# Constructions of Vector-Valued Filters and Vector-Valued Wavelets

Authors: Jianxun He; Shouyou Huang
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

Leta  =(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.


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## Body

## 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. Section2 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. Leta1,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
(1.1)M:a=(a1,a2,…,am)⟼(a1a2a3⋯amama1a2⋯am-1am-1ama1⋯am-2⋮⋮⋮⋮⋮a2a3a4⋯a1)=M(a).
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
(1.2)c=(c1,c2,…,cm)=(a1b1+a2bm+a3bm-1+⋯+amb2,a1b2+a2b1+a3bm+⋯+amb3,…,a1bm+a2bm-1+a3bm-2+⋯+amb1).
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
(1.3)M={M(a):a∈Cm},M*={M(a):a∈Cm,det⁡M(a)≠0}.
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 onLM2(ℝ,ℂm×m)

Firstly, we introduce the basic knowledge of vector-valued functions which can be found in [10, 11]. Let
(2.1)L2(R,Cm×m)={F(t)=(Fl,j(t))m×m:t∈R,Fl,j(t)∈L2(R,C),1≤l,j≤m}
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
(2.2)‖F(t)‖=(∑l,j‍∫R‍|Fl,j(t)|2dt)1/2.
For F, G∈L2(ℝ,ℂm×m), the integral of matrix product F(t)GH(t) is denoted by
(2.3)〈F,G〉=∫R‍F(t)GH(t)dt.
The above operation is not inner product in the common sense; however, it has the linear and commutative properties:
(2.4)〈F,αG1+βG2〉=αH〈F,G1〉+βH〈F,G2〉,〈F,G〉=〈G,F〉H.
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 LM2(ℝ,ℂm×m) be a subspace of L2(ℝ,ℂm×m) which is defined by
(2.5)LM2(R,Cm×m)={x(t)=(x1(t)x2(t)x3(t)⋯xm(t)xm(t)x1(t)x2(t)⋯xm-1(t)xm-1(t)xm(t)x1(t)⋯xm-2(t)⋮⋮⋮⋮⋮x2(t)x3(t)x4(t)⋯x1(t)):x1(t),x2(t),…,xm(t)∈L2(R)}.
Let Z be a set of all integers. A sequence {Φk(t)}k∈Z in LM2(ℝ,ℂm×m) is an orthogonal basis if it is orthogonal, and, for all Φ(t)∈LM2(ℝ,ℂm×m), there is a constant sequence {Ak}k∈Z in M such that
(2.6)Φ(t)=∑k∈Z‍AkΦk(t).
It is obvious that Ak=〈Φ,Φk〉. Let x(t) be a function defined on ℝ. The Fourier transform is
(2.7)x^(f)=∫R‍x(t)e-2iπtfdt.
Suppose that
(2.8)Φ(t)=(ϕ1(t)ϕ2(t)ϕ3(t)⋯ϕm(t)ϕm(t)ϕ1(t)ϕ2(t)⋯ϕm-1(t)ϕm-1(t)ϕm(t)ϕ1(t)⋯ϕm-2(t)⋮⋮⋮⋮⋮ϕ2(t)ϕ3(t)ϕ4(t)⋯ϕ1(t)).
