Abstract

Let a  =(𝑎1,𝑎2,…,𝑎𝑚)∈ℂ𝑚 be an m-dimensional vector. Then, it can be identified with an 𝑚×𝑚 circulant matrix. By using the theory of matrix-valued wavelet analysis (Walden and Serroukh, 2002), we discuss the vector-valued multiresolution analysis. Also, we derive several different designs of finite length of vector-valued filters. The corresponding scaling functions and wavelet functions are given. Specially, we deal with the construction of filters on symmetric matrix-valued functions space.

1. Introduction

Wavelet analysis has been investigated extensively due to its wide applications in pure and applied sciences. Many interesting books and papers on this topic have been published (see [1–7]). The construction of filter banks is very important in applied aspects. The analogous theory can be extended to the cases of vector-valued and matrix-valued function spaces (see [8–11]). For example, Xia and Suter in [11] proposed vector-valued wavelets and vector filter banks and established a sufficient condition on the matrix-valued filters such that the solution of the corresponding two-scale dilation equation is a matrix-valued scaling function for a matrix-valued multiresolution analysis. But they did not give any example of finite length matrix-valued filter. As for this reason, Walden and Serroukh in [9] studied the wavelet analysis of matrix-valued time series and gave the construction of several different finite length matrix-valued (2×2) filters; naturally, how to construct the filters for symmetric matrix case. Possible practical application of this scheme in signal and image processing is numerous. In voice privacy systems, a number of signals may need to be transmitted from one place to another, intermixing of the signals before transmission via the matrix-valued filters, combined with perfect reconstruction, adds greatly to the likelihood of secure communications. In a scalable coding application, the high-quality lower-resolution approximations produced may be transmitted via slower communication channels, while the original can be reproduced using the perfect reconstruction filter banks. Other application areas are in progressive coding scheme, multisatellite measurements of electromagnetic wave fields, analysis of climate-related time series, and analysis of space weather effects, and so on. Here, we shall mention the theory of continuous wavelet transforms for quaternion-valued functions (see [12, 13]). Applying the theory of matrix-valued wavelet analysis, The authors in [8] gave the construction of scaling functions and wavelet functions by identifying the quaternion-valued functions with the complex duplex matrix-valued functions. Also, Bahri in [14] discussed the construction of filter banks of quaternion-valued functions. On the other hand, a quaternion 𝑎+𝑏𝑖+𝑐𝑗+𝑑𝑘(𝑎,𝑏,𝑐,𝑑∈ℝ) can be identified with a matrix 𝑎−𝑏−𝑐−𝑑𝑏𝑎−𝑑𝑐𝑐𝑑𝑎−𝑏𝑑−𝑐𝑏𝑎. Recently, a new work is to construct the filter banks of quaternion-valued functions by using method in [9] with this identification. It is well known that every 𝑚-dimensional vector corresponds to an 𝑚×𝑚 matrix, which is called the circulant matrix. Our purpose of the present paper is to study wavelet analysis of vector-valued time series directly by identifying the vector-valued functions with matrix-valued functions and derive several different designs of finite length vector-valued filters. In order to get more length filter banks, we need to improve the value of parameter 𝜖 in [9]. Also, the corresponding scaling function and wavelet function are given in the paper. Since the scaling and wavelet functions are connected with the vanishing moments and regularity, we will consider this problem in late publication.

This paper is organized as follows. In the remainder of this section, we state some preliminaries. Section 2 will introduce some important results of multiresolution analysis theory in the matrix-valued function cases. In Section 3, we give the construction of finite length vector-valued filters. Followed by several different filter designs, we gain the scaling functions and wavelet functions, respectively. In the last section, we deal with the same problem for the symmetric matrix-valued function cases.

Throughout this paper, the black characters are representation of vectors. Let 𝑎1,𝑎2,…,𝑎𝑛∈ℂ. Then, 𝐚=(𝑎1,𝑎2,…,𝑎𝑚)∈ℂ𝑚 denotes an 𝑚-dimensional vector. The mapping ℳ from ℂ𝑚 to ℂ𝑚×𝑚 is defined by 𝑎ℳ∶𝐚=1,𝑎2,…,𝑎𝑚⟼⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑎1𝑎2𝑎3⋯𝑎𝑚𝑎𝑚𝑎1𝑎2⋯𝑎𝑚−1𝑎𝑚−1𝑎𝑚𝑎1⋯𝑎𝑚−2𝑎⋮⋮⋮⋮⋮2𝑎3𝑎4⋯𝑎1⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠=ℳ(𝐚).(1.1) Clearly, if 𝑚=2, ℳ(𝐚)=𝑎1𝑎2𝑎2𝑎1 is a symmetric matrix, whose diagonal has the same number. If 𝑚≥3, ℳ(𝐚) is not symmetric. For example, let 𝑚=3, we get ℳ(𝐚)=𝑎1𝑎2𝑎3𝑎3𝑎1𝑎2𝑎2𝑎3𝑎1. Let 𝐚=(𝑎1,𝑎2,…,𝑎𝑚), 𝐛=(𝑏1,𝑏2,…,𝑏𝑚), and ℳ(𝐚) is called the circulant matrix. It is a very important class of Toeplitz matrices (see [15, page 201]). And we can verify that ℳ(𝐚)ℳ(𝐛)=ℳ(𝐜), where 𝑐𝐜=1,𝑐2,…,𝑐𝑚=𝑎1𝑏1+𝑎2𝑏𝑚+𝑎3𝑏𝑚−1+⋯+𝑎𝑚𝑏2,𝑎1𝑏2+𝑎2𝑏1+𝑎3𝑏𝑚+⋯+𝑎𝑚𝑏3,…,𝑎1𝑏𝑚+𝑎2𝑏𝑚−1+𝑎3𝑏𝑚−2+⋯+𝑎𝑚𝑏1.(1.2) This is to say that for any 𝐚,𝐛, ℳ(𝐚)ℳ(𝐛) is closed under the matrix multiplication. An 𝑚×𝑚 complex matrix 𝐵 is said to be normal if 𝐵𝐻𝐵=𝐵𝐵𝐻, where 𝐵𝐻 denotes the complex-conjugate transpose of 𝐵. Thus, we can see that, for every 𝐚=(𝑎1,𝑎2,…,𝑎𝑚), ℳ(𝐚) is normal. Let 𝐌={ℳ(𝐚)∶𝐚∈ℂ𝑚},𝐌∗={ℳ(𝐚)∶𝐚∈ℂ𝑚,detℳ(𝐚)≠0}.(1.3) Then, we have the following.

