m and H and G denote the corresponding where h(k), g (k) is r × r is real-value matrix sequence matrix frequency response which defined by H(ω) = k h(k) e−ikω and G (ω) = k g (k) e−ikω , = 1, . . . , m − 1, respectively. H and G are also called matrix low-pass filter and high-pass filter in the application of signal processing. The set {H, G }1≤
m−1
H
k=0 m−1
H
k=0 m−1 k=0
ω + 2kπ ω + 2kπ G∗ m m
G
ω + 2kπ ω + 2kπ H∗ m m
= Ir ,
(4)
= 0r ,
(5)
ω + 2kπ ω + 2kπ G∗n m m
= δ−n Ir ,
0 < , n < m.
(6)
The sequence H which satisfies (4) is called a conjugate quadrature filter (CQF). If the multifilter banks {H, G }1≤
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functions. Paraunitary matrix extension is one of the important approaches for constructing multiwavelets associated with a given multiscaling functions. Let H(z) = k h(k)z−k is a given low-pass filter which generated m-band compactly supported orthonormal multiscaling functions , then H satisfies (4). To obtain m-band compactly supported orthonormal multiwavelets corresponding to , the key problem is seek the high-pass G , = 1, . . . , m − 1 filter satisfied with (5) and (6), which can be reduced to the problem of paraunitary matrix extensions. The polyphase matrix E(z) of multifilter banks {H, G }, = 1, . . . , m − 1 is the block matrix E(z) = [E,n (z)]0≤,n≤m−1 ,
(7)
where the polyphase symbols E,n are defined by 1 E,n (z) = √ g (mk + n)z−k , m k
(8)
where h = g0 . It is not difficult to verify that (4)–(6) are equivalent to the following form: E(z)ET (z−1 ) = Irm .
(9)
From the above theory, we know that H satisfies (4) if and only if matrix [E0,0 (z), E0,1 (z), . . . , E0,m−1 (z)] = P(z) is paraunitary. An s × n(s ≤ n) matrix filter P(z) is called paraunitary if P(z)P T (z−1 ) = Is ,
z = 0,
that is P(eiω ) is a matrix of orthogonal rows for all ω ∈ R. The basic idea of orthogonal extension of paraunitary matrix is that: for a given r × mr matrix P(z), finding (m − 1)r × mr matrix such that [E0 (z), E1 (z), . . . , Em−1 (z)] = Q(z), where Ej (z), j = 0, 1, . . . , m − 1 is (m − 1)r × r Laurent polynomial matrix and such that matrix defined in (7), i.e.,
P(z) E(z) = Q(z)
is paraunitary. From the literature [10], we know that for a given paraunitary matrix P(z), one always can find its paraunitary extensions matrix Q(z) such that (9) holds.
3. Symmetry property of multiwavelets with different symmetric center In this section, we will analyse some symmetric property on multiscaling functions and multiwavelets with their components not having the same symmetric center. Suppose a scalar function f (x) is symmetric about point a, then f (x) = f (2a − x). Analogously, f (x) is antisymmetric about point a if and only if f (x) = −f (2a − x). For multiscaling functions (x) = (φ1 , . . . , φr )T and multiwavelets (x) = (ψ1 , . . . , ψr )T , we want all their components are symmetric or antisymmetric, i.e., φj (x) = sj φj (2aj − x),
ψj (x) = tj ψj (2bj − x),
(10)
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where sj , tj ∈ {−1, 1}, aj , bj ∈ R. That is sj , tj determine whether a function is symmetric or antisymmetric and aj , bj are the centers of symmetry. We say that will have the different symmetric center if there exist ai = aj . The following lemma present the characterization of multiscaling functions and multiwavelets with this symmetry in terms of matrix filter in the Fourier domain [17]. Lemma 1. Assume that H is an FIR matrix filter and = (φ1 , . . . , φr )T is an m-band compactly ˆ supported multiscaling functions with φ(0) = 0. If H satisfies SDa (mω)H(−ω)SDa (−ω) = H(ω)
(11)
for some a = (a1 , . . . , ar ) ∈ Rr , then φj is symmetric/antisymmetric about
φj
aj − x = sj φj (x), m−1
aj , 2(m−1)
i.e.,
1 ≤ j ≤ r,
(12)
where S is the diagonal matrix having coefficients sj on the diagonal, and Da (ω) = diag(e−ia1 ω/(m−1) , e−ia2 ω/(m−1) , . . . , e−iar ω/(m−1) )
(13)
or a vector a = (a1 , . . . , ar ) ∈ Rr . Conversely, if φ is L2 -stable and maj = ai mod(m − 1), 1 ≤ i, j ≤ r, then (12) implies (11). Lemma 2. Assumed H is an FIR matrix filter satisfying (11) for some a = (a1 , . . . , ar ) ∈ Rr , and ˆ is an m-band compactly supported multiscaling functions with φ(0) = 0. Let = (ψ1 , . . . , ψr )T be a vector-valued function defined by (x) = m
g(k)(mx − k)
k
for some FIR matrix filter G. If G satisfies TDb (mω)G(−ω)TDa (−ω) = G(ω)
(14)
for some b = (b1 , . . . , br ) ∈ Rr , then
ψj
bj − x = tj ψj (x), m−1
1 ≤ j ≤ r,
(15)
where T and Db (ω) are defined analogously with S and Da (ω). Conversely, if is L2 -stable and mbj = ai mod(m − 1), 1 ≤ i, j ≤ r, then (15) implies (14).
