0 引言

小波变换具有“数学显微镜”之称, 当利用小波实施时频分析时, 由于同时具有时间和频率的局部特性以及多分辨分析的特性, 使得对离散数据的处理变得相对容易, 起初人们只是研究定义在实数或复数上的小波分析.近几年来, 人们开始热衷于研究局部紧的零维交换群上的小波理论. P-adic Vilenkin群就是一类局部紧的零维交换群, 当p=2时, 即为康托尔二元群.1996年, 文献[1]首次介绍了康托尔群上多分辨分析的概念并提出了构造正交小波基的方法.之后, 文献[2-3]深入研究了康托尔群上的小波理论的创建和小波框架的构造.文献[4-14]主要研究了康托尔群的推广Vilenkin群, 详细介绍了Vilenkin群上的尺度函数、小波函数、小波框架以及多分辨分析理论, 但是都没有具体地研究Vilenkin群上信号的分解与重构算法.在文献[15-17]中, 李万社等人探讨了双向小波的快速分解和重构算法以及a尺度正交双向小波的Mallat算法.文章在此基础上主要是对Vilenkin群上的信号进行了分析, 选取一种比较常见的尺度函数以及相对应的小波函数, 建立相应的多分辨分析, 对Vilenkin群上的信号创建一种分解和重构的算法, 使人们能够通过观察、研究信号的局部信息, 更加明晰地分析、理解和处理信号.相关理论性内容参见文献[18].

1 Vilenkin群

每一个正实数都可以表示成一个p进制的数(最常见的就是二进制数和十进制数), 即∀x∈R, 都有x=$\sum\limits_{i \in {\rm{Z}}} {{x_i}{p^{-i-1}}} $, x_i∈0, 1, …, p-1, 简记为x=(x_i)_i∈Z.用G_p来表示由x=0或者对于x存在一个N(x), 使得∀i < N(x), x_i=0且x_N(x)=1所构成的集合, 定义其上的二元运算为⊖:

$ x \ominus y: = {\left( {\left| {{x_i} - {y_i}} \right|} \right)_{i \in {\bf{Z}}}},\forall x,y \in {G_p}. $

该二元运算满足:

(1) 结合律:x(yz)=(xy)z, ∀x, y, z∈G_p;

(2) 存在幺元素0∈G_p, x⊖0=0⊖x=x, ∀x∈G_p;

(3) 对于每一个x∈G_p, 都存在逆元素x^-1∈G_p, 使得x^-1⊖x=x⊖x^-1=0.

故G_p是一个群, 称之为Vilenkin群, 并且对∀x, y∈G_p, 满足x⊖y=y⊖x, 因此Vilenkin群也是Abel群.定义G_p上的映射‖x‖:=p^-N(x), x∈G_p, 并规定当x=0时, ‖x‖=0.令d(x, y)=‖x⊖y‖, ∀x, y∈G_p, 由于d满足:

(1) d(x, y)≥0, 且d(x, y)=0的充要条件是x=y;

(2) d(x, y)=d(y, x), ∀x, y∈G_p;

(3) d(x, y)≤d(x, z)+d(z, y), 对任意z都成立.

故d可以表示x与y之间的距离, (G_p, d)为一度量空间.进一步, 由于d(x, y)≤max{d(x, z), d(y, z)}, (G_p, d)可被称为超度量空间.以x为中心的圆邻域可以表示成其中元素与x之间的距离, 即

$ {I_n}\left( x \right) = \left\{ {y \in {G_p}\left| {d\left( {x,y} \right) < {p^{ - n}}} \right.} \right\}. $

当n=0时, I_n(x)简记为I(x); 当x=0时, I_n(0) 简记为I_n.综上, I₀(0) 简记为I.定义集合E⊂G_p上的特征函数为I_E, 即

$ I\left( x \right) = \left\{ \begin{array}{l} 1,x \in E,\\ 0,x \in {G_p}\backslash E. \end{array} \right. $

正如前文所述, 可以定义G_p上的映射λ:G_p→[0, ∞), 即

$ \lambda \left( x \right) = \sum\limits_{i \in {\bf{Z}}} {{x_i}{p^{ - 1}},\forall x \in {G_p}} . $

该映射为一一映射.事实上, 若存在$\lambda \left( {{x^1}} \right) = \sum\limits_{i = J_0^1}^{{J^1}} {x_i^1{p^{-i}}} $, $\lambda \left( {{x^2}} \right) = \sum\limits_{i = J_0^2}^{{J^2}} {x_i^2{p^{-i}}} $, 且λ(x¹)=λ(x²), 由于x_i∈{0, 1, …, p-1}, (p-1)(1+p+p²)+…+p^j-1=p^j-1 < p^j, 故J¹=J², J₀¹=J₀², x_i¹=x_i², i∈Z.因此x¹=x².定义G_p上的扩张函数A:G_p→G_p, 即(Ax)_i=x_i+1, ∀x∈G_p, 其逆运算为(A^-1x)_i=x_i-1, 并且当n>0时${A^n} = \underbrace {A \circ A \circ \cdots \circ {A_n}}_n$; 当n < 0时${A^n} = \underbrace {{A^{-1}} \circ {A^{-1}} \circ \cdots \circ {A^{-1}}}_{ - n}$; 当n=0时A⁰为恒等映射A⁰x=x.