Theorem 1.1. 𝐌∗ is a subgroup of 𝐆𝐋(𝑚,ℂ) in the sense of matrix multiplication, where 𝐆𝐋(𝑚,ℂ) is the set of all nonsingular linear transforms on ℂ𝑚.

2. Multiresolution Analysis on 𝐿2𝑀(ℝ,ℂ𝑚×𝑚)

Firstly, we introduce the basic knowledge of vector-valued functions which can be found in [10, 11]. Let 𝐿2ℝ,ℂ𝑚×𝑚=𝐹𝐅(𝑡)=𝑙,𝑗(𝑡)𝑚×𝑚∶𝑡∈ℝ,𝐹𝑙,𝑗(𝑡)∈𝐿2(ℝ,ℂ),1≤𝑙,𝑗≤𝑚(2.1) denote the space of matrix-valued functions defined on ℝ with value in 𝐶𝑚×𝑚. The Frobenius norm on 𝐿2(ℝ,ℂ𝑚×𝑚) is defined by ‖𝐅(𝑡)‖=𝑙,𝑗ℝ||𝐹𝑙,𝑗||(𝑡)2𝑑𝑡1/2.(2.2) For 𝐅, 𝐆∈𝐿2(ℝ,ℂ𝑚×𝑚), the integral of matrix product 𝐅(𝑡)𝐆𝐻(𝑡) is denoted by ⟨𝐅,𝐆⟩=ℝ𝐅(𝑡)𝐆𝐻(𝑡)𝑑𝑡.(2.3) The above operation is not inner product in the common sense; however, it has the linear and commutative properties: ⟨𝐅,𝛼𝐆1+𝛽𝐆2⟩=𝛼𝐻⟨𝐅,𝐆1⟩+𝛽𝐻⟨𝐅,𝐆2⟩,⟨𝐅,𝐆⟩=⟨𝐆,𝐅⟩𝐻.(2.4) For convenience, we also call the operator in (2.3) the “inner product." The concept of orthogonality on 𝐿2(ℝ,ℂ𝑚×𝑚) can be given in natural way: for 𝐅𝑗,𝐅𝑘∈𝐿2(ℝ,ℂ𝑚×𝑚), 𝐅𝑗,𝐅𝑘 is called orthogonal if ⟨𝐅𝑗,𝐅𝑘⟩=𝐈𝑚𝛿𝑗,𝑘, where 𝛿𝑗,𝑘 is the Kronecker delta. Let 𝐿2𝑀(ℝ,ℂ𝑚×𝑚) be a subspace of 𝐿2(ℝ,ℂ𝑚×𝑚) which is defined by 𝐿2𝑀ℝ,ℂ𝑚×𝑚=⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑥𝐱(𝑡)=1(𝑡)𝑥2(𝑡)𝑥3(𝑡)⋯𝑥𝑚𝑥(𝑡)𝑚(𝑡)𝑥1(𝑡)𝑥2(𝑡)⋯𝑥𝑚−1𝑥(𝑡)𝑚−1(𝑡)𝑥𝑚(𝑡)𝑥1(𝑡)⋯𝑥𝑚−2𝑥(𝑡)⋮⋮⋮⋮⋮2(𝑡)𝑥3(𝑡)𝑥4(𝑡)⋯𝑥1⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠(𝑡)∶𝑥1(𝑡),𝑥2(𝑡),…,𝑥𝑚(𝑡)∈𝐿2⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭.(ℝ)(2.5) Let 𝐙 be a set of all integers. A sequence {Φ𝑘(𝑡)}𝑘∈𝐙 in 𝐿2𝑀(ℝ,ℂ𝑚×𝑚) is an orthogonal basis if it is orthogonal, and, for all Φ(𝑡)∈𝐿2𝑀(ℝ,ℂ𝑚×𝑚), there is a constant sequence {𝐀𝑘}𝑘∈𝐙 in 𝐌 such that 𝚽(𝑡)=𝑘∈𝐙𝐀𝑘𝚽𝑘(𝑡).(2.6) It is obvious that 𝐀𝑘=⟨Φ,Φ𝑘⟩. Let 𝑥(𝑡) be a function defined on ℝ. The Fourier transform is ̂𝑥(𝑓)=ℝ𝑥(𝑡)𝑒−2𝑖𝜋𝑡𝑓𝑑𝑡.(2.7) Suppose that ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝜙𝚽(𝑡)=1(𝑡)𝜙2(𝑡)𝜙3(𝑡)⋯𝜙𝑚𝜙(𝑡)𝑚(𝑡)𝜙1(𝑡)𝜙2(𝑡)⋯𝜙𝑚−1(𝜙𝑡)𝑚−1(𝑡)𝜙𝑚(𝑡)𝜙1(𝑡)⋯𝜙𝑚−2𝜙(𝑡)⋮⋮⋮⋮⋮2(𝑡)𝜙3(𝑡)𝜙4(𝑡)⋯𝜙1⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠(𝑡).(2.8) We say that Φ(𝑡) generates a multiresolution analysis for 𝐿2𝑀(ℝ,ℂ𝑚×𝑚) if the sequence of closed subspaces 𝑉𝑗2=span−𝑗/2𝚽𝑡2𝑗−𝑘∶𝑘∈𝐙(2.9) is nested, such that(1)⋯⊂𝑉3⊂𝑉2⊂𝑉1⊂𝑉0⊂𝑉−1⊂𝑉−2⊂⋯,(2)⋃𝑗∈𝐙𝑉𝑗=𝐿2𝑀(ℝ,ℂ𝑚×𝑚), and ⋂𝑗∈𝐙𝑉𝑗={𝟎𝑚}, where 𝟎𝑚 is an 𝑚×𝑚 matrix of zeros,(3)Φ(𝑡)∈𝑉0 if and only if Φ(𝑡−𝑘)∈𝑉0 for all 𝑘∈𝐙,(4)Φ(𝑡)∈𝑉𝑗 if and only if √(1/2)Φ(𝑡/2)∈𝑉𝑗+1,(5){Φ(𝑡−𝑘)∶𝑘∈𝐙} is an orthonormal basis for 𝑉0.

