Яковлев А.Н. Основы Вейвлет-преобразования Сигналов

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<ul><li><p>.. </p><p> - </p><p> - </p><p> 2003 </p></li><li><p> 2</p><p> 621.372(075.8)+004.92(075.8) 474 </p><p>: </p><p> - </p><p> () </p><p>- . , . .. (), . . , . .. () </p><p> () </p><p> .. </p><p> 474 -: . . : - , 2003. 104 . </p><p>ISBN 5-7782-0405-1 </p><p> - ( 1), - ( 2) , - ( 3) . - , -. - Wavelet Toolbox MATLAB 6. </p><p> , , , , , . </p><p> 621.372(075.8)+004.92(075.8) </p><p>ISBN 5-7782-0405-1 , 2003 </p><p> . . , 2003 </p></li><li><p> 3</p><p>50- () </p><p> - -, , - (), . </p><p> (wavelet) (Grossmann) (Morlet) 80- XX -. , (Dobechies), (Meyer), (Mallat), (Farge), (Chui) . </p><p> - () (t x), - (-/). . - () () - , - . - , ( ) - . </p><p> , , , , - , - , -</p></li><li><p> 4</p><p> , , , , , , , . </p><p> . - - , . </p><p> , - (), Mathcad, MATLAB, Mathematica. JPEC-200, MPEG-4 Corel DRAW 9/10 , , , , . -, Analog Devices - ADV6xx (ADV601, ADV601LC, ADV611, ADV612), - . </p><p> . - [15, 19, 20, 26]. </p><p> . [13], [4], . [5] . [6], - [1014]. ( 100 300 ), . </p><p> - , - . </p><p> [7], .. .. 2002 . ( 5000 ., 38 .., 160 .), . - - , MATLAB </p></li><li><p> 5</p><p>6.0/6.1. [8] ( 3000, 28 .., 270 .) , -, Wavelet Toolbox, Wavelet Extension Pack Wavelet Explorer, - () MATLAB 6.0/6.1, Mathcad 2001 Mathe-matica 4. , .. - 20012002 . </p><p> - - , Wavelet Seminar () . - - - , -, . </p><p> , - [22, 25]. </p><p> - , . - , , . </p><p> - . , - (, , ) -. </p><p> , - , , -, , - , , . </p><p> - . . .. . . . . .. , .. MATLAB. </p></li><li><p> 6</p><p> 1 </p><p> - </p><p>1.1. [13, 14], ( )S t , -</p><p> 2</p><p>1</p><p>2[ ( )]t</p><p>tS t dt &lt; , (1.1) </p><p> { ( )}n t , .. 0 0</p><p>0( ) ( ) .... ( ) ... ( )n n n n</p><p>nS t C t C t C t</p><p>== + + + = , (1.2) </p><p> 2</p><p>12</p><p>1 ( ) ( )t</p><p>n ntn</p><p>C S t t dt= , (1.3) </p><p> 2</p><p>1</p><p>2 2|| || ( )t</p><p>n nt</p><p>t dt = (1.4) , ( )n t . , </p></li><li><p> 7</p><p> 1 2[ , ]t t - </p><p> 2</p><p>1</p><p>2|| ||( ) ( )0</p><p>tn</p><p>k nt</p><p>t t dt = </p><p>, ,, .k nk n= (1.5) </p><p> ( )n t , ( 2|| || 1n = ), (), { }( )n t -. , - . </p><p> (1.2), nC (1.3), . </p><p> ( )n nC t , (1.2), - ( )S t , - 0{ ,.., ,..}nC C . 0{ ,.., ,..}nC C - , , - (. 