Causal FIR symmetric paraunitary matrix extension and construction of symmetric tight M-dilated framelets

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Abstract

We consider a causal FIR symmetric paraunitary (PU) matrix extension with parameters, and the constructions of symmetric orthogonal wavelets and symmetric tight framelets with an integer dilation factor M2. Firstly, we propose an algorithm for factorizing a causal FIR symmetric Laurent polynomial vector into the product of order-one paraunitary matrices and constant vector. Secondly, based on the factorization algorithm, we propose a method for a causal symmetric PU extension with parameters of a Laurent polynomial vector. This method enables us to parameterize polyphase matrix whose first row is equal to the Laurent polynomial vector composed of polyphase components of given lowpass filter. And this symmetric PU extension provides a minimal factorization structure. Thirdly, we consider the constructions of symmetric orthogonal wavelets and symmetric tight framelets with integer dilation factor by the causal symmetric PU extension. Finally, several examples are provided to illustrate the construction methods proposed in this paper.

Introduction

In this paper, we study a causal FIR symmetric paraunitary matrix extension with parameters and the construction of symmetric tight framelets with integer dilation factor M2 via unitary extension principle. We begin with some necessary notations and basic concepts. Throughout this paper, i denotes the imaginary unit such that i2=1. R,C,Z,N and C denote the set of real numbers, the set of complex number, the set of integers, the set of natural numbers and the set of complex numbers, respectively. x and x denote the integer floor and ceiling of x. For xZ and yN,y2, xmody:=xx/yy. We denote by Lp(R), for 1p<+, the norm space of all the measurable functions satisfying fLp(R)=(R|f(x)|pdx)1/p<+ and lp(Z) the norm space of all sequences c={cn|nZ} defined on Z such that clp(Z)=(nZ|cn|p)1/p<+. In particular, L2(R) is the Hilbert space with inner product (f,g)L2(R)=Rf(x)g(x)dx and l2(Z) is the Hilbert space with inner product (a,b)l2(Z)=nZanbn. For a function fL1(R), fˆ(ω)=Rf(x)eiωxdx,ωR is called a Fourier transform of function f. It can be extended to functions of L2(R). By l0(Z) we denote the linear space of all sequences h={h(k):kZ}:ZC on Z such that {kZ:h(k)0} is a finite set. Sequences of l0(Z) are called finite impulse response (FIR) filters. For a sequence hl0(Z), the z-transform H(z)=nZh(n)zn,zC\{0} of h is called a symbol of h. For a filter h={h(k):kZ}l0(Z), if h(m)h(n)0 and h(k)=0 for all kZ\[m,n], then we define the filter support and length of h to be fsupp(h):=[m,n] and len(h):=nm. And we will also call the interval [m,n] the coefficient support interval of H(z) and write fsupp(H):=[m,n] to denote the support interval of the coefficient sequence h. The highest power n of z1 in H(z) is called the degree of H(z) and denoted by deg[H(z)]. And a filter h is said to be causal if fsupp(h)[0,). Also, we say that a sequence h:ZC has symmetry about a point c/2 if h(k)=ϵh(ck),kZ with ϵ{1,1},cZ. Here h is said to be symmetric if ϵ=1 and antisymmetric if ϵ=1. For given filter h (or H(z)) and integer M2, we call hk={hk(n)=h(Mn+k):nZ},k=0,1,,M1 (or Hk(z)=nh(Mn+k)zn,k=0,1,,M1) the polyphase components of h (or H(z)). H(z) is expressed as H(z)=k=0M1Hk(zM)zk in terms of its polyphase components. For K,lZ,K1,0l<M, the symbol H0(z) satisfying fsupp(H0)=[0,L1],L:=KM+l has symmetry if and only if polyphase components H0,k(z),k=0,1,M1 of H0(z) satisfyH0,k(z)=ϵzKH0,l1k(z1),0kl1,H0,k(z)=ϵz(K1)H0,M+l1k(z1),lkM1, where ϵ=1 if H0(z) is symmetric and ϵ=1 if H0(z) is antisymmetric (see [34]).