In this case, Φ is called a scaling function. Let Φ∈𝐿2𝑀(ℝ,ℂ𝑚×𝑚). Then, the Fourier transform of Φ is given by ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝜙𝚽(𝑓)=1𝜙(𝑓)2𝜙(𝑓)3𝜙(𝑓)⋯𝑚𝜙(𝑓)𝑚(𝜙𝑓)1(𝜙𝑓)2(𝜙𝑓)⋯𝑚−1(𝜙𝑓)𝑚−1𝜙(𝑓)𝑚𝜙(𝑓)1𝜙(𝑓)⋯𝑚−2𝜙(𝑓)⋮⋮⋮⋮⋮2𝜙(𝑓)3𝜙(𝑓)4𝜙(𝑓)⋯1⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠(𝑓).(2.10) Evidently, Φ∈𝐿2𝑀(ℝ,ℂ𝑚×𝑚) and the Fourier transform of Φ(2𝑡−𝑘) with respect to the variable 𝑡 is 12𝚽𝑓2𝑒−𝑖𝜋𝑘𝑓.(2.11) Notice that Φ∈𝑉0⊂𝑉−1, {Φ(2𝑡−𝑘)∶𝑘∈𝐙} is an orthonormal basis for 𝑉−1, it follows that there exist constant matrices 𝐆𝑘∈𝐌, such that two-scale dilation equation holds: √𝚽(𝑡)=2𝑘∈𝐙𝐆𝑘𝚽(2𝑡−𝑘).(2.12) Let ∑𝐆(𝑓)=𝑘∈𝐙𝐆𝑘𝑒−𝑖2𝜋𝑘𝑓, then we have 1𝚽(𝑓)=√2𝐆𝑓2𝚽𝑓2.(2.13)

By the orthogonality, 𝐑𝚽(𝑡)𝚽𝐻(𝑡−𝑘)𝑑𝑡=𝐈𝑚𝛿𝑘0,𝑘∈𝐙.(2.14) We know that ∑𝑙𝐆𝑙𝐆𝐻2𝑘+𝑙=𝐼𝑚𝛿𝑘0,𝑘∈𝐙. This implies that 𝐆(𝑓)𝐆𝐻1(𝑓)+𝐆𝑓+2𝐆𝐻1𝑓+2=2𝐈𝑚.(2.15) Setting 𝑓=0, we have 𝐆(0)=𝑘∈𝐙𝐆𝑘=√2𝐈𝑚1,𝐆2=𝟎𝑚.(2.16) For 𝑓∈ℝ, we let 𝐇(𝑓)=𝑘∈𝐙𝐇𝑘𝑒−𝑖2𝜋𝑘𝑓(2.17) satisfy 𝐆(𝑓)𝐇𝐻1(𝑓)+𝐆𝑓+2𝐇𝐻1𝑓+2=𝟎𝑚,𝐇(𝑓)𝐇𝐻1(𝑓)+𝐇𝑓+2𝐇𝐻1𝑓+2=2𝐈𝑚.(2.18) Analogous to the proof of in [11, Proposition 1], we can get the following.

Theorem 2.1. Suppose that √Ψ(𝑓)=(1/2)𝐇(𝑓/2)Φ(𝑓/2), then Ψ𝑘(𝑡)=Ψ(𝑡−𝑘),𝑘∈𝐙 constitute an orthonormal basis for 𝑉−1=𝑊0⨁𝑉0, where Ψ is called a wavelet function.

