Causal FIR symmetric paraunitary matrix extension and construction of symmetric tight M-dilated framelets
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 via unitary extension principle. We begin with some necessary notations and basic concepts. Throughout this paper, i denotes the imaginary unit such that . and 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. and denote the integer floor and ceiling of x. For and , . We denote by , for , the norm space of all the measurable functions satisfying and the norm space of all sequences defined on such that . In particular, is the Hilbert space with inner product and is the Hilbert space with inner product . For a function , is called a Fourier transform of function f. It can be extended to functions of . By we denote the linear space of all sequences on such that is a finite set. Sequences of are called finite impulse response (FIR) filters. For a sequence , the z-transform of h is called a symbol of h. For a filter , if and for all , then we define the filter support and length of h to be and . And we will also call the interval the coefficient support interval of and write to denote the support interval of the coefficient sequence h. The highest power n of in is called the degree of and denoted by . And a filter h is said to be causal if . Also, we say that a sequence has symmetry about a point if with . Here h is said to be symmetric if and antisymmetric if . For given filter h (or ) and integer , we call (or ) the polyphase components of h (or ). is expressed as in terms of its polyphase components. For , the symbol satisfying has symmetry if and only if polyphase components of satisfy where if is symmetric and if is antisymmetric (see [34]).
We use bold-faced characters to denote vectors and matrices. denotes the transpose of matrix M. And, and denote the identity matrix, the 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 . More generally, we denote the block diagonal matrix which main diagonals of square matrices are continuously putted on its main diagonal by . For a Laurent polynomial matrix , we define , where denotes the complex conjugate of the transpose of the matrix . For a Laurent polynomial matrix with real coefficients, equality holds. For a Laurent polynomial matrix , K is called the order of . Let . A Laurent polynomial matrix is said to be paraunitary (PU) if . For a filter bank , , let be the polyphase components of . Then is called a polyphase matrix of filter bank . A filter bank , is called a paraunitary filter bank (PUFB) if polyphase matrix of , is paraunitary. When , a filter bank , is said to be critically subsampled. And, the symmetry, causality and FIR property of a filter bank , are defined by the symmetry, causality and FIR property of all filters . In the most general FIR symmetric filterbanks, the length of the jth filter is , where and are integers, , and . In practice, should take the same value to have a symmetry extension implementation for finite-length signals (see [13]). Let be a causal FIR PUFB such that . Then the filter bank is symmetric, i.e., , for if and only if (see [34]) where . 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 of a causal FIR PUFB, the minimum number of delay elements (i.e., elements) required to implement is called the (McMillan) degree of and denoted by (see [35]). A structure (factorization) of is said to be minimal if the number of delay elements required for its implementation is equal to the . For the square polyphase matrix of a critically subsampled causal FIR PUFB, the equality 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 be an integer, and . Then, we call the system an M-dilated wavelet system and M a dilation factor. The M-dilated wavelet system is called a (normalized) wavelet tight frame for if for all . If , then normalized tight frame is an orthonormal basis for (see [9], [30]). A method for constructing normalized wavelet tight frame for is to use an M-refinable function. For an integer , we call an M-refinable function if ϕ satisfies the refinement equation for some is called a mask with a dilation factor M. Identifying symbol of with , we also call a mask of ϕ. The refinement equation can be rewritten in terms of Fourier transform of ϕ as . For a mask of an M-refinable function ϕ with , its symbol satisfies And M-refinable function ϕ is called an orthonormal M-refinable function if the integer shifts of ϕ are orthonormal, that is, , where δ denotes the Dirac sequence such that and for all . A mask of an orthonormal M-refinable function is an M-orthogonal filter, that is, Equality (1.5) can be rewritten in terms of polyphase components of as A mask of an M-refinable function ϕ is symmetric about if and only if ϕ is symmetric about , that is, . And if and only if (see [19]).
For a given , let be the subspace that generated by ϕ, that is, the smallest closed subspace of that contains . Let D be the operator of dilation and set . The function ϕ is said to generate multiresolution analysis (MRA) if the sequence satisfies the following conditions: . The sequence of spaces generated by an M-refinable function 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 , M-dilated wavelet system is said to be MRA-based if there exists an MRA of such that the condition holds. If, in addition, the system is a (tight) frame, then we refer to its elements as (tight) M-dilated framelets. And, are called as generators. In particular, for the case of , if 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 be an PUFB such that is the mask of a compactly supported M-refinable function ϕ with . And, define in terms of its Fourier transform by Then is a normalized tight frame of (see [2], [8], [10], [30]). In addition, if the filter bank is symmetric, then have symmetry. For the case of , if the M-refinable function ϕ is orthonormal, then the tight M-dilated framelets are orthogonal M-dilated wavelets, that is, they form an orthonormal basis of (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 , there exist filters such that is a PUFB if and only if symbol of satisfies the inequality (see [8]) Equality (1.8) can be rewritten in terms of polyphase components as Let consider the construction of M-refinable function. If symbol of a filter satisfies the condition (1.4), then infinite product 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 to the refinement equation (see [5], [9]). If the distribution ϕ belongs to , then ϕ is an M-refinable function. The problem on whether a distribution ϕ defined by (1.10) belongs to is related the convergence of cascade algorithm for refinement equation (1.3) (see [15], [16]). Since is a Laurent polynomial satisfying (1.4), can be expressed as where is the symbol of a FIR filter, satisfying conditions and . The (Sobolev) smoothness exponent of the symbol with respect to the dilation factor M is defined to be , where denotes the spectral radius of the square matrix , where and . For any , (see [9], [15], [16]). It has been proved in [16] that the cascade algorithm converges in if and only if Moreover, and the shifts of the function ϕ are orthonormal if and only if 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 satisfying some desirable properties must be designed and then the symbols of FIR bandpass filters such that filter bank 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 whose first row is equal to the Laurent polynomial vector composed of the polyphase components of .
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 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 . 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 be integers that . Hereafter, we will use the following notation: where m is 0 or 1, and . 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 be integers that and l be an integer that . For given Laurent polynomial vector
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 and be the symbol of a symmetric filter satisfying conditions (1.4), (1.5), (1.11) and where is an even integer for the case of even M. Denote . Let be the polyphase components of and let
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 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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