因为G_p为局部紧群, 其上的Haar测度^[2]具有正则性和平移不变性.规定∫_GpI_Idμ(x)=1, 那么就可以和实数一样定义Vilenkin群上的l次方可积的函数空间L_l(G_p).

2 多分辨分析

令尺度函数φ=I_I, 该函数具有两个重要的性质:(1) 由函数φ平移得到的函数系{φ₀, k:=φ(x⊖ λ^-1(k))}是彼此正交的; (2) 其满足双尺度方程

$ \varphi = \sum\limits_{i = 0}^{p - 1} {{\varphi _{1,i}}} . $

(1)

其中φ_{j, k}=p^2/jφ(A^jx⊖ λ^-1(k)), j∈Z, k∈N.可以证明当j固定时, φ_{j, k}是彼此正交的, 且∀j∈Z, k∈N, ∫_Gpφ_{j, k}dμ(x)=1.设函数空间V_n由{p^n/2φ(Aⁿx), p^n/2φ(Aⁿx⊖ λ^-1(1)), …}组成, 则容易得到以下结论:

(1) 对于函数f(x)∈L₂(G_p), 当且仅当f(A^-n)∈V₀时, f(x)∈V_n;

(2) 对于函数f(x)∈L₂(G_p), 当且仅当f(Aⁿ)∈V_n时, f(x)∈V₀;

(3) 函数系{φ_{n, k}}构成V_n的一组标准正交基.

同时, 空间序列V_n具有递增性质

$ \cdots \subset {V_{ - 2}} \subset {V_{ - 1}} \subset {V_0} \subset {V_1} \subset {V_2} \subset \cdots $

和逼近性质

$ \bigcap\limits_{n \in {\bf{Z}}} {{V_n} = 0} ,\bigcap\limits_{n \in {\bf{Z}}} {{V_n} = {L_2}\left( {{G_p}} \right)} . $

令小波空间W_n=V_n+1-V_n, 则由空间序列V_n的性质可知, L₂(G_p)可被分解成小波空间序列W_n的直和, 即

$ {L_2}\left( {{G_p}} \right) = \mathop \oplus \limits_{n \in {\bf{Z}}} {W_n}. $

由参考文献[5]可知, 相应的小波函数为

$ {\psi ^l} = \sum\limits_{\alpha = 0}^{p - 1} {\exp \left( {2l\alpha {\rm{\pi i/}}p} \right){\varphi _{1,\alpha }}} ,l = 1,2, \cdots ,p - 1, $

(2)

并且$\psi _{j, k}^{\left( l \right)} = {p^{-j/2}}\sum\limits_{s = 0}^{p-1} {\exp \left( {2\pi {\rm{i}}vs/p} \right){\varphi _{j + 1, pk + s}}} $构成L₂(G_p)的一组标准正交基.因此, ∀f∈L₂(G_p), f可以被唯一地表示成

$ f\overset{{{L}_{2}}\left( {{G}_{p}} \right)}{\mathop{\doteq }}\,\sum\limits_{l=1}^{p=1}{\sum\limits_{j\in \mathbf{Z},k\in \mathbf{N}}{{{a}_{j,k}}\psi _{j,k}^{\left( l \right)}}}. $

能量有限空间L₂(G_p)还可分解为如下直和

$ {L_2}\left( {{G_p}} \right) = {V_0} \oplus {W_0}, \oplus {W_1} \oplus {W_2} \oplus \cdots $

因此, ∀f∈L₂(G_p), f还可以被唯一地表示成

$ f \mathop = \limits^{{L_2}} \sum\limits_{k \in {\bf{N}}} {{a_k}{\varphi _{0,k}}} + \sum\limits_{l = 1}^{p = 1} {\sum\limits_{j,k \in {\bf{N}}} {{a_{j,k}}\psi _{j,k}^{\left( l \right)}} } . $

(3)

下面将介绍如何将Vilenkin群上的信号分解成上述形式.

3 信号的分解与重构

基于多分辨分析理论, 对于任意的f∈L₂(G_p), 都存在一个f_n∈V_n, 使得∀ε>0, |f-f_n| < ε.因此, 对f进行分解相当于把V_n中的函数f_n先分解到V_n-1和W_n-1中, 然后对V_n-1进行逐级分解.设

$ {f_n}\left( x \right) = \sum\limits_{k \in {\bf{N}}} {{a_{n,k}}\varphi \left( {{A^n}x \ominus {\lambda ^{ - 1}}\left( k \right)} \right)} , $

其分解的实质就是将表达式中的φ_{n, k}用φ_{n-1, k}与ψ_{n-1, k}^l(l=1, 2, …, p-1) 来表示, 再将φ_{n-1, k}进行类似分解.那么首先需要得到φ_{n, k}与φ_{n-1, k}和ψ_{n-1, k}^l之间的关系.