The matrix filters 𝐆(𝑓) and 𝐇(𝑓) are called matrix quadrature mirror filters (MQMF). Since 𝐆(𝑓) is normal, by the spectral theorem of normal matrices in [16], we can obtain that 𝐆(𝑓) is unitarily equivalent to a diagonal matrix, namely, 𝐆𝜆(𝑓)=𝑈diag1(𝐆(𝑓)),𝜆2(𝐆(𝑓)),…,𝜆𝑚𝑈(𝐆(𝑓))𝐻,(2.19) where 𝑈∈𝐔(𝑚), 𝐔(𝑚) denotes the unitary matrix group of order 𝑚, 𝜆𝑞(𝐆(𝑓))(𝑞=1,2,…,𝑚) are the eigenvalues of 𝐆(𝑓), and “diag" means the diagonal matrix. Generally, even if 𝑚=2, it is possible that 𝜆1(𝑓)≠𝜆2(𝑓). Also, it seems to be true that 𝑈 in (2.19) should belong to 𝐌. However, it is not the case; we shall display this fact. For simplicity, we assume that 𝑚=2. Let 𝑈=𝛼𝛽𝛽𝛼∈𝐔(2)∩𝐌 satisfy the relation 𝐆𝜆(𝑓)=𝑈diag1(𝐆(𝑓)),𝜆2𝑈(𝐆(𝑓))𝐻,(2.20) where 𝜆1(𝐆(𝑓))≠𝜆2(𝐆(𝑓)) are nonzero eigenvalues. Write 𝜆𝑗=𝜆𝑗(𝐆(𝑓)). Since 𝐆(𝑓)∈𝐿2𝑀(ℝ,ℂ2×2), it follows that |𝛼|2𝜆1+|𝛽|2𝜆2=|𝛼|2𝜆2+|𝛽|2𝜆1. But |𝛼|2+|𝛽|2=1. Therefore, we have √|𝛼|=|𝛽|=1/2. On the other hand, from the equality 𝛼𝛽𝜆1+𝛽𝛼𝜆2=𝛼𝛽𝜆2+𝛽𝛼𝜆1, we have (𝛼𝛽−𝛽𝛼)𝜆1=(𝛼𝛽−𝛽𝛼)𝜆2. But (𝛼𝛽+𝛽𝛼)=0, which implies that 𝛼𝛽𝜆1=𝛼𝛽𝜆2. This is a contradiction.

It is natural to ask what is the form of 2×2 unitary matrices in 𝐌? The following theorem will give the answer.

Theorem 2.2. Let 𝑈 be a 2×2 unitary matrix in 𝐌. Then, 𝑈=cos𝜃𝑖sin𝜃𝑖sin𝜃cos𝜃 or 𝑈=𝑖cos𝜃sin𝜃sin𝜃𝑖cos𝜃.

From the discussion for [11, Proposition 2, page 513], we have the following theorem.

Theorem 2.3. If inf|𝑓|≤1/4|𝜆𝑞(𝐆(𝑓))|>0 for any 1≤𝑞≤𝑚, then the solution Φ of the two-scale dilation equation (2.13) is a scaling function for 𝐿2𝑀(ℝ,ℂ𝑚×𝑚), and 𝚿𝑗,𝑘=2−𝑗/2𝚿𝑡2𝑗−𝑘∶𝑗,𝑘∈𝐙(2.21) constitutes an orthonormal basis for the space 𝐿2𝑀(ℝ,ℂ𝑚×𝑚).

From the construction of quaternion-valued filters in [8], we have seen that the estimation of the eigenvalues 𝜆𝑞(𝐆(𝑓)) can be transferred to calculate the value of det𝐆(𝑓), while the latter is easy to be checked in practice. But in the present case, the situation is different; we have to involve the computation for all eigenvalues of 𝐆(𝑓).