(α+1)m−1 (α+1)m−1 In this paper, we always assuming that H(ω) = k=0 h(k) e−ikω , G (ω) = k=0 g (k) e−ikω , 1 ≤ < m is an orthogonal causal FIR multifilter bank generating multiscaling functions and multiwavelets , , = 1, . . . , m − 1. Suppose h(k), g (k) = 0 if k ∈ / [0, (α + 1)m − 1] for some α ∈ α Z+ \{0}, then , , 1 ≤ < m, are supported in 0, 1 + α + m−1 [22]. In the following, we will discuss symmetric orthogonal multifilter banks with corresponding multiscaling functions and multiwavelets with their components not having the same symmetric center. To this end, we first need to find a characterization of this type symmetric in terms of filter sequences. By Lemmas 1 and 2, we may
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consider H(z) =
m(α+1)−1 k=0
h(k)z−k , G (z) =
m(α+1)−1 k=0
g (k)z−k , = 1, . . . , m − 1 satisfy (4)–(6) and
z−((α+1)m−1) diag(S0 zm , s0 )H(z−1 ) diag(S0 , s0 z) = H(z), z−((α+1)m−1) diag(S1 zm , S2 )G (z−1 ) diag(S0 , s0 z) = G (z),
1 ≤ < m,
(16)
where s0 = ±1, S0 , S1 , S2 , 1 ≤ < m are diagonal matrices with diagonal entries 1 or −1. Then we have the following theorem.
Theorem 1. Let {H, G }1≤
(17)
where
0 S0 0 Jm−1 ⊗ 1 0 . JS0 = S0 0 Proof. Let P(z) = [E0,0 (z), E0,1 (z), . . . , E0,m−1 (z)], Q (z) = [E,0 (z), E,1 (z), . . . , E,m−1 (z)],
= 1, . . . , m − 1.
To complete the proof, it suffices to prove that z−α diag(S0 z, s0 )P(z−1 ) diag(JS0 , s0 z) = P(z)
(18)
and z−α diag(S1 z, S2 )Q (z−1 ) diag(JS0 , s0 z) = Q (z),
= 1, . . . , m = 1.
(19)
Denote
Eij (z) =
Eij11 (z) Eij12 (z) Eij21 (z) Eij22 (z)
,
i, j = 0, 1, . . . , m − 1,
where Eij11 (z) and Eij12 (z) are the (r − 1) × (r − 1) and (r − 1) × 1 matrix, Eij21 (z) and Eij22 (z) are the 1 × (r − 1) and 1 × 1 matrix.