根据双尺度方程(1) 和小波函数(2) 可以得到

$ \left( {\begin{array}{*{20}{c}} 1&1&1& \cdots &1\\ 1&{\exp \left( {2{\rm{ \mathsf{ π} i/}}p} \right)}&{\exp \left( {4{\rm{ \mathsf{ π} i/}}p} \right)}& \cdots &{\exp \left( {2\left( {p - 1} \right){\rm{ \mathsf{ π} i/}}p} \right)}\\ 1&{\exp \left( {4{\rm{ \mathsf{ π} i/}}p} \right)}&{\exp \left( {8{\rm{ \mathsf{ π} i/}}p} \right)}& \cdots &{\exp \left( {4\left( {p - 1} \right){\rm{ \mathsf{ π} i/}}p} \right)}\\ 1&{\exp \left( {6{\rm{ \mathsf{ π} i/}}p} \right)}&{\exp \left( {12{\rm{ \mathsf{ π} i/}}p} \right)}& \cdots &{\exp \left( {6\left( {p - 1} \right){\rm{ \mathsf{ π} i/}}p} \right)}\\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 1&{\exp \left( {2\left( {p - 1} \right){\rm{ \mathsf{ π} i/}}p} \right)}&{\exp \left( {4\left( {p - 1} \right){\rm{ \mathsf{ π} i/}}p} \right)}& \cdots &{\exp \left( {2{{\left( {p - 1} \right)}^2}{\rm{ \mathsf{ π} i/}}p} \right)} \end{array}} \right). $

$ \left( {\begin{array}{*{20}{c}} {\varphi \left( {Ax} \right)}\\ {\varphi \left( {Ax \ominus {\lambda ^{ - 1}}\left( 1 \right)} \right)}\\ {\varphi \left( {Ax \ominus {\lambda ^{ - 1}}\left( 2 \right)} \right)}\\ {\varphi \left( {Ax \ominus {\lambda ^{ - 1}}\left( 3 \right)} \right)}\\ \vdots \\ {\varphi \left( {Ax \ominus {\lambda ^{ - 1}}\left( {p - 1} \right)} \right)} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {\varphi \left( x \right)}\\ {{\psi ^1}\left( x \right)}\\ {{\psi ^2}\left( x \right)}\\ {{\psi ^3}\left( x \right)}\\ \vdots \\ {{\psi ^{p - 1}}\left( x \right)} \end{array}} \right), $

解得

$ \left( {\begin{array}{*{20}{c}} {\varphi \left( {Ax} \right)}\\ {\varphi \left( {Ax \ominus {\lambda ^{ - 1}}\left( 1 \right)} \right)}\\ {\varphi \left( {Ax \ominus {\lambda ^{ - 1}}\left( 2 \right)} \right)}\\ {\varphi \left( {Ax \ominus {\lambda ^{ - 1}}\left( 3 \right)} \right)}\\ \vdots \\ {\varphi \left( {Ax \ominus {\lambda ^{ - 1}}\left( {p - 1} \right)} \right)} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {\frac{1}{p}}&{\frac{1}{p}}&{\frac{1}{p}}& \cdots \\ {\frac{1}{p}}&{\frac{{\exp \left( { - 2{\rm{ \mathsf{ π} i/}}p} \right)}}{p}}&{\frac{{\exp \left( { - 4{\rm{ \mathsf{ π} i/}}p} \right)}}{p}}& \cdots \\ {\frac{1}{p}}&{\frac{{\exp \left( { - 4{\rm{ \mathsf{ π} i/}}p} \right)}}{p}}&{\frac{{\exp \left( { - 8{\rm{ \mathsf{ π} i/}}p} \right)}}{p}}& \cdots \\ {\frac{1}{p}}&{\frac{{\exp \left( { - 6{\rm{ \mathsf{ π} i/}}p} \right)}}{p}}&{\frac{{\exp \left( { - 12{\rm{ \mathsf{ π} i/}}p} \right)}}{p}}& \cdots \\ \vdots & \vdots & \vdots & \vdots \\ {\frac{1}{p}}&{\frac{{\exp \left( { - 2\left( {p - 1} \right){\rm{ \mathsf{ π} i/}}p} \right)}}{p}}&{\frac{{\exp \left( { - 4\left( {p - 1} \right){\rm{ \mathsf{ π} i/}}p} \right)}}{p}}& \cdots \end{array}} \right)\left( {\begin{array}{*{20}{c}} {\varphi \left( x \right)}\\ {{\psi ^1}\left( x \right)}\\ {{\psi ^2}\left( x \right)}\\ {{\psi ^3}\left( x \right)}\\ \vdots \\ {{\psi ^{p - 1}}\left( x \right)} \end{array}} \right). $

