832教學(xué)課件第三節(jié)小波變換與多分辨分析

1、數(shù) 字 信 號(hào) 處 理DigitalSignalProcessing主講人：陳后金電子信息小波變換與多分辨分析u 信號(hào)短時(shí)Fourier變換u 小波展開(kāi)與小波變換u 小波變換與多分辨分析u 小波變換分解與重構(gòu)算法u 基于小波變換的信號(hào)處理u 利用實(shí)現(xiàn)信號(hào)的小波分析小波變換與多分辨分析u 信號(hào)空間(signal space)u 尺度函數(shù)(scaling function)j(t)u 小波函數(shù)(wavelet function)y (t)u 多分辨分析(Multiresolution Analysis, MRA)u 尺度函數(shù)系數(shù)與小波函數(shù)系數(shù)小波函數(shù)(wavelet function)y (t)根

2、據(jù)信號(hào)空間的概念，由尺度函數(shù)j(t) 可以定義其對(duì)應(yīng)的小波函數(shù)y(t)，再由小波函數(shù)y(t)經(jīng)過(guò)尺度展縮與 平移得到小波信號(hào)yj,k(t)，即y j,k (t)j, k Zj (t)y (t)y j,k (t) = 2 j / 2y (2 j t - k )小波函數(shù)(wavelet function)y (t)小波信號(hào)yj,k(t)設(shè)計(jì)為尺度信號(hào)jj,k(t)的正交信號(hào)，即存在j,l (t) = j j,k (t) yj j,k (t),yj,l (t)dt = 0j, k,l ZW j = Spany j,k (t)kVjWjV j= Spanj j,k (t)k正交和IV = 0WWjj=

3、 VWVj +1jjUV = Vj +1jj例：Haar小波函數(shù)yH(t)。因?yàn)閖H(t) 與yH(t)正交，所以yH(t)應(yīng)為yH(t)jH(t)11t11/20-1t101/2jH (t), y H (t) = jH (t) y H (t)dt = 0function)y (t)小波函數(shù)(waveletV1 V3V2V0V1 = V0 W0V2 = V1 W1=V0 W0MW1W2V0W0W1L2 =V W W WLV0 W0 W1 W2 0012小波函數(shù)(wavelet function)y (t)x(t) = cj0 ,kj j0 ,k (t) + d j,kyj,k (t)j= j0

4、kk初始尺度 j=j0= VjWj00Wj +1 Wj +2 Wj +3000LL2信號(hào)x(t)可由小波信號(hào)和尺度信號(hào)共同表達(dá)=LW-WW W WLL2初始尺度 j = -12012信號(hào)x(t)也可完全由小波信號(hào)表達(dá)小波函數(shù)(wavelet function)y (t)x(t) = cj0 ,kj j0 ,k (t) + d j,kyj,k (t)j= j0kk通過(guò)將信號(hào)x(t)展開(kāi)為尺度信號(hào)jj,k(t)和小波信號(hào)yj,k(t)線 性表示，可以更有效地表達(dá)信號(hào)x(t)中的不同分量，有利于信號(hào)的分析與處理。尺度信號(hào)jj,k(t)小波信號(hào)yj,k(t)表示粗略信息(coarse informat

5、ion)信號(hào)x(t)表示精細(xì)信息(detail information)小波函數(shù)(wavelet function)y (t)x1(t)利用Haar小波函數(shù)和尺度函數(shù)表示信號(hào)644V = V W22100t0231C0(t)x1(t)=C0(t)+D0(t)x1(t)=3j(t)+6j(t-1)+3j(t-2)-y(t)+y(t-2)633t231D0(t) 1 t321-1小波函數(shù)(wavelet function)y (t)y (t) W0y (t) VW V101由于小波函數(shù)y(t)隸屬于由尺度信號(hào)j(2t-k)張成的信號(hào)空 間V1，表明y(t)可以由j(2t-k)線性表達(dá)，這就是小波函

6、數(shù)y(t) 的MRA方程：y (t) = h1k 2j (2t - k )kh1k稱為小波函數(shù)系數(shù)(wavelet function coefficient)。小波函數(shù)(wavelet function)y (t)根據(jù)MRA方程計(jì)算Haar小波對(duì)應(yīng)的h1kjH(2t)y (t) = h1k 2j (2t - k )tky H (t) = jH (2t) - jH (2t -1)01/21y (t)H112j2j=(2t) -(2t -1)1HH22t101/211h10 =-h11 =22小波函數(shù)(wavelet function)y (t)若尺度函數(shù)j(t)與小波函數(shù)y(t)滿足正交性，則小

7、波函數(shù)系數(shù)h1k與尺度函數(shù)系數(shù)h0k滿足h k = (-1)k h 1- k 10當(dāng)h0k為有限長(zhǎng)序列，且長(zhǎng)度N為偶數(shù)時(shí)，則有h k = (-1)k h N - 1 - k 10多分辨分析(MRA)= VWWWWLL2j0 +1j0 +2j0 +3j0j0x(t) = cj0 ,kj j0 ,k (t) + d j ,kyj ,k (t)j = j0kk對(duì)應(yīng)信號(hào)x(t)中的粗略(coarse)信息由低分辨率的尺度信號(hào)jj ,k(t)表達(dá)0對(duì)應(yīng)信號(hào)x(t)中的精細(xì)(fine)信息由高分辨率的小波信號(hào)yj,k(t)(jj0)表達(dá) d j ,ky j ,k (t)j = j0kcj0 ,kj j0

8、,k (t)k多分辨分析(MRA)展開(kāi)系數(shù)cjk(cj,k)反映了信號(hào)x(t)中的低頻分量的分布情況，而一系列展開(kāi)系數(shù)dj k (dj,k)反映了信號(hào)x(t)中的高頻分量的分布情況，這些展開(kāi)系數(shù)就是信號(hào)的離散小波變換DWT。=LW-WW W W LL2-12012x(t) = d j,ky j,k (t)kj=-這表明信號(hào)x(t)也可以完全由小波信號(hào)表達(dá)。多分辨分析(MRA)1100-1-1k100200300400100200300400jDoppler信號(hào)c k (t)00,k1100-1100200300400-1d k y100200300400(t)y 0,k (t) d0k 11,

9、kkk1100-1-1100k200300400k100200300400d2k y 2,k (t)d3k y 3,k (t) 多分辨分析(MRA)當(dāng)尺度函數(shù)和小波函數(shù)為正交規(guī)范基時(shí)，信號(hào)的小波展開(kāi)系數(shù)cjk和djk由內(nèi)積計(jì)算c j k = c j,k = x(t),j j,k (t) = x(t) j j,k (t)dt | j(t) |2 |y (t) |2dt = 1d j k = d j,k = x(t),y j,k (t) = x(t) ydt = 1j,k (t)dt| x(t) |2 dt = | ck |2 + | d k |2j0jkj = j0k信號(hào)的DWT滿足Parseval能量守恒尺度函數(shù)系數(shù)h0k與小波函數(shù)系數(shù) h1k的特性1. 若j(t) = h0k 2j(2t - k ) ，并且 j(t)dt = 1kh0k =k2. 若 y (t)dt = 0, 則有h1k = 0k2尺度函數(shù)系數(shù)h0k與小波函數(shù)系數(shù) h1k的特性3. 若實(shí)現(xiàn)j(t)的正交性 j(t) j(t - k )dt = d k h0n h0n - 2k = d k 4. 若實(shí)現(xiàn)y(t)的正交性 y (t) y (t - k )dt = d k