第七章小波与信号处理b.doc

文档介绍：§7-4. Signal Representation: A Multiresolution Analysis
Vn and Properties
Vn=the set of function who are constant on , kÎZ
Ÿ Properties
Wn and Properties
Ÿ Properties
不仅是简单的叠加,而是重组,合成向量
Daubechies小波产生的Wn比Haar的要好
Note:Vn and Vm are not orthonormal;
But
position of Signals
From the second property on last section, one concludes that
is an orthonormal basis in L2(R). Hence for any f(t)Î L2(R),
where (Haar basis)
For many practical signals, there often are many small wavelet coefficients. So that the signal can be represented with trancated wavelet series with fewer terms than its Fourier series requires.
[see the above example on s(t)=exp(-10t)sin(100t)]
Ÿ Haar Basis for L2[0,1)
The Haar basis yn,k given on is for L2(R). They can be used to defined periodic wavelets:

Theorem Functions 1 and with n³0, k = 0,1, ...,2n-1 form an orthonormal basis for L2[0,1).
Ÿ Similarly, periodic scaling functions can be generated as
for n³0, and k = 0,1, ...,2n-1.
Haar Transform
Consider a signal . We want to pose f(t) into items in Vn-1 and Wn-1. This is to say: Given , we want pute and such that

where
ŸNote forms an orthonormal basis for Vn. Therefore,
The DE and WE lead to a beautiful set of relations between these coefficients.
Recall

Þ (平移)


Þ (伸缩)


Þ (长度归一化)
Þ

Þ 故有如下迭代公式
Now we obtain
1989年给出的多分辨率分析的方程(MRA)或为
W
此处的W为正交矩阵,即(MRA)矩阵形式,它阐明了正交双方的可逆计算。
A Filter-Bank Implementation of MRA
If we refer index 2k+1 as “the present time instant”, then 2k is an index representing the “immediate past”. So, the equations (MRA see the last section) can be implemented as follows:
The MRA can be carried out with n levels for signals of length N=2n.
This position process is “reversible”:
One can use to perfectly reconstruct the original signal .
Again, we use the (DE) and (WE) to plish the reconstruction:
If k=even:(比较相同基函数的系数应相等)
If k=odd:
此处相当于补零形