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We substituting the following
Ir −1 11 12 11 z Ir E00 E00 · · · E0(m−1) m H(z) = P(z ) .. = 21 22 21 . E00 E00 · · · E0(m−1) z−(m−1) Ir
Ir−1 0 −1 z Ir−1 12 E0(m−1) 0 22 . E0(m−1) . . z−(m−1) Ir−1 0
into (16) leads to
z−mα+1 S0
m−1
j 11 −m )S0 j=0 z E0j (z m−1 j 21 −m −mα−(m−1) z s0 j=0 z E0j (z )S0
=
m−1 −j 11 m z E0j (z ) j=0 m−1 −j 21 m j=0 z E0j (z )
z−mα+2 S0
m−1
j 12 −m )s0 j=0 z E0j (z m−1 j 22 −m −mα−(m−2) z s0 j=0 z E0j (z )s0
m−1
−j 12 m j=0 z E0j (z ) m−1 −j 22 m . j=0 z E0j (z )
Then we can obtain 11 −m 11 z−mα+j+1 S0 E0j (z )S0 = z−(m−j−1) E0(m−j−1) (zm ), j = 1, 2, . . . , m − 1; 12 −m 12 z−mα+j+2 S0 E0j (z )s0 = z−(m−j−2) E0(m−j−2) (zm ), j = 1, 2, . . . , m − 2, 12 12 z−mα+m+1 S0 E0(m−1) (z−m )s0 = z−(m−1) E0(m−1) (zm ); 21 −m 21 z−mα−(m−j−1) s0 E0j (z )S0 = z−(m−j−1) E0(m−j−1) (zm ), j = 1, 2, . . . , m − 1; 22 −m 22 (z )s0 = z−(m−j−2) E0(m−j−2) (zm ), j = 1, 2, . . . , m − 2, z−mα−(m−j−2) s0 E0j 22 22 (z−m )s0 = z−(m−1) E0(m−1) (zm ). z−mα+1 s0 E0(m−1)
That is 11 −m 11 z−mα+m S0 E0j (z )S0 = E0(m−j−1) (zm ), j = 1, 2, . . . , m − 1; 12 −m 12 (z )s0 = E0(m−j−2) (zm ), j = 1, 2, . . . , m − 2, z−mα+m S0 E0j 12 12 z−mα+2m S0 E0(m−1) (z−m )s0 = E0(m−1) (zm ); 21 −m 21 z−mα s0 E0j (z )S0 = E0(m−j−1) (zm ), j = 1, 2, . . . , m − 1; 22 −m 22 (z )s0 = E0(m−j−2) (zm ), j = 1, 2, . . . , m − 2, z−mα s0 E0j 22 22 (z−m )s0 = E0(m−1) (zm ). z−mα+m s0 E0(m−1)
0 1 0 z−1
0 z−(m−1)
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It follows that z−mα diag(S0 zm , s0 )P(z−m ) diag(JS0 , s0 zm ) = P(zm ), which implies (18) holds, where
0 S0 0 Jm−1 ⊗ 1 0 . JS0 = S0 0 By using the similar arguments, we can obtain (19). Combining (18) and (19) together, one can obtain (16). Denote J(z) = diag(JS0 , s0 z), then it is not difficult to obtain the following lemma. Lemma 3.
Tr J(z) =
1 + s0 z if m is even, Tr(S0 ) + s0 z if m is odd.
(20)
Theorem 2. Suppose multifilter bank {H, G }1≤
(21)
Taking the trace of both sides of (21), using the fact that Tr(AB) = Tr(BA) and Lemma 3, one can obtain Tr(S(1)) = Tr(E(1) diag(JS0 , s0 )E−1 (1)) = Tr(E−1 (1)E(1) diag(JS0 , s0 ))
= Tr(diag(JS0 , s0 )) = which implies (i), (ii) and (iii) hold.
1 + s0 if m is even, Tr(S0 ) + s0 if m is odd,
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Theorem 3. Suppose multifilter bank {H, G }1≤
Tr(s0 + S21 + · · · + S2m−1 ) = 1.
(22)
Tr(s0 + S21 + · · · + S2m−1 ) = s0 .
(23)
Tr(s0 + S21 + · · · + S2m−1 ) = Tr(S0 ).
(24)
Tr(s0 + S21 + · · · + S2m−1 ) = s0 .
(25)
(2) When m is even and α is odd, Tr(S0 + S11 + · · · + S1m−1 ) = 1, (3) When m is odd and α is even, Tr(S0 + S11 + · · · + S1m−1 ) = s0 , (4) When m is odd and α is odd, Tr(S0 + S11 + · · · + S1m−1 ) = Tr(S0 ),
Proof. From the proof of Theorem 3, we can obtain
Tr(S(1)) =
1 + s0 if m is even, Tr(S0 ) + s0 if m is odd,
Thus we have
Tr(S(−1)) = (−1)
α
1 − s0 if m is even, Tr(S0 ) − s0 if m is odd.
2 2s0
if m is even and α is even, if m is even and α is odd, Tr(S(1)) + Tr(S(−1)) = 2Tr(S0 ) if m is odd and α is even, 2s if m is odd and α is odd. 0 2s0 2
if m is even and α is even, if m is even and α is odd, Tr(S(1)) − Tr(S(−1)) = 2s0 if m is odd and α is even, 2Tr(S ) if m is odd and α is odd. 0 This lead to (22)–(25).