1.1). </p><p> ( )S t -</p><p> nC ( -) (1.3). </p><p> (1.2) () - N </p><p> 0 00</p><p>( ) ( ) ... ( ) ( )N</p><p>N N n nn</p><p>S t C t C t C t=</p><p>= + + = . (1.2) : { }( )n t N (-), , = </p><p>2</p><p>1</p><p>2( ) ( )</p><p>t</p><p>tS t S t dt =</p><p>2</p><p>1</p><p>2</p><p>0( ) ( )</p><p>t N</p><p>n nnt</p><p>S t C t dt=</p><p> . (1.6) </p><p>nC</p><p>0C1C 2C iC</p><p>0 1 2 i n </p><p>. 1.1 </p></li><li><p> 8</p><p> , - N . (1.2) - . </p><p> / = , (1.7) ( 1 ), - ( )S t , .. </p><p> 2</p><p>1</p><p>2 2 [ ( )]t</p><p>tS S t dt= = . (1.8) </p><p> (1.8) (1.2) </p><p> [ ]21</p><p>2 ( )t</p><p>tS t dt= 22</p><p>0n n</p><p>nC</p><p>== , (1.9) </p><p> { }( )n t 2</p><p>0n</p><p>nC</p><p>== . </p><p>, 2 1T t t= </p><p> [ ]21</p><p>2 2p</p><p>0</p><p>1 1( )t</p><p>n nnt</p><p>P S t dt C </p><p>== = = . (1.10) </p><p> (1.9) (1.10) . </p><p> -. . </p><p> , - , . -, ; . -, - . - , [13, 14]. </p></li><li><p> 9</p><p> , . - , , , . - , , [14]. - [13, 14]. </p><p> , . </p><p>1.2. . wavelet ( ondelette) </p><p> () . - - : , , - , . </p><p>- () </p><p> 1( )abt bt</p><p>aa = , (1.11) </p><p> () ( )t , ( b ) ( a ) (. 1.2). 1/ a - a . </p><p>, a b ( )ab t , </p><p>( )t . . 1.2 -</p><p> () (). ( )ab t </p><p> ( ~ 1/ a ), a </p></li><li><p> 10</p><p> ( )ab t , .. ( )t . </p><p> , - () 0 , .. ; 0 a . </p><p>, , . </p><p>1</p><p>2</p><p>3</p><p>b0b </p><p>. 1.3 -</p><p> ( ) ( ) ( )ab t ( . 1.3) . , - - ( / constb a = = ) - t. </p><p>. 1.2 </p></li><li><p> 11</p><p> , () . , ( - ) . , - . , - ( )S t ( ( )S x ). </p><p>1.3. , -</p><p> , . , - , . . </p><p>. -: </p><p> 2 2( )t dt</p><p> = &lt; . (1.12) </p><p>. - , . , : </p><p> 1( ) (1 )t C t + 1( ) (1 )S C + , 0 &gt; . (1.13) , - ( )t </p><p> - . </p><p> . - ( ) (. . 1.2) </p><p> ( ) 0t dt</p><p> = . (1.14) </p><p> - . </p></li><li><p> 12</p><p> ( )t , .. -, , - ( )S 0= -. a . </p><p> , , n </p><p> ( ) 0nt t dt</p><p> = . (1.15) </p><p> n - - () , . </p><p>. . ( )ab t , </p><p>( )t , ( a ) ( b ). </p><p>1.4. , -</p><p> , . 1.1. -</p><p> ( 20 ( ) exp( / 2))g t t= . , , . </p><p> . 1.4 . 1n = , WAVE- . 2n = MHAT-, - (mexican hat -). . , WAVE-. </p><p> [13]. [30] , </p></li><li><p> 13</p><p>1 4g g -. </p><p> 1.1 </p><p>( )t </p><p>( ) </p><p>: , </p><p>WAVE-, , </p><p>MHAT- -c mexican hat), </p><p> n- , </p><p> 2exp( / 2)t t </p><p>2 2(1 )exp( / 2)t t </p><p>2( 1) exp( / 2)n</p><p>nn</p><p>d tdt</p><p> 2( ) 2 exp( / 2)i </p><p>2 2( ) 2 exp( / 2)i </p><p>2( 1) ( ) 2 exp( / 2)n ni DOG difference of gaussians </p><p>2 22 /80,5t te e 2 2/ 2 22 ( )e e </p><p>LP-Littlewood &amp; Paley 1( ) (sin 2 sin )t t t 1/ 2(2 ) , 2 ,</p><p>0,t</p><p>HAAR- 1, 0 1/ 2,1, 1/ 2 1,0, 0, 0.