进而可以得到

$ \left\{ \begin{array}{l} \varphi \left( {{A^n}x} \right) = \frac{1}{p}\varphi \left( {{A^{n - 1}}x} \right) + \frac{1}{p}{\psi ^1}\left( {{A^{n - 1}}x} \right) + \frac{1}{p}{\psi ^2}\left( {{A^{n - 1}}x} \right) + \cdots + \frac{1}{p}{\psi ^{p - 1}}\left( {{A^{n - 1}}x} \right),\\ \varphi \left( {{A^n}x \ominus {\lambda ^{ - 1}}\left( 1 \right)} \right) = \frac{1}{p}\varphi \left( {{A^{n - 1}}x} \right) + \frac{{\exp \left( { - 2{\rm{ \mathsf{ π} i/}}p} \right)}}{p}{\psi ^1}\left( {{A^{n - 1}}x} \right) + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{{\exp \left( { - 4{\rm{ \mathsf{ π} i/}}p} \right)}}{p}{\psi ^2}\left( {{A^{n - 1}}x} \right) + \cdots + \frac{{\exp \left( { - 2\left( {p - 1} \right){\rm{ \mathsf{ π} i/}}p} \right)}}{p}{\psi ^{p - 1}}\left( {{A^{n - 1}}x} \right),\\ \varphi \left( {{A^n}x \ominus {\lambda ^{ - 1}}\left( 2 \right)} \right) = \frac{1}{p}\varphi \left( {{A^{n - 1}}x} \right) + \frac{{\exp \left( { - 4{\rm{ \mathsf{ π} i/}}p} \right)}}{p}{\psi ^1}\left( {{A^{n - 1}}x} \right) + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{{\exp \left( { - 8{\rm{ \mathsf{ π} i/}}p} \right)}}{p}{\psi ^2}\left( {{A^{n - 1}}x} \right) + \cdots + \frac{{\exp \left( { - 4\left( {p - 1} \right){\rm{ \mathsf{ π} i/}}p} \right)}}{p}{\psi ^2}\left( {{A^{n - 1}}x} \right),\\ \varphi \left( {{A^n}x \ominus {\lambda ^{ - 1}}\left( 3 \right)} \right) = \frac{1}{p}\varphi \left( {{A^{n - 1}}x} \right) + \frac{{\exp \left( { - 6{\rm{ \mathsf{ π} i/}}p} \right)}}{p}{\psi ^1}\left( {{A^{n - 1}}x} \right) + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{{\exp \left( { - 12{\rm{ \mathsf{ π} i/}}p} \right)}}{p}{\psi ^2}\left( {{A^{n - 1}}x} \right) + \cdots + \frac{{\exp \left( { - 6\left( {p - 1} \right){\rm{ \mathsf{ π} i/}}p} \right)}}{p}{\psi ^{p - 1}}\left( {{A^{n - 1}}x} \right),\\ \vdots \\ \varphi \left( {{A^n}x \ominus {\lambda ^{ - 1}}\left( {p - 1} \right)} \right) = \frac{1}{p}\varphi \left( {{A^{n - 1}}x} \right) + \frac{{\exp \left( { - 2\left( {p - 1} \right){\rm{ \mathsf{ π} i/}}p} \right)}}{p}{\psi ^1}\left( {{A^{n - 1}}x} \right) + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{{\exp \left( { - 4\left( {p - 1} \right){\rm{ \mathsf{ π} i/}}p} \right)}}{p}{\psi ^2}\left( {{A^{n - 1}}x} \right) + \cdots + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{{\exp \left( { - 2{{\left( {p - 1} \right)}^2}{\rm{ \mathsf{ π} i/}}p} \right)}}{p}{\psi ^{p - 1}}\left( {{A^{n - 1}}x} \right). \end{array} \right. $

(4)

接着将信号f_n按照其下标k进行如下分离:

$ \begin{array}{l} {f_n}\left( x \right) = \sum\limits_{k \in {\bf{N}}} {{a_{n,k}}\varphi \left( {{A^n}x \ominus {\lambda ^{ - 1}}\left( k \right)} \right)} = \\ \;\;\;\;\;\;\;\;\;\;\;\;\sum\limits_{k \in {\bf{N}}} {{a_{n,pk}}\varphi \left( {{A^n}x \ominus {\lambda ^{ - 1}}\left( {pk} \right)} \right)} + \sum\limits_{k \in {\bf{N}}} {{a_{n,pk + 1}}\varphi \left( {{A^n}x \ominus {\lambda ^{ - 1}}\left( {pk + 1} \right)} \right)} + \cdots + \\ \;\;\;\;\;\;\;\;\;\;\;\;\sum\limits_{k \in {\bf{N}}} {{a_{n,pk}}\varphi \left( {{A^n}x \ominus {\lambda ^{ - 1}}\left( {pk + p - 1} \right)} \right)} . \end{array} $

(5)