In the following, we assume that s0 = 1 and suppose S0 = diag(Is , −Ir−s−1 ),
(26)
for a nonnegative integer s ≤ r − 1. Then we have the following assumption because of the above discussion. (i) When m is even, we may suppose S1 = −Ia ,
S2 = diag(Ib , −Ib )
if 2s ≥ r,
S1 = I−a ,
S2 = diag(Ic , −Ic )
if 2s < r, (27)
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where 2s−r 2s−r + 1 if = 2k and r is odd, or is odd and r is even, 2s−r m−1 m−1 m−1 a = 2s−r 2s−r if 2s−r = −2k and r is even, or is odd and r is odd, m−1 m−1 m−1
b=
r−a 2
,c=
r+a 2
, = 1, . . . , m − 1, k is a nonnegative integer.
(ii) When m is odd, both α and r are even, we may suppose S1 = −Ia ,
S2 = diag(Ir/2 , −Ir/2−a ) if 2s ≥ r,
S1 = I−a ,
S2 = diag(Ir/2+a , −Ir/2 ) if 2s < r,
(28)
where
2s − r , = 1, . . . , m − 1. a= m−1 (ii) When both m and α are odd and r is even, we may suppose S1 = diag(Ir−s , −Ir−s ), S1 = diag(Is , −Is ),
S2 = diag(Is−r/2 , −Is−r/2 ) if 2s ≥ r,
S2 = diag(Ir/2−s , −Ir/2−s ) if 2s < r,
(29)
where s ≤ r, = 1, . . . , m − 1. Here we note that the above assumption is just only fashion. There are have many different form. The following lemma is crucial in the following sections. Lemma 4. Let s × (mk − 1)(s ≥ k − 1) real matrix A satisfies A diag(Ix , −Iy )AT = 0,
(30)
where x + y = mk − 1 and x − y = 1. Then there exists v ∈ O(x) such that
vT = 0. A (Iy , 0)
(31)
Proof. The proof of lemma can be carried out similarly as in [12]. Let n denote the rank of A, then n ≤ x. Let {β1 , β2 , . . . , βn } be an orthogonal basis for the columns of the matrix AT . Write
Y1 , [β1 , . . . , βn ] = Z1
Y1 , Z1 are x × n and y × n matrices.
Then we have Y1T Y1 + Z1T Z1 = In and Y1T Y1 = Z1T Z1 . Thus 1 Y1T Y1 = Z1T Z1 = In . 2
L.-H. Cui / Mathematics and Computers in Simulation 70 (2005) 69–89
Therefore
√ √ 2 2 Y , Z1 1 2 2
are x × n and y × n matrices of orthogonal columns, respectively. Let Y2 , Z2 be
the x × (x − n) and y × (y − n) matrices such that one has
79
√ 2 [Y1 , Y2 ] 2
∈ O(x),
√ 2 [Z1 , Z2 ] 2
∈ O(x) ∈ O(y). Then
Y1T −Z1T β = 0, Y2T −[Z2 , 0]T j
1 ≤ j ≤ n.
Thus
Y1T −Z1T AT = 0. Y2T −[Z2 , 0]T
Thus (31) holds true with v = −2 diag([Z1 , Z2 ], I)[Y1 , Y2 ]T ∈ O(x).
From the proof of Lemma 4, we know that orthogonal matrices v are constructed by the Gram–Schmidt process of the rows of A.
4. Symmetric paraunitary matrix extension related to different centers In this section, we will show how to construct the corresponding multiwavelets with some desired symmetry for a given compactly supported orthonormal multiscaling functions with different symmetric m(α+1)−1 center. In concrete, given H(z) = k=0 h(k)z−k satisfying z−((α+1)m−1) diag(S0 zm , 1)H(z−1 ) diag(S0 , z) = H(z),
S0 = diag(Is , −Ir−s−1 )
(32)
for a nonnegative integer s ≤ r − 1. In this case, by the theory in Section 3, we know that if H generates mα−1 an orthonormal multiscaling functions = (φ1 , . . . , φr )T , then φ1 , . . . , φs are symmetric about 2(m−1) mα−1 while φs+1 , . . . , φr−s−1 are antisymmetric about 2(m−1) , and φr is symmetric about m(α+1)−2 . 2(m−1) We want construct G (z) =
m(α+1)−1 k=0
gk z−k , = 1, . . . , m − 1 satisfy
z−((α+1)m−1) diag(S1 zm , S2 )G (z−1 ) diag(JS0 , z) = G (z),
0 < < m,
or equivalently z−((α+1)m−1) diag(S11 zm , S 1 , . . . , S1m−1 zm , S2m−1 )G(z−1 ) diag(JS0 , z) = G(z), j
(33)
where S , = 1, . . . , m − 1, j = 1, 2 defined by one of the equations (27)–(29). For example, in the case of (27), the corresponding multiwavelets have the following symmetric properties: (1) if 2s ≥ r, then mα−1 ψ1 , . . . , ψa are antisymmetric about 2(m−1) , ψa+1 , . . . , ψ(r+a)/2 and ψ(r+a)/2+1 , . . . , ψr are symmetric and mα−1 antisymmetric about m(α+1)−2 , respectively; (2) if 2s < r, then ψ1 , . . . , ψ−a are symmetric about 2(m−1) , 2(m−1) m(α+1)−2 ψ1−a , . . . , ψr−a/2 and ψ(r−a)/2+1 , . . . , φr are symmetric and antisymmetric about 2(m−1) , respectively. Here, all a and b are consistent with those in (27).
