1 Introduction
During the last three decades, the theory of frames, which generalize the notion of bases by allowing redundancy yet still providing a reconstruction formula, has been growing rapidly, since several new applications such as nonlinear sparse approximation (e.g., image compression), coarse quantization, data transmission with erasures, and wireless communication, have been developed [
1‐
7]. As a special class of frames, the multi-band wavelets have attracted considerable attention due to their richer parameter space, to give better energy compaction than 2-band wavelets [
8‐
16].
Let
H be a separable Hilbert space, and
E an indexing set. A sequence
\(\{f_{l}\}_{l\in E}\) is called a frame in
H if there exist constants
\(0< A\leq B<+\infty\) such that
$$ A\|f\|^{2}\leq\sum_{l\in E}\big|\langle f,f_{l}\rangle\big|^{2}\leq B\|f\|^{2}, \quad\forall f\in H, $$
(1.1)
where
A and
B are called lower and upper frame bounds, respectively. If
\(A=B\), the frame is called tight frame.
When
\(H=L^{2}(R)\), a wavelet
\(\psi(x)\in H\) gives rise to a classical wavelet frame
\(\{a^{-j/2}\psi(a^{-j}x-bk),j,k\in Z\}\) with parameters
\(a>1\),
\(b>0\). Chui and Shi [
17] established the relationship between the parameters and the frame bounds,
$$ A\leq\frac{1}{2b\ln a} \int_{-\infty} ^{+\infty}\frac {|\widehat{\psi}(\omega)|^{2}}{|\omega|}\,d\omega\leq B. $$
(1.2)
Equation (
1.2) shows that the energy of a biorthogonal wavelet transform is controllable although it is not conservative. Moreover,
\(B-A\) or
\(\widetilde{B}-\widetilde{A}\) are smaller, the performance of a biorthogonal wavelet transform may be better, i.e., the energy is amplified in some cases and decreased in other cases. The classical biorthogonal wavelets are not tight frames, so obtaining the exact values of their bounds is difficult. Instead of estimating the bounds in (
1.2), we can try to obtain the upper bound and the lower bound of the norm of the sub-band operator.
This paper is organized as follows. In Sect.
2, we define the sub-band operator, and obtain the limit form of the norm of the sub-band operator. We present a method for computing the upper bound and the lower bound of the norm of the sub-band operator based on the theory of circular matrix. Section
3 gives some examples to illustrate the results proposed in this paper.
2 Sub-band operator and d-circular matrix
Recall the sub-band coding scheme or Mallat algorithm associated to a
d-band real biorthogonal wavelets. There are 2
d filters
\(h=(h_{n})_{n\in Z}\),
\(g^{i}=(g^{i}_{n})_{n\in Z}\) (
\(i=1,2,\ldots,d-1\)),
\(\widetilde{h}=(\widetilde{h}_{n})_{n\in Z}\) and
\(\widetilde{g}^{i}=(\widetilde{g}^{i}_{n})_{n\in Z}\) (
\(i=1,2,\ldots,d-1\)),
\(\{h,g^{1},g^{2},\ldots,g^{d-1}\}\) are used for decomposition and
\(\{\widetilde{h},\widetilde{g}^{1},\widetilde{g}^{2},\ldots,\widetilde {g}^{d-1}\}\) for reconstruction. Starting from a data sequence
\(x=(x_{n})_{n\in Z}\), we convolve with
\(\{h,g^{1},g^{2},\ldots,g^{d-1}\}\),
$$ \begin{gathered} c_{n}=\sum_{k}h_{dn-k}x_{k}, \\ p^{i}_{n}=\sum_{k}g^{i}_{dn-k}x_{k},\quad i=1,2,\ldots,d-1. \end{gathered} $$
(2.1)
The reconstruction operation is
$$ \widetilde{x}_{k}=\sum_{n}\Biggl( \widetilde{h}_{dn-k}c_{n} + \sum^{d-1}_{i=1} \widetilde{g}^{i}_{dn-k}p^{i}_{n}\Biggr). $$
(2.2)
The constraint conditions for biorthogonal
d-band filter banks with perfect reconstruction property are:
a.
the low-pass and high-pass condition
$$\sum h_{k}=\sum\widetilde{h}_{k}= \sqrt{d},\qquad \sum g^{i}_{k}=\sum \widetilde{g}^{i}_{k}=0,\quad i=1,2,\ldots,d-1; $$
b.
the biorthogonal condition
$$\sum_{k} h_{k}\widetilde{h}_{k+dj}= \delta_{j},\qquad h_{k}\widetilde {g}^{i}_{k+dj}=g^{i}_{k} \widetilde{h}_{k+dj}=0,\quad\quad g^{i}_{k}\widetilde {g}^{l}_{k+dj}=\delta_{i-l}\delta_{j}, $$
where
\(\delta_{j}\) denotes the
Dirac sequence such that
\(\delta_{j}=1 \) for
\(j=0\) otherwise
\(\delta_{j}=0 \);
c.
the perfect reconstruction condition \(x=\widetilde{x}\).