</p><p>tt</p><p>t t</p><p> &lt; &gt; </p><p>2/ 2 sin / 4</p><p>/ 4iie </p><p>FHAT-, (French hat ) </p><p>1, 1/3,</p><p>1/ 2, 1/3 1,</p><p>0, 1.</p><p>t</p><p>t</p><p>t</p><p> &gt;</p><p>34 sin /33 / 3</p><p> (Morlet) 20 / 2i t te e 2</p><p>0( ) / 2( ) 2 e (Paul) ( n, - ) </p><p>1( 1) (1 )</p><p>n</p><p>ni</p><p> nn +</p><p>+ ( ) 2 ( )ne </p><p>HAAR-. t-, </p></li><li><p> 14</p><p> - , 1/ . </p><p>LR , , , --, . </p><p> - , - . 0 . - ( )t -- . </p><p> -, . - , , - . (Daubechies), (db4) Mathcad. </p><p> . Wavelet Toolbox 2.0/2.1 (MATLAB 6) -</p><p> ; . - MATLAB waveinfo (type), . - wavemenu Wavelet Display. , - : , ( Name) . </p><p>4 3 2 1 0 1 2 3 43</p><p>2</p><p>1</p><p>0</p><p>1</p><p>2</p><p>g1t( )</p><p>g2t( )</p><p>g3t( )</p><p>g4t( )</p><p>t</p><p>0 0.5 1 1.5 2 2.5 3 3.5 400.5</p><p>11.5</p><p>22.5</p><p>33.5</p><p>44.5</p><p>55.5</p><p>6</p><p>Sg1( )</p><p>Sg2( )</p><p>Sg3( )</p><p>Sg4( )</p><p> . 1.4 </p><p>t </p></li><li><p> 15</p><p> . .1 Wavelet Display db4. </p><p> - [8, . 2.9]. </p><p> ( - ) - . - , -. . </p><p>1.5. - () - </p><p>( WT continuous wavelet transform). ( )ab t -</p><p> ( a ) ( b ) ( )t - a b (1.11). </p><p> () () H (.. ) ( )S t : </p><p> ( ) 1( , ) ( ), ( ) ( )s ab t bW a b S t t S t dtaa</p><p> = = , (1.16) </p><p> 21( ) ( , ) ( )s ab</p><p>dadbS t W a b tC a</p><p> = , (1.17) </p><p> C </p><p> 2 1( ) d </p><p>= &lt; , ( , ) , </p><p>( ) - ( )t . - C = 1. </p></li><li><p> 16</p><p> (1.16) , - ( , )sW a b (wavelet spec-trum, time-scale-spectrum - ) - (single spectrum) : ( ) - , .. , b . </p><p> , 0( , )sW b a ( 0a a= ), 0( , )sW a b - ( </p><p>0b b= ). ( )S t -</p><p> u , - 0t t= , - - ua = , 0b t= . </p><p> () ( , )sW a b . ( , )sW a b - (. . 1.5). - ab (. 1.6), - () ( b ). , - , - (sceleton), . </p><p>1.6. - </p><p> . , , - . . </p><p>. (1.16): </p><p> 1 2[ ( ) ( )]W S t S t + = 1 2( , ) ( , )W a b W a b + . </p></li><li><p> 17</p><p>. 0b - 0b : </p><p> 0 0[ ( )] [ , ]W S t b W a b b = . . () </p><p> () ( , )W a b : </p><p> [ ]00 0 0</p><p>1( ) ,a bW S t a Wa a a</p><p> = . </p><p>: </p><p> [ ] ( 1) ( ) [ ( )]m m mt t abW d S S t d t dt</p><p>= , </p><p> [...]