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From the above, we know that, the key problem to construct a corresponding m-band multiwavelet which have the symmetric property can be reduced to the symmetric paraunitary extension of a given (α+1)m−1 matrix. That is: suppose H(z) = k=0 h(k)z−k is a matrix CQF which satisfies (4) and symmetric conditions (16), then its polyphase matrix P(z) = [E0,0 (z), E0,1 (z), . . . , E0,m−1 (z)] satisfies z−α diag(S0 z, 1)P(z−1 ) diag(JS0 , z) = P(z),
(34)
where
0 S0 0 Jm−1 ⊗ 1 0 . JS0 = S0 0 We want to construct Q(z) = [E0 (z), E1 (z), . . . , Em−1 (z)] such that Q(z) is the paraunitary extension matrix of P(z) and satisfies z−α diag(S11 z, S21 , . . . , S1m−1 z, S2m−1 )Q(z−1 ) diag(JS0 , z) = Q(z),
(35)
where Sj , = 1, . . . , m − 1, j = 1, 2 are defined by one of the equations (27)–(29). Since
Tr JS0 =
1 if m is even, Tr(S0 ) = 2s − r + 1 if m is odd.
Therefore, there exist R ∈ O(mr − 1) defined by A 0 −A √ 0 2 0 if m is even, √ 2 A 0 A R= B 0 −B 2 √ 0 2Ir−1 0 if m is odd, B 0 B
(36)
where A= Jm 2
−1
⊗
1 S0
S0
,
B = J m−1 ⊗ 2
1
S0
such that RJS0 RT = diag(−A, 1, A) holds, where mr
x=
2 (m−1)r 2
if m is even, + s if m is odd,
mr
y=
−1
2 (m−1)r 2
if m is even,
+ r − 1 − s if m is odd.
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From Theorems 2 and 3, we know that multiwavelet systems have two more symmetric filters than antisymmetric filters for all even m and these odd m only when 2s = r. In the following, we consider the symmetric extension for all even m and these odd m only when 2s = r. In this case, mr
x=
2 mr 2
mr
if m is even, if m is odd,
y=
2 mr 2
− 1 if m is even, − 1 if m is odd.
Here, we note that x − y = 1. Let M0 be such a 2r × 2r permutation matrix that M0T diag(−A, 1, A, z)M0 = diag(z, D),
(37)
where D = diag(Ix , −Iy ) ∈ O(2r − 1), x − y = 1. Constructing B(z) = P(z) diag(RT , 1)M0 , then B(z) possessing the following features: (i) B(z) is casual and paraunitary. (ii) P(z) satisfies (18) if and only if B(z) satisfies z−α diag(S0 z, 1)B(z−1 ) diag(z, D) = B(z) holds, where D = diag(Ix , −Iy ). (iii) For α ≥ 1, B(z) has the form of
(38)
0 a0 0 0 0 0 −1 B(z) = 0 0 b0 + 0 0 0 z , c0 d1 d2 0 d1 −d2
(39)
where a0 , b0 are s × x and (r − s − 1) × y matrices, c0 ∈ R and d1 , d2 are 1 × x and 1 × y row vectors. (iv) For α ≥ 2, B(z) has the form of
a0 b0 a1 b1 −1 S0 a0 S0 b1 D −(α−2) B(z) = + z + ··· + z c1 d2 D c0 d0 c1 d1
0 S0 b0 D −(α−1) 00 + z + z−α , c0 d1 D 0 d0 D
(40)
where cj ∈ R, aj and dj are the (r − 1) × 1 and 1 × (mr − 1) vectors, bj is the (r − 1) × (mr − 1) matrix. Now we are prepared to discuss the symmetric extension of B(z). We want to construct a causal filter C(z) is a paraunitary extension of C(z) and satisfies z−α diag(S11 z, S21 , . . . , S1m−1 z, S2m−1 )C(z−1 ) diag(z, D) = C(z). First let us consider the case α = 1. From (39) and the paraunitaryness of B, we have a0 a0T = Is ,
b0 b0T = Ir−s−1 ,
a0 d1T = 0,
b0 d2T = 0,
(41)
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d1 d1T = d2 d2T ,
c02 + 4d1 d2T = 1.