3. Construction of Filters

Let 𝐆(𝑓) be a finite polynomial matrix in 𝑒−𝑖2𝜋𝑓, that is, 𝐆(𝑓)=𝐿−1𝑙=0𝐆𝑙𝑒−𝑖𝑙2𝜋𝑓.(3.1) Suppose that √𝐆(0)=2𝐈𝑚 and satisfies 𝐆(𝑓)𝐆𝐻1(𝑓)+𝐆𝑓+2𝐆𝐻1𝑓+2=2𝐈𝑚.(3.2) If inf|𝑓|≤1/4|𝜆𝑞(𝐆(𝑓))|>0 for all 1≤𝑞≤𝑚, then the solution of the two-scale dilation equation (2.13) is a scaling function in 𝐿2𝑀(ℝ,ℂ𝑚×𝑚), and 𝚿𝑗,𝑘(𝑡)=2−𝑗/2𝚿𝑡2𝑗−𝑘∶𝑗,𝑘∈𝐙(3.3) is an orthonormal basis in 𝐿2𝑀(ℝ,ℂ𝑚×𝑚). In order to get the designs of the vector-valued filters, we need to deduce that 𝐆(𝑓) and 𝐇(𝑓) should satisfy the necessary condition. Firstly, we consider that 𝐆(𝑓) has the form 𝑒𝐆(𝑓)=𝑖2𝜋𝑓𝛾√2𝐈𝑚+𝑒𝜖𝑖2𝜋𝑓𝐑(2𝑓),𝜖∈{−1,1},(3.4) where 𝛾 is an integer, 𝐑(𝑓)∈𝐿2𝑀(ℝ,ℂ𝑚×𝑚) is a paraunitary matrix with unit periodicity, namely, 𝐑(𝑓)𝐑𝐻(𝑓)=𝐈𝑚, 𝐑(𝑓+1)=𝐑(𝑓), 𝐑(0)=𝐈𝑚. Actually, if 𝜖 in (3.4) is taken to be 0 and 𝐑(𝑓) satisfies 𝐑(𝑓)𝐑𝐻(𝑓)=𝐈𝑚, 𝐑(𝑓+1)=−𝐑(𝑓), 𝐑(0)=𝐈𝑚, then we still have 𝐆(𝑓)𝐆𝐻1(𝑓)+𝐆𝑓+2𝐆𝐻1𝑓+2=12𝐈𝑚𝐈+𝐑(2𝑓)𝑚+𝐑𝐻+𝐈(2𝑓)𝑚𝐈+𝐑(2𝑓+1)𝑚+𝐑𝐻=𝟏(2𝑓+1)𝟐𝐈𝑚𝐈+𝐑(2𝑓)𝑚+𝐑𝐻+𝐈(2𝑓)𝑚𝐈−𝐑(2𝑓)𝑚−𝐑𝐻(2𝑓)=2𝐈𝑚.(3.5) Namely, (2.15) and (2.16) are still true. Thus, 𝜆𝑞𝑒(𝐆(𝑓))=𝑖2𝜋𝑓𝛾√21+𝑒𝜖𝑖2𝜋𝑓𝜆𝑞.(𝐑(2𝑓))(3.6) We are now in a position to define 𝐇(𝑓). As shown in [9], the matrix 𝐇(𝑓) can be given in the following: 𝐇(𝑓)=𝑒−2𝑖𝜋𝑓(𝑆−1+𝛿)𝐆1𝑓+2,(3.7) where 𝐿 is the design length of the filter 𝐆𝑙, and 𝛿∈{0,1}, such that 𝐿−1+𝛿 is odd. Actually, it is deduced that 𝐇(𝑓)=𝐿−1+𝛿𝑚=𝛿(−1)𝐿−1+𝛿−𝑚𝐆𝐻𝐿−1+𝛿−𝑚𝑒−𝑖2𝜋𝑓𝑚.(3.8) In particular, if 𝐿 is even, then we take 𝛿=0. Thus, 𝐇(𝑓)=𝐿−1𝑙=0𝐇𝑙𝑒−𝑖2𝜋𝑓𝑙,(3.9) where 𝐇𝑙=(−1)𝑙+1𝐆𝐻𝐿−𝑙−1,𝑙=0,1,…,𝐿−1. If 𝐿 is odd, then take 𝛿=1 and let 𝐆𝐿=𝟎𝑚, we add the filter length of 𝐆(𝑓) to 𝐿+1. Suppose that 𝐆(𝑓)=𝑙=0𝐿+1𝐆𝑙𝑒−𝑖𝑙2𝜋𝑓.(3.10) The similar design can be realized.

Design 1
Let √𝐆(𝑓)=(1/2)(𝐈2+𝑒𝑖2𝜋𝑓diag(𝑒−𝑖4𝜋𝑓,𝑒−𝑖4𝜋𝑓)), 𝐆(𝑓) has the same eigenvalues √𝜆(𝐆(𝑓))=(1/2)(1+𝑒−𝑖2𝜋𝑓). It is obvious that inf|𝑓|≤1/4√|𝜆(𝐆(𝑓))|=1+cos2𝜋𝑓>0. At the same time, √𝐆(0)=2𝐈2,𝐆(1/2)=𝟎2, 𝐆(𝑓+1)=𝐆(𝑓). Consequently, we have 𝐆0=𝐆1=1√2𝐈2,𝐇01=−√2𝐈2,𝐇1=1√2𝐈2.(3.11) If 𝐆(𝑓) is defined in the form √𝐆(𝑓)=(1/2)(𝐈2+𝑒−𝑖2𝜋𝑓diag(𝑒𝑖4𝜋𝑓,𝑒𝑖4𝜋𝑓)), we have 𝐆0=𝐆−1=1√2𝐈2,𝐇01=−√2𝐈2,𝐇−1=1√2𝐈2.(3.12) This is a simple case. In the following, we shall deal with the case that 𝐆(𝑓) has different eigenvalues. For a diagonal matrix diag(𝜆1,𝜆2), if 𝜆1≠𝜆2, 𝜌𝜃=cos𝜃sin𝜃−sin𝜃cos𝜃 is a rotation transform on ℝ2, when 𝜃=𝜋/4, 𝜌𝜃diag(𝜆1,𝜆2)𝜌𝐻𝜃∈𝐌. We state it as follows.

Theorem 3.1. Let diag(𝜆1,𝜆2) be an arbitrary diagonal matrix. Then, √√2/2−√2/2√2/22/2diag(𝜆1,𝜆2)√√2/2−√2/2√2/22/2∈𝐌.

Depending on this, we give the following designs.

Design 2
Let 𝜌𝜃=√√2/2−√2/2√2/22/2. Then, ⎛⎜⎜⎜⎜⎝√𝐑(2𝑓)=22√22−√22√22⎞⎟⎟⎟⎟⎠diag1,𝑒𝑖4𝜋𝑓⎛⎜⎜⎜⎜⎝√22−√22√22√22⎞⎟⎟⎟⎟⎠=⎛⎜⎜⎜⎝12+12𝑒𝑖4𝜋𝑓−12+12𝑒𝑖4𝜋𝑓−12+12𝑒𝑖4𝜋𝑓12+12𝑒𝑖4𝜋𝑓⎞⎟⎟⎟⎠=⎛⎜⎜⎜⎝12121212⎞⎟⎟⎟⎠𝑒𝑖4𝜋𝑓+⎛⎜⎜⎜⎝12−12−1212⎞⎟⎟⎟⎠.(3.13)