In order to define (
2.1) and (
2.2) as operators, namely, sub-band operators, we assume that the input signal
\(x\in l^{2}(-\infty,+\infty)\). Now consider the separable Hilbert space
\(l^{2}(-\infty,+\infty)\). Define
$$(x,y)=\sum_{n=-\infty}^{+\infty}x_{i} \overline{y}_{i} $$
as the usual inner product on
\(l^{2}(-\infty,+\infty)\), where
\(x,y\in l^{2}(-\infty,+\infty)\), and
\(\overline{y}_{i}\) denotes the conjugate of the complex number
\(y_{i}\).
Throughout this paper, we assume \(h,g^{1},g^{2},\ldots,g^{d-1},\widetilde{h},\widetilde{g}^{1},\widetilde {g}^{2},\ldots,\widetilde{g}^{d-1}\) have only finitely many nonzero elements.
The proof of Theorem
2.1 is trivial.
As is well known, a bounded linear operator on
\(l^{2}(-\infty,+\infty)\) can be expressed by an infinite-dimensional matrix. Using matrix notations, we have a more helpful expression of the operator
T. Let
$$s_{n,k}=\left [ \textstyle\begin{array}{c} h_{k-dn} \\ g^{1}_{k-dn} \\ g^{2}_{k-dn} \\ \vdots\\ g^{d-1}_{k-dn} \end{array}\displaystyle \right ],\qquad \widetilde{s}_{n,k}=\left [ \textstyle\begin{array}{c} \widetilde{h}_{k-dn} \\ \widetilde{g}^{1}_{k-dn} \\ \widetilde{g}^{2}_{k-dn} \\ \vdots\\ \widetilde{g}^{d-1}_{k-dn} \end{array}\displaystyle \right ]. $$
Then
\(A= [s_{n,k}]\) and
\(\widetilde{A}= [\widetilde{s}_{n,k}]\) (
\(-\infty< n,k<+\infty\)) are infinite block circular matrices along four directions (up, down, left and right). At this time,
where
x and
y are doubly infinite column vectors to fit the matrix operation. Hence
T can be viewed as an infinite matrix
A, i.e.,
\(T=A\). If the sub-band decomposition has the perfect reconstruction condition, then it is obvious that
A and
à should satisfy
$$ A\widetilde{A}^{*}=\widetilde{A}^{*}A=I, $$
(2.3)
where
I denotes the infinite identity matrix and
\(A^{*} \) denotes the transpose of complex conjugate of
A. Thus,
\(T^{-1}=\widetilde{A}^{*}\).
Since \((Ax,y)=(x,A^{*}y)\), the adjoint operator of T is \(T^{*}=A^{*}\). Let \(Q=T^{*}T\), we have the following lemma.
\(Q=T^{*}T\) is called a frame operator in general [
2]. Let
\(\nu_{n}\) and
\(\widetilde{\nu}_{n}\) denote the
nth rows of
A and
Ã, respectively. Clearly,
\((\nu_{j} ,\widetilde{\nu}_{k})=\delta_{j-k}\), where
\(\delta_{j}\) is the Dirac sequence, i.e.,
\(\delta_{j}=1 \) for
\(j=0\) otherwise
\(\delta_{j}=0 \). Therefore,
\(\{ \nu_{n}\}\) and
\(\{ \widetilde{\nu}_{n}\}\) are dual biorthogonal bases in
\(l^{2}(-\infty,+\infty)\). Let
\(e_{n}\in l^{2}(-\infty,+\infty)\), its
nth component be 1 and otherwise be 0. Then
\(T\widetilde{\nu}_{n}=e_{n}=\widetilde{T}\nu_{n}\). For an arbitrary
\(x\in l^{2}(-\infty,+\infty)\),
$$\begin{gathered} x=\sum_{n=-\infty}^{+\infty}(x,\nu_{n}) \widetilde{\nu}_{n}=\sum_{n=-\infty }^{+\infty}(x, \widetilde{\nu}_{n})\nu_{n}, \\ Qx=\sum_{n=-\infty}^{+\infty}(x,\nu_{n})Q \widetilde{\nu}_{n}=\sum_{n=-\infty }^{+\infty}(x, \nu_{n})T^{*}e_{n}=\sum_{n=-\infty}^{+\infty}(x, \nu _{n})T^{*}\widetilde{T}\nu_{n} =\sum _{n=-\infty}^{+\infty}(x,\nu_{n})\nu_{n}.\end{gathered} $$
Let
m and
M denote the lower bound and the upper bound of
\(\|T\|\), respectively.