/m m mtd d dt= , 1m . , -, , - ( )S t - , . , , - , . </p><p>- . , - - . </p><p> ( a - - ( )ab t ) - (), . - - , - () , ( ) . </p><p> -- . - -</p></li><li><p> 18</p><p> . b - , - a , , , . - (., , [1]). </p><p>1.7. - -</p><p> . , - () (), -, , , - . </p><p> . </p><p>1.7.1. - Mathcad </p><p>(1.11), (1.16), (1.17). -</p><p> Mathcad . - . - . </p><p> , a b ; , , , - . a b - , - . </p></li><li><p> 19</p><p> 1.1. ( ) sin( )s t A t= , 1A = , 2 / 2 / 50T = = , 0 = . </p><p> : </p><p> : 256N = , 2</p><p>22( ) : exp( / 2)</p><p>dMHAT t tdt</p><p>= . </p><p>: 1( , , ) t ba b t MHATaa = . </p><p>-: </p><p> : 1..30a = , : 0..50b = , ( , ) : ( , , ) ( )N</p><p>N</p><p>W a b a b t s t dt</p><p>= , : ( , )abN W a b= ( , )abN W a b= . 1.5 </p><p> , . 1.6 (a,b). , ( , )W a b 0a a= ( )s t ; 0 1/a : . 0( , )W a b 0b b= . </p><p>. 1.5 </p></li><li><p> 20</p><p> . 1.6 </p><p> 1.2. : 1 1 2 2( ) : sin( ) sin( )s t A t A t= + , </p><p> 1 2 1A A= = , 1 12 /T = , 2 22 /T = , 1 50T = , 2 10T = . </p><p>22 / 22( ) :</p><p>tdMHAT t edt</p><p> = , N:=256, </p><p> ( , , ) : t ba b t MHATa = , ( , ) : ( , , ) ( )</p><p>N</p><p>N</p><p>W a b a b t f t dt</p><p>= , a := 130, b : = 050, : ( , )abN W a b= . </p><p> ( , )W a b . 1.7 , . 1.8 (a,b). </p><p>. 1.7 </p><p>0 25 50 75 100 125 150 175 200 225 25021.5</p><p>10.5</p><p>00.5</p><p>11.5</p><p>22</p><p>2</p><p>s t( )</p><p>2500 t</p></li><li><p> 21</p><p> . 1.8 </p><p> . 1.9, - </p><p> . a , .. 1 2a = ( 1 21/a : ), , () - . a - [( ) / ]t b a , , , .. - . 2 15a = ( 2 11/a : ) . - a - ( 25a &gt; ), - . </p><p>. 1.9 </p></li><li><p> 22</p><p> 1.9, - ( , )W a b , 1 13b = 2 17b = . </p><p>, , - (. 1.1) - a ( 1..3a : ), - , .. 2 2sin( )A t . </p><p> 1.3. </p><p> U 5:= t0 20:= 60:=s t( ) U t0 t t0 +if</p><p>0 otherwise</p><p>:=</p><p>0 20 40 60 80 100 12010123456</p><p>s t( )</p><p>t</p><p>MHAT t( ) 2texp</p><p>t22</p><p>d</p><p>d</p><p>2:=</p><p>N :=128 </p><p>( , , ) : t ba b t MHATa = , ( , ) : ( , , ) ( )</p><p>N</p><p>N</p><p>W a b a b t f t dt</p><p>= , : 1..50a = , : 0..100b = , : ( , )baN W a b= . </p><p> . 1.10 </p></li><li><p> 23</p><p>- . 1.10. . 1.10, - -</p><p> 20b = 80b = ( 1..5a : ). , </p><p>Mathcad . - . Intel Celeron (667 128 ) 5 . MATLAB. </p><p>1.7.2. - MATLAB Wavelet Toolbox MATLAB </p><p> - - . - ; ( ), , a , - . -. </p><p> Continuous Wavelet 1-D Complex Continuous Wavelet 1-D Wavelet Toolbox - . </p><p> . 