1、 “小波分析”是分析原始信号各种变化的特性，进一步用于数据压缩、噪声去除、特征选择等。例如歌唱信号：是高音还是低音，发声时间长短、起伏、旋律等。从平稳的波形发现突变的尖峰。小波分析是利用多种“小波基函数”对“原始信号”进行分解。时间A时间B参考：M.Vetterli,”Wavelets and Subband Coding“,Prentice Hall PTR,1995 p.11 MOMMOwf,log2傅里叶变换(Fourier)基小波基时间采样基t(t)(-tj-defFdeFf-tj)(21(t)tb)-(t(t)b),(tj-degfWFgtjbb)-(t(t)egg,令：(t)(t)
2、,t(t)(t)b),(b-b,ggfdgfWF则：t|(t)|tt|(t)|)t-(t|(t)1D2222022tdgdgg|dGdGG|2222022|)(|21|)(|)-(|)(1D0(t)t|t|glim1t|(t)|)(|21|(t)|222dgGg0t|(t)|t(t)1t220dg|g|0|)(|)(1220dG|G|d|G|dg|2-222t)(21t|(t)tDD)(j(t)(t|(t)221t(t)t22Ggdg|d|g|2t(t)(t)t|dgg|22(t)t21|dg|22-2t(t)21|(t)t21dgg41海森堡测不准原理为某一常数a ,(t)a(t)tgg为一
3、个解。21(t)222t-egt)ab-t(t)|a|b)(a,21-dgffTgt)ab-t(t)|a|b)(a,(21-dffW本小波小波原型或母小波或基：(t)小波函数，简称小波：Rb 0;a R,a ),ab-t(|a|(t)1/2-ba,尺度伸缩参数时间平移参数归一化因子(t)(t),t(t)(t)b)(a,(ba,ba,fdffW一般可以简记为：(t)(t),t(t)(t)b)(a,(ba,ba,fdffW)(),(21)()(21b)(a,(ba,ba,FdFfW others 01t1/2 1,-1/2t0 1,(t),2t-2412)t-(132(t)/e2-ttj2(t)/
4、ee(t)(t),t(t)(t)b)(a,(ba,ba,fdffWbaa(t)b)(a,(C(t)2ba,1-ddfWf d|2)(C 其中：(t)(t),Cbab)(a,b)()(a,(a Parseval(R)L(t)(t)-2-2gfddgWfWgf 定理证明：利用，、假设-1/2-bj-tj-1/2-ba,)(a|at)ab-t(|a|)(|ededF|FffW)(a)(2|a)(),(21(t)(t),b)(a,(-1/2ba,ba,baa)(tb)(a,(C)(t ),t-(t(t)20ba,1-00ddfWfg 代入有：令d|2)(C-0t(t)(0)d即：(t)(t),t(t)
5、(t)b)(a,(ba,ba,fdffWbaa(t)b)(a,(C(t)2ba,1-ddfWf d|)()(C 其中：互为对偶关系k)-t(a|a|(t)(t)-j0-j/20ka,akj,j0j0小波函数：(t)(t),)ka,)(a(C kj,j0j0kj,ffW变换系数：2jk2kj,2|B,|AfffZm l,k,j,mk,j,lml,kj,kj,kj,，有：的对偶此时存在jkkj,kj,(t),(t)(t)fff：有唯一的小波级数展开使得任意的分析小波合成小波2jk2kj,|A,ffjkkj,kj,-1(t),A(t)ff分析小波的对偶是它本身，类似于正交变换。)21-,23()21-,23(-1)(0,321eee231j2j23,|vevv有：容易证明对于任意矢量。但他们不是标准正交基jkkj,kj,(t),(t)ff2mnnm,kj,2kj,|,|2jmknknjmnm,kj,4kj,|,|时时jkkj,kj,(t)C(t)f重构信号：(t)(t),C kj,kj,f变换系数：k)-t(22(t)-j-j/2kj,