We use bold-faced characters to denote vectors and matrices. MT denotes the transpose of matrix M. And, IM,JM and 0M denote the M×M identity matrix, the M×M reversal matrix and the null matrix, respectively. The dimension subscript M will be omitted if the dimension is clear from the context. We denote the diagonal matrix which vector a is putted on its main diagonal by diag(a). More generally, we denote the block diagonal matrix which main diagonals of square matrices Ai,i=1,2,,n are continuously putted on its main diagonal by diag(A1,A2,,An). For a Laurent polynomial matrix M(z)=kZMkzk, we define M˜(z):=kZMkzk, where Mk denotes the complex conjugate of the transpose of the matrix Mk. For a Laurent polynomial matrix M(z) with real coefficients, equality M˜(z)=MT(z1) holds. For a Laurent polynomial matrix M(z)=k=0KMkzk,MK0, K is called the order of M(z). Let M,NN,NM2. A N×M Laurent polynomial matrix M(z) is said to be paraunitary (PU) if M˜(z)M(z)=IM. For a filter bank {h0,h1, ,hN1}, let Hj,k(z),k=0,1,,M1 be the polyphase components of Hj(z). Then E(z)=(Hj,k(z))j=0,1,,N1;k=0,1,,M1 is called a polyphase matrix of filter bank {h0,h1,,hN1}. A filter bank {h0,h1, ,hN1} is called a paraunitary filter bank (PUFB) if polyphase matrix E(z) of {h0,h1, ,hN1} is paraunitary. When N=M, a filter bank {h0,h1, ,hN1} is said to be critically subsampled. And, the symmetry, causality and FIR property of a filter bank {h0,h1, ,hN1} are defined by the symmetry, causality and FIR property of all filters hj,j=0,1,,N1. In the most general FIR symmetric filterbanks, the length of the jth filter is Lj=KjM+lj, where Kj and lj are integers, 0KjK, and 0ljM1. In practice, lj should take the same value to have a symmetry extension implementation for finite-length signals (see [13]). Let {h0,h1,,hN1} be a causal FIR PUFB such that fsupp(hj)=[0,Lj1],Lj=KjM+l,0lM,j=0,,N1. Then the filter bank {h0,h1,,hN1} is symmetric, i.e., hj(n)=ϵjhj(Lj1n), n=0,1,,Lj1,ϵj{1,1} for j=0,1,,N1 if and only if (see [34])E(z)=DZ(z)E(z1)J(z), where D=diag(ϵ0,ϵ1,,ϵN1),J(z)=diag(z1Jl,JMl),Z(z)=diag(zK01,zK11,,zKN1). The polyphase matrix of a causal FIR PUFB has different factorizations and a factorization provides an efficient structure for optimal design and fast implementation (see [12], [33], [35]). For the polyphase matrix E(z) of a causal FIR PUFB, the minimum number of delay elements (i.e., z1 elements) required to implement E(z) is called the (McMillan) degree of E(z) and denoted by degE(z) (see [35]). A structure (factorization) of E(z) is said to be minimal if the number of delay elements required for its implementation is equal to the degE(z). For the square polyphase matrix E(z) of a critically subsampled causal FIR PUFB, the equality deg[detE(z)]=degE(z) holds (see [35]). A FIR symmetric PUFB is parameterized by factoring its polyphase matrix and designed by determining parameters for the PUFB to satisfy several desirable conditions. Efficient factorizations to design the FIR symmetric PUFBs have been studied by many researchers. The efficiency of the factorizations is that it provides a fast implementation structure such as lifting or lattice structure. On the other hand, the construction of symmetric orthogonal wavelets or symmetric tight framelets are closely related with the design of FIR symmetric PUFBs. Let consider MRA-based constructions of orthogonal wavelets and tight framelets. Let M2 be an integer, {ψj:j=1,2,,N1}L2(R) and ψj,k,n(x):=Mk/2ψj(Mkxn),k,nZ,j=1,2,,N1. Then, we call the system {ψj,k,n(x):k,nZ,j=1,2,,N1} an M-dilated wavelet system and M a dilation factor. The M-dilated wavelet system {ψj,k,n(x)} is called a (normalized) wavelet tight frame for L2(R) if j=1N1kZnZ|(f,ψj,k,n)|2=fL2(R)2 for all fL2(R). If ψj=1,j=1,2,,N1, then normalized tight frame {ψj,k,n(x)} is an orthonormal basis for L2(R) (see [9], [30]). A method for constructing normalized wavelet tight frame for L2(R) is to use an M-refinable function. For an integer M2, we call ϕL2(R) an M-refinable function if ϕ satisfies the refinement equationϕ(x)=MkZh0(k)ϕ(Mxk) for some h0l2(Z) is called a mask with a dilation factor M. Identifying symbol H0(z) of h0 with h0, we also call H0(z) a mask of ϕ. The refinement equation can be rewritten in terms of Fourier transform of ϕ as ϕˆ(ω)=M1H0(z)ϕˆ(ω/M),z=eiω/M. For a mask h0 of an M-refinable function ϕ with ϕˆ(0)=1, its symbol H0(z) satisfiesH0(1)=M. And M-refinable function ϕ is called an orthonormal M-refinable function if the integer shifts of ϕ are orthonormal, that is, (ϕ(k),ϕ)L2(R)=δk, where δ denotes the Dirac sequence such that δ0=1 and δk=0 for all k0. A mask h0 of an orthonormal M-refinable function is an M-orthogonal filter, that is,k=0M1|H0(ei2kπ/Mz)|2=M(|z|=1). Equality (1.5) can be rewritten in terms of polyphase components H0,k(z),k=0,1,,M1 of H0(z) ask=0M1|H0,k(z)|2=1(|z|=1). A mask h0 of an M-refinable function ϕ is symmetric about c/2 if and only if ϕ is symmetric about c/(2(M1)), that is, ϕ(x)=ϕ(c/(M1)x). And fsupp(h0)=[m,n] if and only if supp(ϕ)=[m/(M1),n/(M1)] (see [19]).