We say that Φ(t) generates a multiresolution analysis for LM2(ℝ,ℂm×m) if the sequence of closed subspaces
(2.9)Vj=span⁡{2-j/2Φ(t2j-k):k∈Z}
is nested, such that(1)
⋯⊂V3⊂V2⊂V1⊂V0⊂V-1⊂V-2⊂⋯,(2)
⋃j∈ZVj¯=LM2(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∈LM2(ℝ,ℂm×m). Then, the Fourier transform of Φ is given by
(2.10)Φ^(f)=(ϕ^1(f)ϕ^2(f)ϕ^3(f)⋯ϕ^m(f)ϕ^m(f)ϕ^1(f)ϕ^2(f)⋯ϕ^m-1(f)ϕ^m-1(f)ϕ^m(f)ϕ^1(f)⋯ϕ^m-2(f)⋮⋮⋮⋮⋮ϕ^2(f)ϕ^3(f)ϕ^4(f)⋯ϕ^1(f)).
Evidently, Φ^∈LM2(ℝ,ℂm×m) and the Fourier transform of Φ(2t-k) with respect to the variable t is
(2.11)12Φ^(f2)e-iπkf.
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:
(2.12)Φ(t)=2∑k∈Z‍GkΦ(2t-k).
Let G(f)=∑k∈Z‍Gke-i2πkf, then we have
(2.13)Φ^(f)=12G(f2)Φ^(f2).By the orthogonality,(2.14)∫R‍Φ(t)ΦH(t-k)dt=Imδk0,k∈Z.
We know that ∑l‍GlG2k+lH=Imδk0,k∈Z. This implies that
(2.15)G(f)GH(f)+G(f+12)GH(f+12)=2Im.
Setting f=0, we have
(2.16)G(0)=∑k∈Z‍Gk=2Im,G(12)=0m.
For f∈ℝ, we let
(2.17)H(f)=∑k∈Z‍Hke-i2πkf
satisfy
(2.18)G(f)HH(f)+G(f+12)HH(f+12)=0m,H(f)HH(f)+H(f+12)HH(f+12)=2Im.
Analogous to the proof of in [11, Proposition 1], we can get the following.Theorem 2.1.
Suppose thatΨ^(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 filtersG(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,
(2.19)G(f)=Udiag⁡(λ1(G(f)),λ2(G(f)),…,λm(G(f)))UH,
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
(2.20)G(f)=Udiag⁡(λ1(G(f)),λ2(G(f)))UH,
where λ1(G(f))≠λ2(G(f)) are nonzero eigenvalues. Write λj=λj(G(f)). Since G(f)∈LM2(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 α¯βλ1+β¯αλ2=α¯βλ2+β¯αλ1, we have (α¯β-β¯α)λ1=(α¯β-β¯α)λ2. But (α¯β+β¯α)=0, which implies that α¯βλ1=α¯βλ2. This is a contradiction.It is natural to ask what is the form of2×2 unitary matrices in M? The following theorem will give the answer.Theorem 2.2.
LetU 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.
Ifinf|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 LM2(ℝ,ℂm×m), and
(2.21){Ψj,k=2-j/2Ψ(t2j-k):j,k∈Z}
constitutes an orthonormal basis for the space LM2(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