将式(4) 中的x用x⊖ λ-1(p^1-nk)替换并代入式(5) 得

$ \begin{array}{l} {f_n}\left( x \right) = \sum\limits_{k \in {\bf{N}}} {{a_{n,pk}}\left[ {\frac{1}{p}\varphi \left( {{A^{n - 1}}x \ominus {\lambda ^{ - 1}}\left( k \right)} \right) + \frac{1}{p}{\psi ^1}\left( {{A^{n - 1}}x \ominus {\lambda ^{ - 1}}\left( k \right)} \right) + } \right.} \\ \left. {\;\;\;\;\;\;\;\;\;\;\;\;\frac{1}{p}{\psi ^2}\left( {{A^{n - 1}}x \ominus {\lambda ^{ - 1}}\left( k \right)} \right) + \cdots + \frac{1}{p}{\psi ^p} - 1\left( {{A^{n - 1}}x \ominus {\lambda ^{ - 1}}\left( k \right)} \right)} \right] + \\ \;\;\;\;\;\;\;\;\;\;\;\;\sum\limits_{k \in {\bf{N}}} {{a_{n,pk + 1}}\left[ {\frac{1}{p}\varphi \left( {{A^{n - 1}}x \ominus {\lambda ^{ - 1}}\left( k \right)} \right) + \frac{{\exp \left( { - 2{\rm{ \mathsf{ π} i}}/p} \right)}}{p}{\psi ^1}\left( {{A^{n - 1}}x \ominus {\lambda ^{ - 1}}\left( k \right)} \right) + } \right.} \\ \;\;\;\;\;\;\;\;\;\;\;\;\left. {\frac{{\exp \left( { - 4{\rm{ \mathsf{ π} i}}/p} \right)}}{p}{\psi ^2}\left( {{A^{n - 1}}x \ominus {\lambda ^{ - 1}}\left( k \right)} \right) + \cdots + \frac{{\exp \left( { - 2\left( {p - 1} \right){\rm{ \mathsf{ π} i}}/p} \right)}}{p}{\psi ^p} - 1\left( {{A^{n - 1}}x \ominus {\lambda ^{ - 1}}\left( k \right)} \right)} \right] + \\ \;\;\;\;\;\;\;\;\;\;\;\;\sum\limits_{k \in {\bf{N}}} {{a_{n,pk + 2}}\left[ {\frac{1}{p}\varphi \left( {{A^{n - 1}}x{\lambda ^{ - 1}}\left( k \right)} \right) + \frac{{\exp \left( { - 4{\rm{ \mathsf{ π} i}}/p} \right)}}{p}{\psi ^1}\left( {{A^{n - 1}}x \ominus {\lambda ^{ - 1}}\left( k \right)} \right) + } \right.} \\ \;\;\;\;\;\;\;\;\;\;\;\;\left. {\frac{{\exp \left( { - 8{\rm{ \mathsf{ π} i}}/p} \right)}}{p}{\psi ^2}\left( {{A^{n - 1}}x{\lambda ^{ - 1}}\left( k \right)} \right) + \cdots + \frac{{\exp \left( { - 4\left( {p - 1} \right){\rm{ \mathsf{ π} i}}/p} \right)}}{p}{\psi ^2}\left( {{A^{n - 1}}x{\lambda ^{ - 1}}\left( k \right)} \right)} \right] + \cdots + \\ \;\;\;\;\;\;\;\;\;\;\;\;\sum\limits_{k \in {\bf{N}}} {{a_{n,pk + p - 1}}\left[ {\frac{1}{p}\varphi \left( {{A^{n - 1}}x{\lambda ^{ - 1}}\left( k \right)} \right) + \frac{{\exp \left( { - 2\left( {p - 1} \right){\rm{ \mathsf{ π} i}}/p} \right)}}{p}{\psi ^1}\left( {{A^{n - 1}}x{\lambda ^{ - 1}}\left( k \right)} \right) + } \right.} \\ \;\;\;\;\;\;\;\;\;\;\;\;\left. {\frac{{\exp \left( { - 4\left( {p - 1} \right){\rm{ \mathsf{ π} i}}/p} \right)}}{p}{\psi ^2}\left( {{A^{n - 1}}x{\lambda ^{ - 1}}\left( k \right)} \right) + \cdots + \frac{{\exp \left( { - 2{{\left( {p - 1} \right)}^2}{\rm{ \mathsf{ π} i}}/p} \right)}}{p}{\psi ^{p - 1}}\left( {{A^{n - 1}}x{\lambda ^{ - 1}}\left( k \right)} \right)} \right] = \\ \;\;\;\;\;\;\;\;\;\;\;\;\sum\limits_{k \in {\bf{N}}} {\left[ {\left( {{a_{n,pk}} + {a_{n,pk + 1}}\exp \left( { - 2{\rm{ \mathsf{ π} i}}/p} \right) + {a_{n,pk + 2}}\exp \left( { - 4{\rm{ \mathsf{ π} i}}/p} \right) + \cdots + } \right.} \right.} \\ \;\;\;\;\;\;\;\;\;\;\;\;\left. {\left. {{a_{n,pk + p - 1}}\exp \left( { - 2\left( {p - 1} \right){\rm{ \mathsf{ π} i}}/p} \right)} \right)/p} \right]{\psi ^1}\left( {{A^{n - 1}}x \ominus {\lambda ^{ - 1}}\left( k \right)} \right) + \\ \;\;\;\;\;\;\;\;\;\;\;\;\sum\limits_{k \in {\bf{N}}} {\left[ {\left( {{a_{n,pk}} + {a_{n,pk + 1}}\exp \left( { - 4{\rm{ \mathsf{ π} i}}/p} \right) + {a_{n,pk + 2}}\exp \left( { - 8{\rm{ \mathsf{ π} i}}/p} \right) + \cdots + } \right.} \right.} \\ \;\;\;\;\;\;\;\;\;\;\;\;\left. {\left. {{a_{n,pk + p - 1}}\exp \left( { - 4\left( {p - 1} \right){\rm{ \mathsf{ π} i}}/p} \right)} \right)/p} \right]{\psi ^2}\left( {{A^{n - 1}}x \ominus {\lambda ^{ - 1}}\left( k \right)} \right) + \cdots + \\ \;\;\;\;\;\;\;\;\;\;\;\;\sum\limits_{k \in {\bf{N}}} {\left[ {\left( {{a_{n,pk}} + {a_{n,pk + 1}}\exp \left( { - 2\left( {p - 1} \right){\rm{ \mathsf{ π} i}}/p} \right) + {a_{n,pk + 2}}\exp \left( { - 4\left( {p - 1} \right){\rm{ \mathsf{ π} i}}/p} \right) + \cdots + } \right.} \right.} \\ \;\;\;\;\;\;\;\;\;\;\;\;\left. {\left. {{a_{n,pk + p - 1}}\exp \left( { - 2{{\left( {p - 1} \right)}^2}{\rm{ \mathsf{ π} i}}/p} \right)} \right)/p} \right]{\psi ^2}\left( {{A^{n - 1}}x \ominus {\lambda ^{ - 1}}\left( k \right)} \right) = \\ \;\;\;\;\;\;\;\;\;\;\;\;{f_{n - 1}} + g_{n - 1}^1 + g_{n - 1}^2 + \cdots + g_{n - 1}^{p - 1}. \end{array} $

(6)

上式中第一项是f_n在V_n-1中的分量, 后p-1项是在W_n-1中的分量.由此可以得到下面的分解定理.