For a given ϕL2(R), let V0 be the subspace that generated by ϕ, that is, the smallest closed subspace of L2(R) that contains {ϕ(xn):nZ}. Let D be the operator of dilation (Df)(x)=Mf(Mx) and set Vk=DkV0,kZ. The function ϕ is said to generate multiresolution analysis (MRA) {Vk:kZ} if the sequence satisfies the following conditions: (1)VkVk+1,kZ,(2)kZVk=L2(R),kZVk={0}. The sequence of spaces {Vk:kZ} generated by an M-refinable function ϕL2(R) satisfies the first condition of MRA but does not satisfy the second condition generally. If an M-refinable function ϕ has a compact support, then it generates MRA (see [4]). For given {ψj:j=1,2,,N1}L2(R), M-dilated wavelet system {ψj,k,n(x):k,nZ,j=1,2,,N1} is said to be MRA-based if there exists an MRA {Vk:kZ} of L2(R) such that the condition {ψj}V1 holds. If, in addition, the system {ψj,k,n} is a (tight) frame, then we refer to its elements as (tight) M-dilated framelets. And, ψj:j=1,2,,N1 are called as generators. In particular, for the case of N=M, if {ψj,k,n} is an orthonormal basis, then we refer to its elements as orthogonal M-dilated wavelets.