LetG(f) be a finite polynomial matrix in e-i2πf, that is,
(3.1)G(f)=∑l=0L-1‍Gle-il2πf.
Suppose that G(0)=2Im and satisfies
(3.2)G(f)GH(f)+G(f+12)GH(f+12)=2Im.
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 LM2(ℝ,ℂm×m), and
(3.3){Ψj,k(t)=2-j/2Ψ(t2j-k):j,k∈Z}
is an orthonormal basis in LM2(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
(3.4)G(f)=ei2πfγ2(Im+eϵi2πfR(2f)),ϵ∈{-1,1},
where γ is an integer, R(f)∈LM2(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
(3.5)G(f)GH(f)+G(f+12)GH(f+12)=12[(Im+R(2f))(Im+RH(2f))+(Im+R(2f+1))(Im+RH(2f+1))]=12[(Im+R(2f))(Im+RH(2f))+(Im-R(2f))(Im-RH(2f))]=2Im.
Namely, (2.15) and (2.16) are still true. Thus,
(3.6)λq(G(f))=ei2πfγ2{1+eϵi2πfλq(R(2f))}.
We are now in a position to define H(f). As shown in [9], the matrix H(f) can be given in the following:
(3.7)H(f)=e-2iπf(S-1+δ)G(f+12),
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
(3.8)H(f)=∑m=δL-1+δ‍(-1)L-1+δ-mGL-1+δ-mHe-i2πfm.
In particular, if L is even, then we take δ=0. Thus,
(3.9)H(f)=∑l=0L-1‍Hle-i2πfl,
where Hl=(-1)l+1GL-l-1H,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
(3.10)G(f)=∑L+1l=0‍Gle-il2πf.
The similar design can be realized.Design 1
LetG(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
(3.11)G0=G1=12I2,H0=-12I2,H1=12I2.
If G(f) is defined in the form G(f)=(1/2)(I2+e-i2πfdiag(ei4πf,ei4πf)), we have
(3.12)G0=G-1=12I2,H0=-12I2,H-1=12I2.
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)ρθH∈M. We state it as follows.Theorem 3.1.
Letdiag(λ1,λ2) be an arbitrary diagonal matrix. Then, (2/22/2-2/22/2)diag(λ1,λ2)(2/2-2/22/22/2)∈M.Depending on this, we give the following designs.Design 2
Letρθ=(2/22/2-2/22/2). Then,
(3.13)R(2f)=(2222-2222)diag⁡(1,ei4πf)(22-222222)=(12+12ei4πf-12+12ei4πf-12+12ei4πf12+12ei4πf)=(12121212)ei4πf+(12-12-1212).Clearly,R(f)RH(f)=I2, R(f+1)=R(f), R(0)=I2. Taking γ=-1, ϵ=-1 in (3.4), we obtain
(3.14)G(f)=e-i2πf2(I2+(12121212)ei2πf+(12-12-1212)e-i2πf).
Consequently, we have
(3.15)λ1(G(f))=e-i2πf2(1+ei2πf),λ2(G(f))=e-i2πf2(1+e-i2πf).
It is easy to verify that
(3.16)inf⁡|f|≤1/4|λq(G(f))|=inf⁡|f|≤1/41+cos⁡(2πf)>0
for q=1,2. Therefore, we get
(3.17)G(f)=12[(12121212)+I2e-i2πf+(12-12-1212)e-i4πf],G0=122(1111),G1=12I2,G2=122(1-1-11).
Let G3=02 and L=4, from the relation (3.9), we then find that
(3.18)H0=02,H1=122(1-1-11),H2=-12I2,H3=122(1111).
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
Letρθ=(2/22/2-2/22/2). Setting α=(e2iπf+e-2iπf), β=(e6iπf+e2iπf+e-2iπf+e-i6πf). Then,
(3.19)R(2f)=(2222-2222)diag⁡(12α,14β)(22-222222)=12(12α+14β-12α+14β-12α+14β12α+14β)=18(1111)ei6πf+18(3-1-13)ei2πf+18(3-1-13)e-i2πf+18(1111)e-i6πf.
Taking γ=-3, ϵ=0 in (3.4), we obtain R(f)RH(f)=Im, R(f+1)=-R(f), R(0)=Im, and
(3.20)inf⁡|f|≤1/4|λ1(G(f))|=12inf⁡|f|≤1/41+cos⁡(2πf)>0,inf⁡|f|≤1/4|λ2(G(f))|=12inf⁡|f|≤1/41+12(cos⁡(2πf)+cos⁡(6πf))>0.
Therefore, we get
(3.21)G(f)=182(1111)+182(3-1-13)e-i4πf+12I2e-6iπf+182(3-1-13)e-i8πf+182(1111)e-i12πf.
Consequently,
(3.22)G0=182(1111),G1=02,G2=182(3-1-13),(3.23)G3=12I2,G4=182(3-1-13),G5=02,G6=182(1111).
Here, L=7 is odd, so we let G7=02. Also, from (3.4), we have
(3.24)H0=02,H1=182(1111),H2=02,H3=182(3-1-13),H4=-12I2,H5=182(3-1-13),H6=02,H7=182(1111).
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).The graph of the scale functionΦ for Design 3.