Clearly, 𝐑(𝑓)𝐑𝐻(𝑓)=𝐈2, 𝐑(𝑓+1)=𝐑(𝑓), 𝐑(0)=𝐈2. Taking 𝛾=−1, 𝜖=−1 in (3.4), we obtain 𝑒𝐆(𝑓)=−𝑖2𝜋𝑓√2⎛⎜⎜⎜⎝𝐈2+⎛⎜⎜⎜⎝12121212⎞⎟⎟⎟⎠𝑒𝑖2𝜋𝑓+⎛⎜⎜⎜⎝12−12−1212⎞⎟⎟⎟⎠𝑒−𝑖2𝜋𝑓⎞⎟⎟⎟⎠.(3.14) Consequently, we have 𝜆1𝑒(𝐆(𝑓))=−𝑖2𝜋𝑓√21+𝑒𝑖2𝜋𝑓,𝜆2𝑒(𝐆(𝑓))=−𝑖2𝜋𝑓√21+𝑒−𝑖2𝜋𝑓.(3.15) It is easy to verify that inf|𝑓|≤1/4||𝜆𝑞||(𝐆(𝑓))=inf|𝑓|≤1/4√1+cos(2𝜋𝑓)>0(3.16) for 𝑞=1,2. Therefore, we get 1𝐆(𝑓)=√2⎡⎢⎢⎢⎣⎛⎜⎜⎜⎝12121212⎞⎟⎟⎟⎠+𝐈2𝑒−𝑖2𝜋𝑓+⎛⎜⎜⎜⎝12−12−1212⎞⎟⎟⎟⎠𝑒−𝑖4𝜋𝑓⎤⎥⎥⎥⎦,𝐆0=12√2⎛⎜⎜⎝⎞⎟⎟⎠1111,𝐆1=1√2𝐈2,𝐆2=12√2⎛⎜⎜⎝⎞⎟⎟⎠.1−1−11(3.17) Let 𝐆3=02 and 𝐿=4, from the relation (3.9), we then find that 𝐇0=𝟎2,𝐇1=12√2⎛⎜⎜⎝⎞⎟⎟⎠1−1−11,𝐇21=−√2𝐈2,𝐇3=12√2⎛⎜⎜⎝⎞⎟⎟⎠1111.(3.18) To change the coefficients of filter bank 𝐆(𝑓), it is valid to produce a type of unitary matrices 𝑈 and 𝑈𝐻 in Theorem 2.2 on left and right sides of 𝐑(2𝑓).

In the following, we shall give a construction of more length filter banks.

Design 3
Let 𝜌𝜃=√√2/2−√2/2√2/22/2. Setting 𝛼=(𝑒2𝑖𝜋𝑓+𝑒−2𝑖𝜋𝑓), 𝛽=(𝑒6𝑖𝜋𝑓+𝑒2𝑖𝜋𝑓+𝑒−2𝑖𝜋𝑓+𝑒−𝑖6𝜋𝑓). Then, ⎛⎜⎜⎜⎜⎝√𝐑(2𝑓)=22√22−√22√22⎞⎟⎟⎟⎟⎠1diag21𝛼,4𝛽⎛⎜⎜⎜⎜⎝√22−√22√22√22⎞⎟⎟⎟⎟⎠=12⎛⎜⎜⎜⎝121𝛼+41𝛽−21𝛼+4𝛽−121𝛼+4𝛽121𝛼+4𝛽⎞⎟⎟⎟⎠=18⎛⎜⎜⎝⎞⎟⎟⎠𝑒1111𝑖6𝜋𝑓+18⎛⎜⎜⎝⎞⎟⎟⎠𝑒3−1−13𝑖2𝜋𝑓+18⎛⎜⎜⎝⎞⎟⎟⎠𝑒3−1−13−𝑖2𝜋𝑓+18⎛⎜⎜⎝⎞⎟⎟⎠𝑒1111−𝑖6𝜋𝑓.(3.19) Taking 𝛾=−3, 𝜖=0 in (3.4), we obtain 𝐑(𝑓)𝐑𝐻(𝑓)=𝐈𝑚, 𝐑(𝑓+1)=−𝐑(𝑓), 𝐑(0)=𝐈𝑚, and inf|𝑓|≤1/4||𝜆1||=1(𝐆(𝑓))√2inf|𝑓|≤1/4√1+cos(2𝜋𝑓)>0,inf|𝑓|≤1/4||𝜆2||=1(𝐆(𝑓))√2inf|𝑓|≤1/411+2(cos(2𝜋𝑓)+cos(6𝜋𝑓))>0.(3.20) Therefore, we get 1𝐆(𝑓)=8√2⎛⎜⎜⎝⎞⎟⎟⎠+111118√2⎛⎜⎜⎝⎞⎟⎟⎠𝑒3−1−13−𝑖4𝜋𝑓+1√2𝐈2𝑒−6𝑖𝜋𝑓+18√2⎛⎜⎜⎝⎞⎟⎟⎠𝑒3−1−13−𝑖8𝜋𝑓+18√2⎛⎜⎜⎝⎞⎟⎟⎠𝑒1111−𝑖12𝜋𝑓.(3.21) Consequently, 𝐆0=18√2⎛⎜⎜⎝⎞⎟⎟⎠1111,𝐆1=𝟎2,𝐆2=18√2⎛⎜⎜⎝⎞⎟⎟⎠𝐆3−1−13,(3.22)3=1√2𝐈2,𝐆4=18√2⎛⎜⎜⎝⎞⎟⎟⎠3−1−13,𝐆5=𝟎2,𝐆6=18√2⎛⎜⎜⎝⎞⎟⎟⎠1111.(3.23) Here, 𝐿=7 is odd, so we let 𝐆7=𝟎2. Also, from (3.4), we have 𝐇0=𝟎2,𝐇1=18√2⎛⎜⎜⎝⎞⎟⎟⎠1111,𝐇2=𝟎2,𝐇3=18√2⎛⎜⎜⎝⎞⎟⎟⎠,𝐇3−1−1341=−√2𝐈2,𝐇5=18√2⎛⎜⎜⎝⎞⎟⎟⎠3−1−13,𝐇6=𝟎2,𝐇7=18√2⎛⎜⎜⎝⎞⎟⎟⎠.1111(3.24) The corresponding 2×2 matrix functions Φ and Ψ of this design are plotted in Figures 1 and 2, respectively. Notice that we take the values of 𝜙1(3) and 𝜙2(3) as 3/2 and −1/2, respectively, in the figure plotting. However, this design does not determine the values of 𝜙1(3) and 𝜙2(3). Thus, in practical application, we have the freedom to choose the values of 𝜙1(3) and 𝜙2(3).