$$\begin{gathered} Tx=\sum_{n=-\infty}^{+\infty}(x,\nu_{n})T \widetilde{\nu}_{n}=\sum_{n=-\infty }^{+\infty}(x, \nu_{n})e_{n}, \\ \|Tx\|^{2}=\sum_{n=-\infty}^{+\infty}\big|(x, \nu_{n})\big|^{2}.\end{gathered} $$
Then
$$ m^{2}\|x\|^{2}\leq\sum_{n=-\infty}^{+\infty}\big|(x, \nu _{n})\big|^{2}\leq M^{2}\|x\|^{2}. $$
(2.4)
Similarly,
$$ \widetilde{m}^{2}\|x\|^{2}\leq\sum _{n=-\infty}^{+\infty }\big|(x,\widetilde{\nu}_{n})\big|^{2} \leq\widetilde{M}^{2}\|x\|^{2} , $$
(2.5)
where
m̃ and
M̃ are the lower bound and the upper bound of
\(\|\widetilde{T}\|\), respectively. Therefore, (
2.4) and (
2.5) are the counterparts of (
1.1) in
\(l^{2}(-\infty,+\infty)\).
Now we define a finite matrix
\(A_{n}\) as the partial matrix of the infinite matrix
A, its row index and column index are finite with the following form:
$$A_{n}=\left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad }c@{\quad}c@{\quad}c@{\quad}c} h_{0} & h_{1} & h_{2} &\ldots& h_{d-1} & \cdots& 0 & 0 & 0 & 0\\ g^{1}_{0} & g^{1}_{1} & g^{1}_{2} &\ldots& g^{1}_{d-1} & \cdots& 0 & 0 & 0 & 0\\ g^{2}_{0} & g^{2}_{1} & g^{2}_{2} & \ldots&g^{2}_{d-1} & \cdots& 0 & 0 & 0 & 0\\ g^{d-1}_{0} &g^{d-1}_{1} & g^{d-1}_{2} &\ldots& g^{d-1}_{d-1} & \cdots& 0 & 0 & 0 & 0\\ h_{-d} & \ldots& h_{-3} & h_{-2} & h_{-1}& \cdots&0 & 0 & 0 & 0\\ g^{1}_{-d} &\ldots& g^{1}_{-3} & g^{1}_{-2} & g^{1}_{-1}& \cdots& 0 & 0 & 0 & 0 \\ g^{2}_{-d} & \ldots&g^{2}_{-3} & g^{2}_{-2} & g^{2}_{-1}& \cdots& 0 & 0 & 0 & 0 \\ g^{d-1}_{-d}& \ldots& g^{d-1}_{-3} & g^{d-1}_{-2} & g^{d-1}_{-1}& \cdots& 0 & 0 & 0 & 0 \\ \cdots&\cdots& \cdots& \cdots& \cdots& \cdots& \cdots& \cdots& \cdots&\cdots\\ 0 & 0 & 0 & 0 & \cdots& h_{0} & h_{1} & h_{2} & \ldots&h_{d-1} \\ 0 & 0 & 0 & 0 & \cdots&g^{1}_{0} & g^{1}_{1} & g^{1}_{2} &\ldots& g^{1}_{d-1}\\ 0 & 0 & 0 & 0 & \cdots& g^{2}_{0} & g^{2}_{1} & g^{2}_{2} &\ldots& g^{2}_{d-1}\\ 0 & 0 & 0 & 0 & \cdots& g^{d-1}_{0} & g^{d-1}_{1} & g^{d-1}_{2} &\ldots& g^{d-1}_{d-1}\\ 0 & 0 & 0 & 0& \cdots& h_{-d} &\ldots& h_{-3} & h_{-2} & h_{-1} \\ 0 & 0 & 0 & 0 & \cdots&g^{1}_{-d}&\ldots& g^{1}_{-3} & g^{1}_{-2} & g^{1}_{-1}\\ 0 & 0 & 0 & 0 & \cdots&g^{2}_{-d} &\ldots& g^{2}_{-3} & g^{2}_{-2} & g^{2}_{-1}\\ 0 & 0 & 0 & 0 & \cdots&g^{d-1}_{-d}&\ldots& g^{d-1}_{-3} & g^{d-1}_{-2} & g^{d-1}_{-1} \end{array}\displaystyle \right ]. $$
Theoretically, Theorem
2.2 gives an exact value for
\(\|Q\|\) and
\(\|T\|=\sqrt{\|Q\|}\) is used to compute the norm of
T. However, the eigenvalues are not easy to compute for generalized block Toeplitz matrices. We shall use the theory of circular matrix to compute the norm of