1.11 - , - - . : ( ) , . - - ( ). </p><p> cwt, (..2.2), : COEF = cwt(S, SCALES, wname PLOTMODE, XLIM), S , SKALES - a , wname () , </p></li><li><p> 24</p><p> . 1.11 </p><p> PLOTMODE : lvl -, glb , absvil lvlabs , XLIM . </p><p> 1.4. function garm t = 0:0.00001:0.0004; A1 = 1; F1 = 10000; a1 = 45; s = A1*soc(2*pi*F1*t+a1); figure (1); plot(t,s); axis([0 0.0004 -3 3]); grid on; subplot(211), plot(t,s); title(' S(t)') subplot(212), c = cwt(s, 1:2:32, 'mexh', 'abs1vl', [0 10]); title('- '); xlabel(' , b'); ylabel(' , a'); end , </p><p> . 1.12, , ( ) ( ). </p></li><li><p> 25</p><p> . 1.12 </p><p> 1.5. ( )S t </p><p> : </p><p>function binar t = 0:0.000001:0.0004; A1 = 1; A2 = 1; F1 = 10000; F2 = 2*F1; a = 90; b = 90; a1 = a*0.0174533; a2 = b*0.0174533; s = A1*sin(2*pi*F1*t-a1) + A2*sin(2*pi*F2*t-a2); figure (1); plot(t,s); axis([0 0.0004 -3 3]); grid on; subplot(211), plot(t,s); title(' S(t)') </p><p> . 1.13 </p></li><li><p> 26</p><p>subplot(212), c = cwt(s,1:2:50,'mexh','abs1v',[0 1]); title('- S(t)'); xlabel(' , b'); ylabel(' , a'); end </p><p> ( )S t ( , )W a b . 1.13. - , a ; , . </p><p> 1.6. ( )x t ( )S t </p><p> ( )n t m = 0 - g . </p><p>function bigarm_rauch t = 0:0.000001:0.001; A1 = 1; A2 = 1; F1 = 10000; F2 = 2*F1; a = 90; b = 90; a1 = a*0.0174533; a2 = b*0.0174533; s1(1:200) = 0; t2 = 0.0002:0.000001:0.0007; s2 = A1*sin(2*pi*F1*t2-a1) + A2*sin(2*pi*F2*t2-a2); s3(1:300) = 0; s = [s1 s2 s3]; randn('state',0); g = 0.5; n =g *randn(size(t)); x = s+n; figure (1); subplot(211), plot(t,x,'k'); t itle(' x(t)'); grid on; gtext('F=10, 1=2=1, g=0.5 B'); </p><p>. 1.14 </p></li><li><p> 27</p><p>subplot(212), c = cwt(x,1:124,'mexh','absglb',[0 50]); title('- W(a,b)'); xlabel(' , b'); ylabel(' , a'); end . 1.14 . -</p><p> - ( 1 5a : ), a , ; . - , . </p><p> 1.7. function pr_rauch_wav t = 0:0.000001:0.000300; A1 = 2; F1 = 0; s1(1:75) = 0; t2 = 0.000075:0.000001:0.000175; s2 = A1*cos(2*pi*F1*t2); s3(1:125) = 0; s = [s1 s2 s3]; randn('state',0); g = 0.5; n = g*randn(size(t)); x = s + n; figure (1); subplot(211), plot(t,x,'k'); title(' x(t)'); grid on; subplot(212), c = cwt(x,1:27,'mexh','absglb',[0 10]); title('-'); xlabel(' , b'); ylabel(' , a'); end (. 1.15) -</p><p> , ( 20a &gt; ) . () . </p><p> . 