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Figure 1 
The graph of the scale function for Design 3.
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Figure 2 
The graph of the wavelet function for Design 3.

Next, we are going to construct the filters for the case of 𝑚≥3. We need to use the methods of discrete Fourier transform matrix (DFT). Let 𝜔𝑚=𝑒−2𝑖𝜋/𝑚=cos(2𝜋/𝑚)−𝑖sin(2𝜋/𝑚). The parameter 𝜔𝑚 is an 𝑚th root of unity due to 𝜔𝑚𝑚=1. Write 𝛽𝑗𝑘=𝜔𝑚(𝑗−1)(𝑘−1). Then, 𝐅𝑚=(𝛽𝑗𝑘)𝑚×𝑚 is called the discrete Fourier transform matrix of order 𝑚. For example, if 𝑚=2, the discrete Fourier transform matrix 𝐅2 of order 2 is 111−1, 111−1diag(𝜆1(𝐆(𝑓)), 𝜆2(𝐆(𝑓)))111−1∈𝐌. We can construct the filters by substituting 111−1 for √(1/2)11−11. Now, we describe another case. Let 𝑚=4, the discrete Fourier transform matrix 𝐅4 of order 𝑚 is given by 𝐅4=⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠11111−𝑖−1𝑖1−11−11𝑖−1−𝑖,(3.25) and the inverse matrix of 𝐅4 is 𝐅4−1=14⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠11111𝑖−1−𝑖1−11−11−𝑖−1𝑖.(3.26) We have known that the discrete Fourier transform is an extremely important tool in applied mathematics and engineering, specially in signal processing. Let us define the downshift permutation 𝐒4 by 𝐒4=⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠0001100001000010(3.27) and a vector 𝐛=(𝑎1,𝑎2,𝑎3,𝑎4)𝑇, where 𝑇 means the transpose. Then, ℳ𝐛𝑇=𝐛,𝐒4𝐛,𝐒24𝐛,𝐒34𝐛=𝐅4−1𝐅diag4𝐛𝐅4.(3.28) Actually, we can verify that 𝐅4−1diag(𝜆1(𝐆(𝑓)), 𝜆2(𝐆(𝑓)), 𝜆3(𝐆(𝑓))=𝜆4(𝐆(𝑓)))𝐅4∈𝐌. According to this theory, we can do some designs of the filters for the case 𝑚=4. Other cases can be done analogously.

4. Symmetric Matrix Case

In the above examples, we deal with the constructions of filter banks for 𝐿2𝑀(ℝ,ℂ2×2). As an application of this theory, we will discuss the same problem on the space 𝐿2𝑆ℝ,ℂ2×2=⎧⎪⎨⎪⎩⎛⎜⎜⎝𝑥1(𝑡)𝑥2𝑥(𝑡)2(𝑡)𝑥3(⎞⎟⎟⎠𝑡)∶𝑥1(𝑡),𝑥2(𝑡),𝑥3(𝑡)∈𝐿2⎫⎪⎬⎪⎭(ℝ).(4.1) Obviously, 𝐿2𝑀(ℝ,ℂ2×2)⊂𝐿2𝑆(ℝ,ℂ2×2). Since ⎛⎜⎜⎝⎞⎟⎟⎠⎛⎜⎜⎝⎞⎟⎟⎠⎛⎜⎜⎝⎞⎟⎟⎠=⎛⎜⎜⎝cos𝜃sin𝜃−sin𝜃cos𝜃𝑎𝑏𝑏𝑎cos𝜃−sin𝜃sin𝜃cos𝜃𝑎+2𝑏cos𝜃sin𝜃𝑏cos2𝜃−sin2𝜃𝑏cos2𝜃−sin2𝜃⎞⎟⎟⎠=⎛⎜⎜⎝⎞⎟⎟⎠,𝑎−2𝑏cos𝜃sin𝜃𝑎+𝑏sin2𝜃𝑏cos2𝜃𝑏cos2𝜃𝑎−𝑏sin2𝜃(4.2) we can see that 𝜌𝜃𝑎𝑏𝑏𝑎𝜌𝐻𝜃∉𝐌; however, 𝜌𝜃𝑎𝑏𝑏𝑎𝜌𝐻𝜃 is symmetric with different elements in diagonal if 𝜃≠0,𝜋/2,𝜋,3/2𝜋,2𝜋. From this, we can give some designs of filter banks in 𝐿2𝑆(ℝ,ℂ2×2).