Q. The block circular matrix is defined as follows [
19]:
$$B_{n}=\left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad }c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} h_{0} & h_{1} & h_{2} &\ldots& h_{d-1} & \cdots& h_{-d} &\ldots& h_{-3} & h_{-2} & h_{-1} \\ g^{1}_{0} & g^{1}_{1} & g^{1}_{2} &\ldots& g^{1}_{d-1} & \cdots& g^{1}_{-d} &\ldots& g^{1}_{-3} & g^{1}_{-2} & g^{1}_{-1} \\ g^{2}_{0} & g^{2}_{1} & g^{2}_{2} &\ldots& g^{2}_{d-1}& \cdots& g^{2}_{-d}&\ldots& g^{2}_{-3} &g^{2}_{-2} & g^{2}_{-1}\\ g^{d-1}_{0} & g^{d-1}_{1} &\ldots& g^{d-1}_{2} & g^{d-1}_{d-1} & \cdots& g^{d-1}_{-d} &\ldots& g^{d-1}_{-3} & g^{d-1}_{-2} &g^{d-1}_{-1}\\ h_{-d}&\ldots& h_{-3} &h_{-2} & h_{-1}& \cdots&h_{-2d}&\ldots& h_{-7} & h_{-6} & h_{-5}\\ g^{1}_{-d}&\ldots& g^{1}_{-3} & g^{1}_{-2} & g^{1}_{-1} &\cdots& g^{1}_{-2d}&\ldots& g^{1}_{-7} & g^{1}_{-6} & g^{1}_{-5}\\ g^{2}_{-d}&\ldots& g^{2}_{-3} & g^{2}_{-2} & g^{2}_{-1} &\cdots& g^{2}_{-2d}&\ldots& g^{2}_{-7} & g^{2}_{-6} & g^{2}_{-5}\\ g^{d-1}_{-d}&\ldots& g^{d-1}_{-3} & g^{d-1}_{-2} & g^{d-1}_{-1} &\cdots& g^{d-1}_{-2d}&\ldots& g^{d-1}_{-7} & g^{d-1}_{-6} & g^{d-1}_{-5}\\ \cdots&\cdots& \cdots& \cdots& \cdots& \cdots& \cdots& \cdots& \cdots& \cdots& \cdots\\ h_{2d}& h_{2d+1} & h_{2d+2} &\ldots& h_{3d-1}& \cdots&h_{d} & h_{d+1} & h_{d+2} &\ldots& h_{2d-1} \\ g^{1}_{2d} & g^{1}_{2d+1} &g^{1}_{2d+2} &\ldots& g^{1}_{3d-1}& \cdots& g^{1}_{d} & g^{1}_{d+1} &g^{1}_{d+2} &\ldots& g^{1}_{2d-1}\\ g^{2}_{2d} & g^{2}_{2d+1} &g^{2}_{2d+2} &\ldots& g^{2}_{3d-1} & \cdots& g^{2}_{d} & g^{2}_{d+1} &g^{2}_{d+2} &\ldots& g^{2}_{2d-1} \\ g^{d-1}_{2d} & g^{d-1}_{2d+1} &g^{d-1}_{2d+2} &\ldots& g^{d-1}_{3d-1} & \cdots& g^{d-1}_{d} & g^{d-1}_{d+1} &g^{d-1}_{d+2} &\ldots& g^{d-1}_{2d-1}\\ h_{d} & h_{d+1} & h_{d+2} &\ldots& h_{2d-1} & \cdots& h_{0} & h_{1} & h_{2} &\ldots& h_{d-1}\\ g^{1}_{d} & g^{1}_{d+1} &g^{1}_{d+2} &\ldots& g^{1}_{2d-1} & \cdots& g^{1}_{0} & g^{1}_{1} & g^{1}_{2} &\ldots& g^{1}_{d-1}\\ g^{2}_{d} & g^{2}_{d+1} &g^{2}_{d+2} &\ldots& g^{2}_{2d-1} & \cdots& g^{2}_{0} & g^{2}_{1} & g^{2}_{2} &\ldots& g^{2}_{d-1}\\ g^{d-1}_{d} & g^{d-1}_{d+1} &g^{d-1}_{d+2} &\ldots& g^{d-1}_{2d-1} & \cdots& g^{d-1}_{0} & g^{d-1}_{1} &g^{d-1}_{2} &\ldots& g^{d-1}_{d-1} \end{array}\displaystyle \right ]. $$
Clearly,
\(A_{n}\) is different from
\(B_{n}\). Let
\(C_{n}=B_{n}-A_{n}\), then only finitely many (fixed) elements in
\(C_{n}\) are not 0 no matter how large the dimension 4
n is. We have
$$ A_{n}^{*}A_{n}=B_{n}^{*}B_{n}-C_{n}^{*}B_{n}-B_{n}^{*}C_{n}+C_{n}^{*}C_{n}. $$
(2.6)
We have Theorem
2.3.