Thus we know a0 , b0 are s × x and (r − s − 1) × y matrices of orthonormal rows. Let θ be such a real number that cos θ = c0 ,
sin θ =
1 − c02 .
Then d1 , d2 can be written as d1 =
1 sin θu0 , 2
d2 =
1 sin θv0 , 2
(42)
where u0 and v0 are 1 × x and 1 × y row vectors such that
a0 u0
and
b0 v0
are (s + 1) × x and (r − s) × y matrices of orthonormal to rows. If sin θ = 0, then d1 = 0, d2 = 0 and 2d1 2d2 unit vectors u0 , v0 orthonormal to rows of a0 , b0 , respectively, will do. If sin θ = 0, u0 = sin , v0 = sin . θ θ Consider the case 2s ≥ r. Choose (x − s − 1) × x, (2s − r) × y and (x − s − 1) × y matrices u˜ , v˜ 1 , v˜ such that [a0T , uT0 , u˜ T ] ∈ O(x),
[b0T , vT0 , v˜ T , v˜ T1 ] ∈ O(y),
where u0 , v0 are the vectors satisfying (42). Then C(z) defined by
0 0 2˜v1 0 0 0 −2 sin θ cos θu0 cos θv0 0 cos θu0 − cos θv0 1 1 0 u˜ v˜ u˜ −˜v C(z) = + 0 z−1 2 2 0 −u0 0 u0 v0 v0 0 u˜ v˜ 0 −˜u v˜ is a symmetric paraunitary extension of B(z) with C(z) satisfying (41) for α = 1. For the case 2s < r, choose (r − 2s) × x, (y − r + s) × x and (y − r + s) × y matrices u˜ 1 , u˜ and v˜ such that [a0T , uT0 , u˜ T , u˜ T1 ] ∈ O(x),
[b0T , vT0 , v˜ T ] ∈ O(y),
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where u0 , v0 are the vectors satisfying (42). Then C(z) defined by
0 2u˜ 1 0 0 0 0 −2 sin θ cos θu0 cos θv0 0 cos θu0 − cos θv0 1 1 0 u˜ v˜ u˜ −˜v C(z) = + 0 z−1 2 2 0 −u0 0 u v v 0 0 0 0 u˜ v˜ 0 −˜u v˜ is a symmetric paraunitary extension of B(z) with C(z) satisfying (41) for α = 1. Theorem 4. Suppose B(z) is a casual paraunitary matrix satisfies (32) for α = 1. Then C(z) constructed above is a symmetric paraunitary extension of B(z) satisfying (41) α = 1. In the following, we discuss the case α ≥ 2. We first define
0 0 1 2 0 W(z) = 2 0 w1 0 −w1
0 0 2w2 0 1 0 0 + −Iy 2 0 w1 0 w1 Iy
0 0 −1 z Iy Iy
(43)
for
w1 ∈ O(x), w= w2 where w1 , w2 are y × x and 1 × x matrices, respectively. Then by a direct calculation, one has the following lemma. Lemma 5. Let W(z) be the matrix defined by (43) with some w ∈ O(x). Then (i) W(z)W(z−1 )T = Imr , (ii) z−1 diag(z−1 , D)W(z−1 ) diag(z, D) = W(z). From (40) and the paraunitaryness of B(z), we have
a0 b0 c0 d0
00 0 d0 D
T
= 0,
which leads to
b0 D[b0T , d0T ] = 0. d0
a0 b0 c0 d0
0 S0 b0 D c0 d1 D
T
a1 b1 + c1 d1
00 0 d0 D
T
= 0,
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84
By Lemma 3, we can construct w ∈ O(x) satisfying
b0 d0
wT = 0. (Iy , 0)
(44)
Denote
w1 w= , w2 where w1 , w2 are y × x and 1 × x matrices, respectively. From (44), we have
wT2 wT1 d0 D 0 −Iy
= 0.