定理1  (分解定理)假设f_l∈V_l, 并且f_l可以表示为

$ {f_l}\left( x \right) = \sum\limits_{k \in {\bf{N}}} {{a_{l,k}}\varphi \left( {{A^n}x \ominus {\lambda ^{ - 1}}\left( k \right)} \right)} , $

则f_l(x)可以分解为

$ {f_l}\left( x \right) = {f_{l - 1}}\left( x \right) + g_{n - 1}^1 + g_{n - 1}^2 + \cdots + g_{n - 1}^{p - 1}. $

其中

$ \begin{array}{*{20}{c}} {{f_{l - 1}}\left( x \right) = \sum\limits_{k \in {\bf{N}}} {{a_{n - 1,k}}\varphi \left( {{A^{l - 1}}x \ominus {\lambda ^{ - 1}}\left( k \right)} \right)} \in {V_{l - 1}},}\\ {g_{l - 1}^v = \sum\limits_{k \in {\bf{N}}} {a_{l - 1,k}^v\psi \left( {{A^{l - 1}}x \ominus {\lambda ^{ - 1}}\left( k \right)} \right)} \in {W_{l - 1}},v = 1,2, \cdots ,p - 1,} \end{array} $

满足

$ \begin{array}{l} {a_{n - 1,k}} = \frac{{{a_{n,pk}} + {a_{n,pk + 1}}\exp \left( { - 2{\rm{\pi i}}/p} \right) + {a_{n,pk + 2}}\exp \left( { - 4{\rm{\pi i}}/p} \right) + \cdots + {a_{n,pk + 2 - 1}}\exp \left( { - 2\left( {p - 1} \right){\rm{\pi i}}/p} \right)}}{p},\\ a_{l - 1,k}^v = \frac{{{a_{n,pk}} + {a_{n,pk + 1}}\exp \left( { - 2v{\rm{\pi i}}/p} \right) + {a_{n,pk + 2}}\exp \left( { - 4v{\rm{\pi i}}/p} \right) + \cdots + {a_{n,pk + p - 1}}\exp \left( { - 2v\left( {p - 1} \right){\rm{\pi i}}/p} \right)}}{p}. \end{array} $

按照上面的分解算法, 一直分解下去就可以得到如同式(3) 的表达形式.

特别地, 当p=2时, G₂为康托尔二元群, 与尺度函数相对应的小波函数只有ψ¹.若f_l∈V_l, 并且

$ {f_l} = \sum\limits_{k \in {\bf{N}}} {{a_{l,k}}\varphi \left( {{A^l}x \ominus {\lambda ^{ - 1}}\left( k \right)} \right)} , $

则f_l可以分解为f_l=f_l-1+g_l-1.其中

$ \begin{array}{l} {f_{l - 1}} = \sum\limits_{k \in {\bf{Z}}} {{a_{l - 1,k}}\varphi \left( {{A^{l - 1}}x \ominus {\lambda ^{ - 1}}\left( k \right)} \right.} \in {V_{l - 1}},\\ {g_{l - 1}} = \sum\limits_{k \in {\bf{Z}}} {a_{l - 1,k}^1{\psi ^1}\left( {{A^{l - 1}}x \ominus {\lambda ^{ - 1}}\left( k \right)} \right.} \in {W_{l - 1}}, \end{array} $

表达式中的系数分别为

$ {a_{l - 1,k}} = \frac{{{a_{l,2k}} + {a_{l,2k + 1}}}}{2},a_{l - 1,k}^1 = \frac{{{a_{l,2k}} - {a_{l,2k + 1}}}}{2}. $

在根据需要对信号进行分解、处理后, 需要对信号进行重新恢复, 也就是所谓的信号的重构.下面讨论信号的重构问题.

定理2  (重构定理)假设

$ \begin{array}{*{20}{c}} {{f_l}\left( x \right) = {f_0}\left( x \right) + g_0^1\left( x \right) + g_0^2\left( x \right) + \cdots + g_0^{p - 1}\left( x \right) + g_1^1\left( x \right) + \cdots + }\\ {g_1^{p - 1}\left( x \right) + \cdots + g_{l - 1}^1\left( x \right) + \cdots + g_{l - 1}^{p - 1}\left( x \right),} \end{array} $

其中

$ \begin{array}{*{20}{c}} {{f_0}\left( x \right) = \sum\limits_{k \in {\bf{Z}}} {{a_{0,k}}\varphi \left( {x \ominus {\lambda ^{ - 1}}\left( k \right)} \right)} \in {V_0},}\\ {g_j^v\left( x \right) = \sum\limits_{k \in {\bf{Z}}} {a_{j,k}^v\varphi \left( {x \ominus {\lambda ^{ - 1}}\left( k \right)} \right)\left( {v = 1,2, \cdots ,p - 1} \right)} \in {W_j}.} \end{array} $

则f_l(x)可以表示成

$ {f_l}\left( x \right) = \sum\limits_{k \in {\bf{Z}}} {{a_{l,k}}\varphi \left( {x \ominus {\lambda ^{ - 1}}\left( k \right)} \right)} \in {V_l}. $

其中系数a_{l, k}可以由下面的公式递推得到:

$ {a_{s,n}} = \left\{ \begin{array}{l} {a_{s - 1,k}} + \sum\limits_{v = 1}^{p - 1} {a_{s - 1,k}^v} ,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;n = pk,\\ {a_{s - 1,k}} + \sum\limits_{v = 1}^{p - 1} {a_{s - 1,k}^v\exp \left( {2v{\rm{\pi i}}/p} \right)} ,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;n = pk + 1,\\ {a_{s - 1,k}} + \sum\limits_{v = 1}^{p - 1} {a_{s - 1,k}^v\exp \left( {4v{\rm{\pi i}}/p} \right)} ,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;n = pk + 2,\\ \vdots \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \vdots \\ {a_{s - 1,k}} + \sum\limits_{v = 1}^{p - 1} {a_{s - 1,k}^v\exp \left( {2\left( {p - 1} \right)v{\rm{\pi i}}/p} \right)} ,\;\;\;\;\;n = pk + p - 1. \end{array} \right. $

证明  只需要证明由f_l-1∈V_l-1和g_l-1^v∈W^l-1可以得到f_l∈V_l的表达式即可.设f_l=f_l-1(x)+g_l-1¹(x)+g_l-1²(x)+…+g_l-1^p-1(x), 其中

$ \begin{array}{*{20}{c}} {{f_{l - 1}}\left( x \right) = \sum\limits_{k \in {\bf{Z}}} {{a_{l - 1,k}}\varphi \left( {{A^{l - 1}}x \ominus {\lambda ^{ - 1}}\left( k \right)} \right)} \in {V_{l - 1}},}\\ {g_{l - 1}^v = \sum\limits_{k \in {\bf{Z}}} {a_{l - 1,k}^v{\psi ^v}\left( {{A^{l - 1}}x \ominus {\lambda ^{ - 1}}\left( k \right)} \right) \in {W_{l - 1}},v = 1,2, \cdots ,p - 1.} } \end{array} $

由式(1)~(2) 可得

$ \left( {\begin{array}{*{20}{c}} {\varphi \left( {{A^{n - 1}}x} \right)}\\ {{\psi ^1}\left( {{A^{n - 1}}x} \right)}\\ {{\psi ^2}\left( {{A^{n - 1}}x} \right)}\\ {{\psi ^3}\left( {{A^{n - 1}}x} \right)}\\ \vdots \\ {{\psi ^p} - 1\left( {{A^{n - 1}}x} \right)} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 1&1&1& \cdots &1\\ 1&{\exp \left( {2{\rm{ \mathsf{ π} i/}}p} \right)}&{\exp \left( {4{\rm{ \mathsf{ π} i/}}p} \right)}& \cdots &{\exp \left( {2\left( {p - 1} \right){\rm{ \mathsf{ π} i/}}p} \right)}\\ 1&{\exp \left( {4{\rm{ \mathsf{ π} i/}}p} \right)}&{\exp \left( {8{\rm{ \mathsf{ π} i/}}p} \right)}& \cdots &{\exp \left( {4\left( {p - 1} \right){\rm{ \mathsf{ π} i/}}p} \right)}\\ 1&{\exp \left( {6{\rm{ \mathsf{ π} i/}}p} \right)}&{\exp \left( {12{\rm{ \mathsf{ π} i/}}p} \right)}& \cdots &{\exp \left( {6\left( {p - 1} \right){\rm{ \mathsf{ π} i/}}p} \right)}\\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 1&{\exp \left( {2\left( {p - 1} \right){\rm{ \mathsf{ π} i/}}p} \right)}&{\exp \left( {4\left( {p - 1} \right){\rm{ \mathsf{ π} i/}}p} \right)}& \cdots &{\exp \left( {2{{\left( {p - 1} \right)}^2}{\rm{ \mathsf{ π} i/}}p} \right)} \end{array}} \right) $

$\left( {\begin{array}{*{20}{c}} {\varphi \left( {{A^n}x} \right)}\\ {\varphi \left( {{A^n}x \ominus {\lambda ^{ - 1}}\left( 1 \right)} \right)}\\ {\varphi \left( {{A^n}x \ominus {\lambda ^{ - 1}}\left( 2 \right)} \right)}\\ {\varphi \left( {{A^n}x \ominus {\lambda ^{ - 1}}\left( 3 \right)} \right)}\\ \vdots \\ {\varphi \left( {{A^n}x \ominus {\lambda ^{ - 1}}\left( {p - 1} \right)} \right)} \end{array}} \right)$

(7)