(a)
(b)
(c)
(d)The graph of the wavelet functionΨ for Design 3.
(a)
(b)
(c)
(d)Next, we are going to construct the filters for the case ofm≥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), (111-1)diag(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
(3.25)F4=(11111-i-1i1-11-11i-1-i),
and the inverse matrix of F4 is
(3.26)F4-1=14(11111i-1-i1-11-11-i-1i).
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
(3.27)S4=(0001100001000010)
and a vector b=(a1,a2,a3,a4)T, where T means the transpose. Then,
(3.28)M(bT)=(b,S4b,S42b,S43b)=F4-1diag⁡(F4b)F4.
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 forLM2(ℝ,ℂ2×2). As an application of this theory, we will discuss the same problem on the space
(4.1)LS2(R,C2×2)={(x1(t)x2(t)x2(t)x3(t)):x1(t),x2(t),x3(t)∈L2(R)}.
Obviously, LM2(ℝ,ℂ2×2)⊂LS2(ℝ,ℂ2×2). Since
(4.2)(cos⁡θsin⁡θ-sin⁡θcos⁡θ)(abba)(cos⁡θ-sin⁡θsin⁡θcos⁡θ)=(a+2bcos⁡θsin⁡θb(cos⁡2⁡θ-sin2θ)b(cos⁡2⁡θ-sin2θ)a-2bcos⁡θsin⁡θ)=(a+bsin⁡2θbcos⁡2θbcos⁡2θa-bsin⁡2θ),
we can see that ρθ(abba)ρθH∉M; however, ρθ(abba)ρθH 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 LS2(ℝ,ℂ2×2).Design 4
Multiplying matrices(1/2-3/23/21/2) and (1/23/2-3/21/2) on the left and right sides of G(f), respectively, in Design 3, we get the coefficients of the filter banks:
(4.3)G0=1162(2-3-1-12+3),G1=02,G2=1162(6+3116-3),G3=12I2,G4=1162(6+3116-3),G5=02,G6=1162(2-3-1-12+3),H0=02,H1=1162(2-3-1-12+3),H2=02,H3=1162(6+3116-3),H4=-12I2,H5=1162(6+3116-3),(4.4)H6=02,H7=1162(2-3-1-12+3).Notice that in this design the scaling functionΛ and the wavelet function Υ are
(4.5)Λ=(ϕ1-32ϕ2-12ϕ2-12ϕ2ϕ1+32ϕ2),Υ=(ψ1-32ψ2-12ψ2-12ψ2ψ1+32ψ2).
The corresponding 2×2 matrix functions Λ and Υ of this design are plotted in Figures 3 and 4, respectively.The graph of the scaling functionΛ for Design 4.
(a)
(b)
(c)
(d)The graph of the wavelet functionΥ for Design 4.
(a)
(b)
(c)
(d)It is a well-known fact that every rotation operatorρα,β,γ on ℝ3 can be described by three Euler's angles which are given by
(4.6)ρα,β,γ=(cos⁡γsin⁡γ0-sin⁡γcos⁡γ0001)(1000cos⁡βsin⁡β0-sin⁡βcos⁡β)(cos⁡αsin⁡α0-sin⁡αcos⁡α0001),
where α,γ∈[0,2π],β∈[0,π]. Obviously, ρα,β,γdiag(λ1,λ2,λ3)ρα,β,γH is also symmetric. The description of the rotation operator on ℝn is analogous. Thus, we can easily construct the filter banks on LS2(ℝ,ℂn×n).As a final example of this paper, we give a construction in casem=4 in space LS2(ℝ,ℂ4×4).Design 5
LetU=(1/2)(1111-11-11-111-111-1-1). Then,
(4.7)R(2f)=14(1111-11-11-111-111-1-1)diag⁡(1,1,e-i4πf,e-i4πf)(1-1-1111111-11-111-1-1)=14((2002022002202002)+(200-202-200-220-2002)e-i4πf).
Taking γ=-1,ϵ=1 in (3.4), we obtain
(4.8)inf⁡|f|≤1/4|λq(G(f))|=inf⁡|f|≤1/41+cos⁡(2πf)>0
for q=1,2, and
(4.9)G(f)=e-i2πf2(I4+14[(2002022002202002)ei2πf+(200-202-200-220-2002)e-2iπf])=(122(1001011001101001)+12I4e-i2πf+122(100-101-100-110-1001)e-4iπf).
Thus, we get
(4.10)G0=122(1001011001101001),G1=12I4,G2=122(100-101-100-110-1001).
Let G3=02 and L=4, from the relation (3.9), we then obtain
(4.11)H0=04,H1=122(100-101-100-110-1001),H2=-12I4,H3=122(1001011001101001).


## 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.



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*Source: 130939-2012-07-09.xml*