(45)
˜ by Let Wα (z) be the matrix defined by (43) with w = wα . Defined B ˜ B(z) = B(z)Wα (z−1 ). ˜ Then (44) and (45) imply that P˜ is casual. Since Wα (z) is paraunitary and satisfies (ii) of Lemma 5, B(z) is also paraunitary and satisfies (38) with α = 1. In this way, we construct wα−1 , . . . , w2 ∈ O(r) similarly such that B(z) can be written as ˜ α (z) = · · · = B1 (z)W2 (z) · · · Wα (z), B(z) = BW where Wj is defined by (43) with w = wj and B1 (z) is an r × mr matrix satisfying (38) with α = 1. By Theorem 4, we can construct a casual filter C1 (z) such that C1 (z) is a symmetric paraunitary extension of B1 (z). Let C(z) = C1 (z)W2 (z) · · · Wα (z). Then C(z) is a symmetric extension of B(z) satisfying (41). Define Q(z) = C(z)M0T diag(R, 1). Then Q(z) is a symmetric paraunitary extension of P(z) with Q(z) satisfying (35). Theorem 5. Suppose P(z) = [E0,0 (z), E0,1 (z), . . . , E0,m−1 (z)] is an r × mr paraunitary matrix satisfying (18). Then [E0 (z), E1 (z), . . . , Em−1 (z)] = Q(z)
(46)
obtained by the above algorithm is a symmetric paraunitary extension of P(z) satisfying (35). Furthermore the filter length of Q(z) is not greater than α.
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Let G(z) be the matrix defined by (8). Then G(z) is casual, satisfies (33) and len(G) ≤ m(α + 1) − 1. Corollary 1. Assumed that the casual FIR H generated an orthonormal multiscaling = (φ1 , . . . , φr )T mα−1 with φ1 , . . . , φs and φs+1 , . . . , φr−s−1 symmetric and antisymmetric about 2(m−1) and φr is symmetric about m(α+1)−2 . Let G be the casual matrix filter obtained by the above algorithm. Then = (ψ1 , . . . , ψr )T 2(m−1) defined by (3) possess the following symmetric properties: mα−1 (i) For even m, ψ1 , . . . , ψa are antisymmetric about 2(m−1) , ψa+1 , . . . , ψ(r+a)/2 and ψ(r+a)/2+1 , . . . , ψr are symmetric and antisymmetric about m(α+1)−2 , respectively, for the case 2s ≥ r; and ψ1 , . . . , ψ−a 2(m−1) mα−1 are symmetric about 2(m−1) , ψ1−a , . . . , ψ(r−a)/2 and ψ(r−a)/2+1 , . . . , ψr are symmetric and antisymmetric about m(α+1)−2 , respectively, for the case 2s < r. 2(m−1) (ii) For odd m and r = 2s, ψ1 , . . . , ψr/2 and ψr/2+1 , . . . , ψr are symmetric and antisymmetric about m(α+1)−2 , respectively. 2(m−1)
5. Examples In this section, we will provide some examples by applying the above theory. Example 1. The following H(z) = √ 4 1 1 3 √ 3 2 h(0) = , 2 13 − 42 − 61
1 0 h(2) = 1 2 3+
√ 2 4
0 − 61
,
3 k=0
h(k)z−k is a matrix CQF, where
1
1 3 h(1) = 1 2 3+
1 0 h(3) = 1 2 3−
√ 2 4
√ 2 4
0 0
0 − 16
,
.
We will construct the corresponding symmetric multiwavelets with high-pass filter g(k), k = 0, 1, 2, 3 by the above algorithm. It is not hard to verify that H(z) =
3 k=0
h(k)z−k satisfying symmetric condition provided by (32), that
is z−3 diag(z2 , 1)H(z−1 ) diag(1, z) = H(z), in this case S0 = (1), and
1
1 JS0 = . 1
L.-H. Cui / Mathematics and Computers in Simulation 70 (2005) 69–89
86
Thus, we have
1 0 −1 √ √ 2 0 2 0 , R= 2 10 1
00 0I M0 = 2 10
1 0 . 0
Let E0,0 , E0,1 be the filter defined by (8). Then E0,0
√ = 2(h0 + h2 z−1 ) =
E0,1
√ 2 1 − 2 3
√ = 2(h1 + h3 z−1 ) =
√ 2 4
√ 2 1 + 2 3
+ √ 2 4
1 3 1 3
√ 2 z−1 4
+
1 3 1 3
√
−
2 4
4 3
− 16 (1 + z−1 ) 0
z−1 1
,
.
Thus √ 4 2 0 3 B(z) = [E0,0 , E0,1 ] diag(RT , 1)M0 = 2 1 − 16
√
2 √3 2 3
√ 0 0 0 0 2 √ + z−1 . 2 0 − 16 32 21 − 21
0
Furthermore, by the scheme provided in Theorem 4, we obtain symmetric extension C(z) of B(z):
C(z) =
√ √ − 2 − 62 (1 + z−1 )
1 2 0
− 13 (1 − z−1 )
2 (1 + z−1 ) 3 √ 2 2 (1 − z−1 ) 3
−
√
2 (1 2
− z−1 )
−(1 + z−1 )
.