把式(7) 代入f_l的表达式中得

$ \begin{array}{l} {f_l}\left( x \right) = \sum\limits_{k \in {\bf{N}}} {{a_{l - 1,k}}\varphi \left( {{A^{l - 1}}x \ominus {\lambda ^{ - 1}}\left( k \right)} \right)} + \sum\limits_{v = 1}^{p - 1} {\sum\limits_{k \in {\bf{N}}} {a_{l - 1,k}^v{\psi ^v}\left( {{A^{l - 1}}x \ominus {\lambda ^{ - 1}}\left( k \right)} \right)} } = \\ \;\;\;\;\;\;\;\;\;\;\;\sum\limits_{k \in {\bf{N}}} {{a_{l - 1,k}}\left[ {\varphi \left( {{A^{l - 1}}x \ominus {\lambda ^{ - 1}}\left( {pk} \right)} \right) + \varphi \left( {{A^{l - 1}}x \ominus {\lambda ^{ - 1}}\left( {pk + 1} \right)} \right) + \cdots + } \right.} \\ \;\;\;\;\;\;\;\;\;\;\;\left. {\varphi \left( {{A^{l - 1}}x \ominus {\lambda ^{ - 1}}\left( {pk + p - 1} \right)} \right)} \right] + \sum\limits_{v = 1}^{p - 1} {\sum\limits_{k \in {\bf{N}}} {a_{l - 1,k}^v\left[ {\varphi \left( {{A^l}x \ominus {\lambda ^{ - 1}}\left( {pk} \right)} \right) + } \right.} } \\ \;\;\;\;\;\;\;\;\;\;\;\exp \left( {2v{\rm{\pi i/}}p} \right)\varphi \left( {{A^l}x \ominus {\lambda ^{ - 1}}\left( {pk + 1} \right)} \right) + \cdots + \\ \;\;\;\;\;\;\;\;\;\;\;\exp \left( {2\left( {p - 1} \right)v{\rm{\pi i/}}p} \right)\varphi \left( {{A^l}x \ominus {\lambda ^{ - 1}}\left( {pk + p - 1} \right)} \right) = \\ \;\;\;\;\;\;\;\;\;\;\;\sum\limits_{k \in {\bf{N}}} {\left\{ {\left[ {{a_{l - 1}},k + \sum\limits_{v = 1}^{p - 1} {a_{l - 1,k}^v} } \right]\varphi \left( {{A^l}x \ominus {\lambda ^{ - 1}}\left( {pk} \right)} \right) + } \right.} \\ \;\;\;\;\;\;\;\;\;\;\;\left[ {{a_{l - 1,k}} + \sum\limits_{v = 1}^{p - 1} {a_{l - 1,k}^v\exp \left( {2v{\rm{\pi i/}}p} \right)} } \right]\varphi \left( {{A^l}x \ominus {\lambda ^{ - 1}}\left( {pk + 1} \right)} \right) + \cdots + \\ \;\;\;\;\;\;\;\;\;\;\;\left. {\left[ {{a_{l - 1,k}} + \sum\limits_{v = 1}^{p - 1} {a_{l - 1,k}^v\exp \left( {2\left( {p - 1} \right)v{\rm{\pi i/}}p} \right)} } \right]\varphi \left( {{A^l}x \ominus {\lambda ^{ - 1}}\left( {pk + p - 1} \right)} \right)} \right\} = \\ \;\;\;\;\;\;\;\;\;\;\;\sum\limits_{k \in {\bf{N}}} {{a_{l,n}}\varphi \left( {{A^l}x \ominus {\lambda ^{ - 1}}\left( n \right)} \right)} . \end{array} $

其中

$ {a_{l,n}} = \left\{ \begin{array}{l} {a_{l - 1,k}} + \sum\limits_{v = 1}^{p - 1} {a_{l - 1,k}^v} ,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;n = pk\\ {a_{l - 1,k}} + \sum\limits_{v = 1}^{p - 1} {a_{l - 1,k}^v\exp \left( {2v{\rm{\pi i/}}p} \right)} ,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;n = pk + 1,\\ {a_{l - 1,k}} + \sum\limits_{v = 1}^{p - 1} {a_{l - 1,k}^v\exp \left( {4v{\rm{\pi i/}}p} \right)} ,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;n = pk + 2,\\ \vdots \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \vdots \\ {a_{l - 1,k}} + \sum\limits_{v = 1}^{p - 1} {a_{l - 1,k}^v\exp \left( {2\left( {p - 1} \right)v{\rm{\pi i/}}p} \right)} ,\;\;\;\;\;\;n = pk + p - 1. \end{array} \right. $

当p=2时, 若已知f_l=f₀+g₀¹+g₁¹+g₂¹+…+g_l-1¹, 其中

$ \begin{array}{l} {f_0}\left( x \right) = \sum\limits_{k \in {\bf{N}}} {{a_{0,k}}\left( {x \ominus {\lambda ^{ - 1}}\left( k \right)} \right) \in {V_0}} ,\\ g_j^1\left( x \right) = \sum\limits_{k \in {\bf{N}}} {a_{j,k}^1\left( {x \ominus {\lambda ^{ - 1}}\left( k \right)} \right) \in {W_j}} . \end{array} $

则有重构定理可以得到

$ {f_l}\left( x \right) = \sum\limits_{k \in {\bf{N}}} {{a_{l,k}}\left( {{A^l}x \ominus {\lambda ^{ - 1}}\left( k \right)} \right) \in {V_l}} , $

其系数a_{l, k}可以由

$ {a_{s,n}}\left\{ \begin{array}{l} {a_{s - 1,k}} + a_{s - 1,k}^1,n = 2k,\\ {a_{s - 1,k}} - a_{s - 1,k}^1,n = 2k + 1. \end{array} \right. $

得到

3 结束语

重新定义了Vilenkin群上的二元运算, 根据经典的尺度函数和小波函数建立了Vilenkin群上的多分辨分析, 并得到了信号的分解和重构定理.使得信号能够通过分解展现出一些局部信息, 帮助人们更好地理解和处理信号, 同时为恢复处理过后的信号提供了精确的算法.文中只采用了一种经典的小波函数, 用以对信号进行分解与重构的算法分析.该理论还可辩证地运用到其他多分辨分析系统中, 以更好地分析和处理信号.