Tight M-dilated framelets and orthogonal M-dilated wavelets can be constructed via unitary extension principle (UEP) as follows. Let {h0,h1,,hN1}l0(Z) be an PUFB such that h0 is the mask of a compactly supported M-refinable function ϕ with ϕˆ(0)=1. And, define {ψj:j=1,2,,N1}L2(R) in terms of its Fourier transform byψˆj(ω)=M1Hj(z)ϕˆ(ω/M),z=eiω/M,j=1,2,,N1. Then {ψj,k,n(x):k,nZ,j=1,2,,N1} is a normalized tight frame of L2(R) (see [2], [8], [10], [30]). In addition, if the filter bank {h0,h1,,hN1} is symmetric, then ψj,j=1,2,,N1 have symmetry. For the case of N=M, if the M-refinable function ϕ is orthonormal, then the tight M-dilated framelets ψj,k,n(x),k,nZ,j=1,2,,M1 are orthogonal M-dilated wavelets, that is, they form an orthonormal basis of L2(R) (see [30]). The following result related to the existence of PUFB whose lowpass filter is the mask of an M-refinable function holds. For a given mask h0l0(Z), there exist filters hjl0(Z),j=1,2,,N1 such that {h0,h1,,hN1} is a PUFB if and only if symbol H0(z) of h0 satisfies the inequality (see [8])k=0M1|H0(ei2kπ/Mz)|2M(|z|=1). Equality (1.8) can be rewritten in terms of polyphase components ask=0M1|H0,k(z)|21(|z|=1). Let consider the construction of M-refinable function. If symbol H0(z) of a filter h0l0(Z) satisfies the condition (1.4), then infinite productϕˆ(ω)=j=1M1H0(eiω/Mj) is an entire function of exponential type and ϕ defined via the inverse Fourier transform of the limit function (1.10) is a unique compactly supported distributional solution satisfying ϕˆ(0)=1 to the refinement equation (see [5], [9]). If the distribution ϕ belongs to L2(R), then ϕ is an M-refinable function. The problem on whether a distribution ϕ defined by (1.10) belongs to L2(R) is related the convergence of cascade algorithm for refinement equation (1.3) (see [15], [16]). Since H0(z) is a Laurent polynomial satisfying (1.4), H0(z) can be expressed asH0(z)=MPM(z)mQ(z),PM(z):=(1+z1+z2++z(M1))/M, where Q(z) is the symbol of a FIR filter, satisfying conditions Q(1)=1 and (1+z1++z(M1))Q(z). The (Sobolev) smoothness exponent of the symbol H0 with respect to the dilation factor M is defined to be ν(H0,M)=1/2logMρ(H0,M), where ρ(H0,M) denotes the spectral radius of the square matrix (q(Mjk))pj,kp, where p:=NM1 and Q(z)Q(z1):=k=NNq(k)zk. For any τ<ν(H0,M), R|ϕˆ(ω)|2(1+|ω|2)τdω< (see [9], [15], [16]). It has been proved in [16] that the cascade algorithm converges in L2(R) if and only ifν(H0,M)>0. Moreover, ϕL2(R) and the shifts of the function ϕ are orthonormal if and only if H0(z) satisfies conditions (1.5) and (1.11) (see [9], [15], [16]).

As considered above, in order to be constructed compactly supported symmetric orthogonal wavelets and symmetric tight framelets, a FIR symmetric refinement mask H0(z) satisfying some desirable properties must be designed and then the symbols of FIR bandpass filters Hl(z),l=1,2,,N1 such that filter bank {H0(z),H1(z),,HN1(z)} is a FIR symmetric PUFB must be derived by using the unitary extension principle (see [2], [8], [10], [30]). Here, construction problem of FIR bandpass filters is a FIR symmetric PU matrix extension problem, that is, the design problem of FIR symmetric PU polyphase matrix E(z) whose first row is equal to the Laurent polynomial vector (H0,0(z),,H0,M1(z)) composed of the polyphase components of H0(z).

The study on this problem was initially done by using a factorization-based parameterization of FIR symmetric PUFBs (see [12], [13], [25], [27], [33], [34], [35]). This parameterization-based method reduces FIR symmetric PU matrix extension problem to the problem of finding parameters for a lowpass filter of the filter bank to be a refinement mask satisfying several desirable conditions. But the problem of finding parameters leads to a nonlinear optimization problem with many variables and several constraints and the bigger dilation factor is, the harder to solve the optimization problem is. Therefore, several different algorithms for FIR symmetric PU matrix extension were proposed. FIR symmetric PU matrix extensions with M=2 were discussed in [7], [11], [17], [18], [20], [22], [26], [29] and references therein. In [1], [15], [21], authors studied FIR symmetric PU matrix extension with M=4. In [14], [18], [21], [28], authors proposed a factorization-based FIR symmetric PU matrix extension method that factorize a symmetric Laurent polynomial vector into a product of matrix factors and constant vector and then extend the constant vector to a symmetric unitary matrix. In [6], [17], [23], [24], several step-by-step algorithms for FIR symmetric PU matrix extension are proposed. In page 209 of [6], authors mentioned as follows: However, the available algorithms in the literature to obtain the extension matrix are often quite complicated and such algorithms only work for dimension one. If the symmetry property is required, then the corresponding algorithms for the matrix extension with symmetry generally are much more involved and there are many special cases to be considered. Therefore, it is necessary to study simple and efficient algorithms for symmetric PU matrix extension. On the other hand, from the viewpoint of implementation and practical application of filterbanks, the factorization algorithms for parameterizing polyphase matrix have some advantages in comparison with the existing PU matrix extension algorithms. First of all, it is a simple and minimal implementation structure. Factors of the factorization are composed of the matrices that resemble the famous butterfly matrix in FFT implementation, the unitary matrices such as DCT, DST and Housholder matrix, and the diagonal matrices whose diagonal elements are 1 or delay (see [25], [33], [34]). Such structure of the factors provides fast and minimal implementation of filterbanks. Also, factorization methods for parameterizing polyphase matrix have the advantage that practical conditions such as coding gain and stopband suppression condition can be imposed. Hence, the problem of finding symmetric PU matrix extension algorithm having the advantages of the factorization algorithms for parameterizing polyphase matrix arises.