Design 4
Multiplying matrices √1/2−√3/23/21/2 and √1/2−√3/23/21/2 on the left and right sides of 𝐆(𝑓), respectively, in Design 3, we get the coefficients of the filter banks: 𝐆0=1√162⎛⎜⎜⎝√2−√3−1−12+3⎞⎟⎟⎠,𝐆1=𝟎2,𝐆2=1√162⎛⎜⎜⎝√6+√3116−3⎞⎟⎟⎠,𝐆3=1√2𝐈2,𝐆4=1√162⎛⎜⎜⎝√6+√3116−3⎞⎟⎟⎠,𝐆5=𝟎2,𝐆6=1√162⎛⎜⎜⎝√2−√3−1−12+3⎞⎟⎟⎠,𝐇0=𝟎2,𝐇1=1√162⎛⎜⎜⎝√2−√3−1−12+3⎞⎟⎟⎠,𝐇2=𝟎2,𝐇3=1√162⎛⎜⎜⎝√6+√3116−3⎞⎟⎟⎠,𝐇41=−√2𝐈2,𝐇5=1√162⎛⎜⎜⎝√6+√3116−3⎞⎟⎟⎠,𝐇(4.3)6=𝟎2,𝐇7=1√162⎛⎜⎜⎝√2−√3−1−12+3⎞⎟⎟⎠.(4.4)

Notice that in this design the scaling function Λ and the wavelet function 𝚼 are ⎛⎜⎜⎜⎝𝜙𝚲=1−√32𝜙2−12𝜙2−12𝜙2𝜙1+√32𝜙2⎞⎟⎟⎟⎠⎛⎜⎜⎜⎝𝜓,𝚼=1−√32𝜓2−12𝜓2−12𝜓2𝜓1+√32𝜓2⎞⎟⎟⎟⎠.(4.5) The corresponding 2×2 matrix functions Λ and 𝚼 of this design are plotted in Figures 3 and 4, respectively.

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Figure 3 
The graph of the scaling function for Design 4.
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Figure 4 
The graph of the wavelet function for Design 4.

It is a well-known fact that every rotation operator 𝜌𝛼,𝛽,𝛾 on ℝ3 can be described by three Euler's angles which are given by 𝜌𝛼,𝛽,𝛾=⎛⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎠,cos𝛾sin𝛾0−sin𝛾cos𝛾00011000cos𝛽sin𝛽0−sin𝛽cos𝛽cos𝛼sin𝛼0−sin𝛼cos𝛼0001(4.6) where 𝛼,𝛾∈[0,2𝜋],𝛽∈[0,𝜋]. Obviously, 𝜌𝛼,𝛽,𝛾diag(𝜆1,𝜆2,𝜆3)𝜌𝐻𝛼,𝛽,𝛾 is also symmetric. The description of the rotation operator on ℝ𝑛 is analogous. Thus, we can easily construct the filter banks on 𝐿2𝑆(ℝ,ℂ𝑛×𝑛).

As a final example of this paper, we give a construction in case 𝑚=4 in space 𝐿2𝑆(ℝ,ℂ4×4).

Design 5
Let 𝑈=(1/2)1111−11−11−111−111−1−1. Then, 1𝐑(2𝑓)=4⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠1111−11−11−111−111−1−1diag1,1,𝑒−𝑖4𝜋𝑓,𝑒−𝑖4𝜋𝑓⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠=11−1−1111111−11−111−1−14⎛⎜⎜⎜⎜⎜⎜⎝⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠+⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠𝑒2002022002202002200−202−200−220−2002−𝑖4𝜋𝑓⎞⎟⎟⎟⎟⎟⎟⎠.(4.7) Taking 𝛾=−1,𝜖=1 in (3.4), we obtain inf|𝑓|≤1/4||𝜆𝑞||(𝐆(𝑓))=inf|𝑓|≤1/4√1+cos(2𝜋𝑓)>0(4.8) for 𝑞=1,2, and 𝑒𝐆(𝑓)=−𝑖2𝜋𝑓√2⎛⎜⎜⎜⎜⎜⎜⎝𝐈4+14⎡⎢⎢⎢⎢⎢⎢⎣⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠𝑒2002022002202002𝑖2𝜋𝑓+⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠𝑒200−202−200−220−2002−2𝑖𝜋𝑓⎤⎥⎥⎥⎥⎥⎥⎦⎞⎟⎟⎟⎟⎟⎟⎠=⎛⎜⎜⎜⎜⎜⎜⎝12√2⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠+11001011001101001√2𝐈4𝑒−𝑖2𝜋𝑓+12√2⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠𝑒100−101−100−110−1001−4𝑖𝜋𝑓⎞⎟⎟⎟⎟⎟⎟⎠.(4.9) Thus, we get 𝐆0=12√2⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠1001011001101001,𝐆1=1√2𝐈4,𝐆2=12√2⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠100−101−100−110−1001.(4.10) Let 𝐆3=𝟎2 and 𝐿=4, from the relation (3.9), we then obtain 𝐇0=𝟎4,𝐇1=12√2⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠100−101−100−110−1001,𝐇21=−√2𝐈4,𝐇3=12√2⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠1001011001101001.(4.11)

5. Conclusion

In this work, we discuss the problem of construction for vector-valued filters. By using the theory of matrix-valued wavelet analysis and technique of discrete Fourier transform matrix, we get some designs of vector-valued filters (circulant matrix space). And the corresponding scaling functions of multiresolution analysis and wavelet functions are obtained. Also, the analogous problem on the symmetric matrix space can be solved thoroughly.

Acknowledgments

The authors of this paper would like to thank Dr. Bo Yu for bringing the reference [15] and some helpful discussions. The work for this paper is supported by the National Natural Science Foundation of China (no. 10971039) and the Doctoral Program Foundation of the Ministry of China (no. 200810780002).