Again by Q(z) = [E10 (z), E11 (z)] = C(z)M0T diag(R, 1), we obtain
√
√
√
√
√ − 2 + 23 + 22 + 23 z−1 − 13 (1 + z−1 ) 22 + 23 + − 22 + 23 z−1 −2 2 2 √ √ √ √ √ Q(z) = . 4 −1 + 2 3 2 + −1 − 2 3 2 z−1 − 32 (1 − z−1 ) 1 + 2 3 2 + 1 − 2 3 2 z−1 0 So we have √ 3 I2 2 2 2 = g(k)z−k , [E10 (z ), E11 (z )] −1 G(z) = z I2 2 k=0
where
g(0) =
√ 4−3 2 24 √ −3+2 2 24
1 − 12 √
− 122
,
g(1) =
√ 4+3 2 24√ 3+2 2 24
− 21 0
,
L.-H. Cui / Mathematics and Computers in Simulation 70 (2005) 69–89
√ 4+3 2 −1 24 √ √ 12 2 −3−2 2 24 12
g(2) =
g(3) =
,
√ 4−3 2 24√ 3−2 2 24
0 0
87
.
The first and the second components of the corresponding = (ψ1 , ψ2 )T is symmetric and antisymmetric about 1, respectively. Example 2. An important multiwavelet system was constructed by J. Geronimo, D. Hardin and P. Massopust (GHM). The GHM multiscaling function (x) = (φ1 (x), φ2 (x))T satisfies the following equation [9]: (x) = 2
3
h(k)(2x − k),
k=0
where
√ 1 12 16 2 √ , h(0) = 40 − 2 −6
1 0√ 0 h(2) = , 40 9 2 −6
1 12√ 0 h(1) = , 40 9 2 20
1 0√ 0 h(3) = . 40 − 2 0
We will construct the corresponding symmetric multiwavelets with high-pass filter g(k), k = 0, 1, 2, 3 by the above algorithm. It is not hard to verify that H(z) = 3k=0 h(k)z−k satisfying symmetric condition provided by (32), that is z−3 diag(z2 , 1)H(z−1 ) diag(1, z) = H(z), in this case S0 = (1), and
1
1 JS0 = . 1 Thus, we have
1 0 −1 √ √ 2 0 2 0 , R= 2 10 1
00 M0 = 0 I2 10
1 0 . 0
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Let E0,0 , E0,1 be the filter defined by (8). Then
E0,0
√ = 2(h0 + h2 z−1 ) =
E0,1
√ = 2(h0 + h2 z−1 ) =
√ 3 2 10
√
4 5
1 − 20 (1 − 9z−1 ) − 3202 (1 + z−1 )
√ 3 2 10 1 (9 20
0
− z−1 )
√ 2 2
,
.
Thus
B(z) = [E0,0 , E0,1 ] diag(RT , 1)M0 =
4 3 5√ √5 √ 2 − 3202 52 2
0
0 −
√ 2 4
+
0
0
√
0 − 3202
0 0
√ √ 2 2 5 4
z−1 .
Furthermore, by the scheme provided in Theorem 4, we obtain symmetric extension C(z) of B(z):
C(z) =
√ √ − 2 − 3102
1 2 0
− 35
√ √ 2 2 2 5 2 4 1 5
+
√
0 − 3102
1 2 0
3 5
√ 2 2 5 − 45
−
√ 2 2
1
z−1 .
Again by Q(z) = [E10 (z), E11 (z)] = C(z)M0T diag(R, 1), we obtain
Q(z) =
1 (9 − z−1 ) √20 2 (1 + 9z−1 ) 20
√
− 3202 (1 + z−1 ) 3 − 10 (1 − z−1 )
1 (1 + 9z−1 ) 20 √ 2 (9 + z−1 ) 20
−
√ 2 2
0
.
So we have √ 3 I2 2 2 2 G(z) = [E10 (z ), E11 (z )] −1 = g(k)z−k , z I2 2 k=0
where √
9 2 −6
1 √ , g(0) = 40 18 −6 2 √ 1 − 2 −6 √ , g(2) = 6 2 40 2
√ 1 − 2 −20 g(1) = , 40 −2 0 √ 1 9 20 g(3) = . 40 −18 0
The first and the second components of the corresponding = (ψ1 , ψ2 )T is symmetric and antisymmetric about 1, respectively.
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89
Acknowledgement The work is supported by Study Foundation for Young Teacher of Beijing University of Chemical Technology QN.0414.
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