This motivates us to consider the problem of finding a causal FIR symmetric PU extension algorithm with parameters and with minimal structure via a factorization whose factors are like that of the factorization for parameterizing polyphase matrix.

Therefore, in this paper, we consider a causal FIR symmetric paraunitary matrix extension with parameters and construction of symmetric orthogonal M-dilated wavelets and symmetric tight M-dilated framelets.

The paper is organized as follows. In Section 2, we consider a factorization of a causal FIR symmetric Laurent polynomial vector into the product of matrix factors of order one and constant vector. In Section 3, we propose a causal symmetric PU extension with parameters of a causal FIR symmetric Laurent polynomial vector based on factorization algorithms formulated in Section 2. In Section 4, we consider the constructions of symmetric orthogonal M-dilated wavelets and symmetric tight M-dilated framelets by using the causal symmetric PU extension proposed in Section 3. In Section 5, we provide some examples to illustrate the construction methods proposed in Section 4.

Section snippets

A factorization of causal Laurent polynomial vector

In this section we consider a factorization method of Laurent polynomial vector into the product of matrix factors of order one and constant vector.

Let N,K be integers that N>1,K1. Hereafter, we will use the following notation:P=diag(IN/2,JN/2),Λ(m)(z)=diag(IN/2+m,z1IN/2m),W=12(IN/20N/2×nIN/20n×N/22In×n0n×N/2IN/20N/2×nIN/2),I(z)=diag(IN/2,z1In,IN/2)I(m):=diag(IN/2,(1m)I(N2N/2),IN/2), where m is 0 or 1, and n=N2N/2. It follows from direct calculation

Symmetric paraunitary matrix extension of Laurent polynomial vector

In order to construct symmetric PUFB whose lowpass filter equals to a given symmetric filter, symmetric polyphase matrix whose first row equals to the Laurent polynomial vector composed of the polyphase components of the symmetric filter must be designed. Hence, in this section, we consider symmetric paraunitary matrix extension problem of Laurent polynomial vector.

Let N,K be integers that N>1,K1 and l be an integer that 0l<N. For given N×1 Laurent polynomial vector h(z)=k=0Kbkzk,bK0

Constructions of symmetric orthogonal M-dilated wavelets and symmetric tight M-dilated framelets

In this section, we consider constructions of symmetric orthogonal M-dilated wavelets and symmetric tight M-dilated framelets via symmetric paraunitary extension method considered in Section 3.

Theorem 4.1

Let M be an integer with M>2 and H0(z) be the symbol of a symmetric filter satisfying conditions (1.4), (1.5), (1.11) andfsupp(H0)=[0,L1],LM, where l:=LmodM is an even integer for the case of even M. Denote K:=L/M. Let H0,k(z),k=0,1,,M1 be the polyphase components of H0(z) and leth(z):=(H0,0(z),H0,1(

Examples

In this section, we shall present several examples to illustrate our symmetric paraunitary matrix extension method for constructing symmetric orthogonal M-dilated wavelets and symmetric tight M-dilated framelets. Firstly, we shall show the construction examples of symmetric orthogonal M-dilated wavelets by Theorem 4.1 of Section 4. For this, we use the symmetric real-valued M-orthogonal filters introduced in [19].

Example 5.1

Let H(z) be the symbol of the secondary symmetric real-valued 3-orthogonal filter

Acknowledgements

The authors would like to thank the anonymous reviewers for their valuable comments and helpful suggestions that improved the presentation of this paper.

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