1.15 </p></li><li><p> 28</p><p> 1.8. mtlb 200 : function ss load mtlb; v=mtlb(1:200); lv = length(v); subplot(211), plot(v); title(' '); set(gca, 'Xlim',[0 200]); [c,l] = wavedec(v,5,'sym2'); cfd = zeros(5,lv); subplot(212) ccfs = cwt(v,1:128,'sym4','plot'); title('-') colormap(pink(32)); xlabel(' , b'); ylabel(' ,a'); end </p><p>. 1.16 - (. 1.16) </p><p> : , ( -, ). </p><p> , -</p><p> - ( , - , ..). </p></li><li><p> 29</p><p>1.8. () -</p><p> - . - . . </p><p> - . , . - , , . , - , - + . , . (, , ..) - , , , -. - . - - , , - . , -, - . </p><p> - : </p><p> ( , ) ( ) ( ) j tS b S t w t b e dt </p><p> = , ( )S t ( )w t b ; - (, , .. </p><p>( ) 1w t = 0 t ( ) 0w t = 0t &lt; t &gt; ), - t (. 1.17) -</p></li><li><p> 30</p><p> b . , .. - . </p><p>0 b b+ t </p><p>s+t </p><p>W(tb)</p><p>. 1.17 , . 1.17, </p><p>( ) ( )S t , . - , . 1.18, . , 2 / . </p><p>, , - , - ( ) . - ( const = ) , . ( ) ( ). </p><p> , , , - - (. 1.18, ), - . </p><p> . -- , (. 1.18, ): a , - . </p></li><li><p> 31</p><p> . - -, . - . </p><p>t tt </p><p>t t </p><p>t </p><p>ffab(t) </p><p>. 1.18 </p><p> : - . - , , . </p><p> -, , - -, , , , , -, . - , - . - . </p></li><li><p> 32</p><p> 2 </p><p>2.1. - a b -</p><p> - -. ( )ab t . . , , [13]: </p><p>2ma = , 2mb k= , 1 1( ) (2 )2</p><p>mmk m</p><p>t bt t kaa</p><p> = = , (2.1) </p><p> m k . ,a b - ,m k . m . </p><p> . (dyadic) . </p><p>. 2.1 - ,a b . m . - m 1 . m - ( )mk t 2mb k= , - m , 0m = . </p></li><li><p> 33</p><p>0 2 4 6 8</p><p>1. </p><p>1. </p><p>0 2 4 6 8</p><p>1</p><p>0 2 4 6 8</p><p>1 </p><p>0 2 4 6 8</p><p>1 </p><p>30(t) </p><p>10(t) </p><p>21(t) 20(t) </p><p>13(t) 11(t) </p><p>(t) 01(t) 07(t) </p><p>t </p><p>t</p><p>t </p><p>t</p><p>. 2.1 -</p><p> : </p><p> ( )( ), ( ) ( ) ( )mk mk mkc S t t S t t dt</p><p>= = , (2.2) </p><p> ,</p><p>( ) ( )mk mkm k</p><p>S t c t= . (2.3) </p></li><li><p> 34</p><p> , mkc (2.3) </p><p>( , )sW a b </p><p> ( )2 , 2m mmkc W k= . (2.4) (2.2) (2.4), , - mkc </p><p> , - ,m k (); - m k . </p><p> (2.3) , ( )S t mkc . - (2.3) , , . , - . </p><p> . , - , .. [8], : , - . </p><p> e. a , b </p><p> 1/ 2 ja = , / 2 jb k= , ( ) 2 (2 )j jjk t t k = , (2.1) .. j a , </p><p>( )jk t . (2.1) m a , .. ( )mk t . </p><p>. , - . - ( 2ma = ), . -, , , . - (). , - . </p></li><li><p> 35</p><p>2.2. , , </p><p> , -- ( DWT). a b , . </p><p> ( ) ( )S t , mf , - { }iS , 0,1, ..., 1i N= , - t : 1/ 2 mt f = , 1/ 2 mf t f= = , (2.5) t...</p></li></ul>