A so-called
d-circular matrix [
20], which is generated by the filters
\(h,g^{1},g^{2},\ldots,g^{d-1}\), is denoted as
\(M_{n} \). For
\(d=4\),
\(M_{3} \) is as follows:
$$M_{3}=\left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad }c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} h_{0} & h_{1} & h_{2} & h_{3} & 0 & 0 & 0 &0 &h_{-4} & h_{-3} & h_{-2} & h_{-1}\\ h_{-4} & h_{-3} &h_{-2} & h_{-1} & h_{0} & h_{1}&h_{2} & h_{3}& 0 & 0& 0 & 0\\ 0 & 0 & 0 & 0 & h_{-4} & h_{-3} &h_{-2} & h_{-1} & h_{0} & h_{1} & h_{2} & h_{3} \\ g^{1}_{0} & g^{1}_{1} & g^{1}_{2} & g^{1}_{3} & 0 & 0 & 0 & 0& g^{1}_{-4} & g^{1}_{-3} & g^{1}_{-2} & g^{1}_{-1}\\ g^{1}_{-4} & g^{1}_{-3} & g^{1}_{-2} & g^{1}_{-1}& g^{1}_{0} & g^{1}_{1} & g^{1}_{2} & g^{1}_{3}& 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & g^{1}_{-4} & g^{1}_{-3} & g^{1}_{-2} & g^{1}_{-1}& g^{1}_{0} & g^{1}_{1} & g^{1}_{2} & g^{1}_{3}\\ g^{2}_{0} & g^{2}_{1} & g^{2}_{2} & g^{2}_{3} & 0 & 0 & 0 & 0& g^{2}_{-4} & g^{2}_{-3} &g^{2}_{-2} & g^{2}_{-1}\\ g^{2}_{-4} & g^{2}_{-3} & g^{2}_{-2} & g^{2}_{-1}& g^{2}_{0} & g^{2}_{1} & g^{2}_{2} & g^{2}_{3} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & g^{2}_{-4} & g^{2}_{-3} & g^{2}_{-2} & g^{2}_{-1}& g^{2}_{0} & g^{2}_{1} & g^{2}_{2} & g^{2}_{3}\\ g^{3}_{0} & g^{3}_{1} & g^{3}_{2} & g^{3}_{3} & 0 & 0 & 0 & 0&g^{3}_{-4}& g^{3}_{-3} & g^{3}_{-2} &g^{3}_{-1}\\ g^{3}_{-4} & g^{3}_{-3} & g^{3}_{-2} & g^{3}_{-1}& g^{3}_{0} & g^{3}_{1} & g^{3}_{2} & g^{3}_{3} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0& g^{3}_{-4} & g^{3}_{-3} & g^{3}_{-2} & g^{3}_{-1}& g^{3}_{0} & g^{3}_{1} & g^{3}_{2} & g^{3}_{3} \end{array}\displaystyle \right ]. $$
Clearly, \(B_{n} \) and \(M_{n} \) are not so different. One can be obtained by exchanging the places of some rows of another, i.e., there exists an orthonormal matrix \(E_{n} \) such that \(M_{n}=E_{n}B_{n} \). The purpose is to facilitate the calculation of the eigenvalues.
Similarly, let
\(\widetilde{M}_{n} \) be a
d-circular matrix generated by
\(\widetilde{h},\widetilde{g}^{1},\widetilde{g}^{2},\ldots,\widetilde{g}^{d-1}\). The perfect reconstruction condition is that there exists an integer
\(N_{0}\), such that, for all
\(n\geq N_{0}\) (in what follows,
n sufficiently large is in this sense),
$$ M_{n}{\widetilde{M}}_{n}^{*}=I_{dn}, $$
(2.7)
where
\(I_{dn}\) is a
\(dn\times dn\) identity matrix.
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