Wavelet Analys

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<p>1. (1145).2. - (1146).2.1. . 2.2. . 2.3. -. 2.4. - .3. - (1150).3.1. . 3.2. . 3.3. .4. - (1153).4.1. . 4.2. -. 4.3. -.5. - (1157).5.1. . 5.2. .5.3. .6. (1162).6.1. - (). 6.2. 500 -. 6.3. . 6.4. . 6.5. .7. (1169). (1170).1. "" ( ) (Grossman &amp; Morlet) 80- [1]. , 1, ; , ,; ( - , , - , ..); ; () - .- - , - () . .. . ,117810 , . 84/32. (095) 333-21-45E-mail: ast@nat.iki.rssi.ru, ast@iki.rssi.ru 23 1996 ., 18 1996 . 1996 . 166, 11 -: .. - , - . , , - , . (, , ) ( , -). - , , . , : . . , () ( ); , - , .PACS numbers: 02.30.f, 02.90.+p, 92.60.e, 92.60.Ry#.. 19961 , , . [17] " " " " , , . (. )1 , . 166, 11() , (). , - -- , -. - (, ) . . single spectrum , - -, time-scale spectrum, wavelet spectrum. , , -. , - , - - . - , , , . - . - , - , - . , . - - , - . - " " . "" - - , , , - . - - . - . - , . , - . -, - . , - - . 2 , -. 3 - , -, . 4 --, -. , , [2 4] (Ingrid Daubechies) [5] (MarieFarge) [6]. 5 - - ; 6 - - .2. - . - -, - . - ( , -) , - sin t, cos t ( exp(it) = cos t i sin t, i =</p> <p>1_ ) .- , . , [4], --, , -., , . - . , , , -, . ( ) .1146 .. [ 19962.1. , . L2(0Y 2p) - ()_2p0f (t)2dt ` Y t (0Y 2p) X (1) - f(t). R(Y ) , f(t) = f(t 2p) Y t RX f(t) 2p-- :f(t) =</p> <p>cn exp(int) X (2)K cn (2) cn = (2p)1_2p0f(t) exp(int) dt Y (3) (2) f(t):limMY N_2p0f(t) </p> <p>NMcn exp(int)2dt = 0 X, wn(t) = exp(int) Y n = F F F Y 1Y 0Y 1Y F F F (4) L2(0Y 2p), w(t) = exp(it) , wn(t) = w(nt)., 2p- - - w(t) = exp(it) = cos t i sin t, .. ( -, )., (2p)1_2p0f(t)2dt =</p> <p>[cn[2X (5)2.2. L2(R) f(t), - R(Y ) - ()Ef =_f(t)2dt ` X (6) L2(0Y 2p) L2(R) . , L2(R) . L2(R), , , wn - L2(R). - L2(R)."", L2(R), , . - " " ( wavelet). L2(0Y 2p), w(t), - L2(R) c(t)., (frequency bands). . R(Y )? , () . , .. c(t k). . - ee : c(2jt k), j k ( jY k I). , (1a2j) (ka2j) , - c(t). :|p|2 = pY p)1a2YpY q) =_p(t)q+(t) dt( ). -,__c(2jt k)__2 = 2ja2__c(t)__2 Y.. c(t) L2(R) , cjk cjk(t) = 2ja2c(2jt k) Y jY k I (7) , .. |cjk|2 = |c|2 = 1. c L2(R) , (7) cjk - - L2(R), ..cjkY clm) = djldkm f L2(R) f(t) =</p> <p>jY k=cjkcjk(t) Y (8) L2(R) , limM1Y N1Y M2Y N2_____f </p> <p>N2M2</p> <p>N1M1cjkcjk_____2= 0 X1*. 166, 11] - 1147 HAAR-, - (Haar), -cH(t) =1 Y 0 4t ` 1a2 Y1 Y 1a2 4t ` 1 Y0 Y t ` 0 Y t 51 X___ (9) , cHjk, cHlm, - (7) 1a2j, 1a2l ka2j,ma2l, . - L2(R) - c(t) - a b:cab(t) = [a[1a2c_t ba_Y aY b RY c L2(R) X (10)H --:_Wc f(aY b) = [a[1a2_f(t)c+_t ba_dt ==_f(t)c+ab(t) dt X (11) , cjk = fY cjk) (8) - f -:cjk =_Wc f_12j Y k2j_X (12) _Wc f(aY b) - () - - W(aY b) Wc f, W[ f [., L2(R) , .. "- " ( , (, ) ()). -( -o ) - [2 5], - , .. , , ,.. , - - . -(11) ; , - - a b, - , .2.3. - L2(0Y 2p), -. L2(R), , , ( aY b). , L2(R) , . - , [2 5], - -. , - "" , .. , " ". , " " . , - -, : (7), (1a2jY ka2j), jY k I, (10), (aY b),aY b R. (aY b), aY b R - (10), :f(t) = C1c___Wc f(aY b) cab(t) da dba2 Y (13)Cc ( - (2p)1a2, ):Cc =_c(o)2[o[1do ` ( -). Cc c(t) L2(R), - . ,, c o = 0 , , :_c(t) dt = 0 X , .. a b 0; , -, Cc = 2_0c(o)2o1do = 2_0c(o)2o1do ` X - - .1148 .. [ 1996 c L2(R) R-, cjk, (7), (Riesz) , A B, 0 ` A4B ` , -A__cjk__224_____</p> <p>j=</p> <p>k=cjkcjk_____224B__cjk__22 (, - ) cjk:__cjk__22 =</p> <p>j=</p> <p>k=[cjk[2` X R- cjk "" cjk ( , cjkY clm) == djldkm), - f(t) =</p> <p>jY k= fY cjk) cjk(t) X (14) c cjk - , cjk cjk (14) . c , - R- (dyadic wavelet), c+, - cjk (7):cjk(t) = c+jk(t) = 2ja2c+(2jt k) Y jY k I X (15) (14) - , c+ cjk -, (10).2.4. -a - - . - . (, ) - L2(R). f(t) , | f |2, :</p> <p>f(o) =_f(t) exp(iot) dt X . , - , , ; -, -. , , , , , , - ( 5). , , . , ; - . - . - (- ) - . -- , , " " - . , - - - , . -o -? c, - c , " " "" "", . z(t) L2(R)(, tz(t) L2(R)), t) Dz t) = 1|z|22_tz(t)2dt YDz = 1|z|2_ __t t)_2z(t)2dt_1a2Y 2Dz. t), Dc, o), D</p> <p>c c - c . - (11) " "_wint =_b at) 2aDcY b at) 2aDcY (16).. b at) 4aDc. Z(o) = c_o o)_, D</p> <p>c. fY g) = fY g)a2p, -(11) - f W(aY b) = [a[1a2_</p> <p>f(o) exp(ibo)Z+_a_o o)a__doX(17) , -, (17) f(o) f(t) "- "_wino =_o)a 1a D</p> <p>cY o)a 1a D</p> <p>c_X (18) o)aa , 2D</p> <p>caa.. 166, 11] - 1149, ,o)a_2D</p> <p>ca_1= o)2D</p> <p>cY , -o [wint[ [wino[, - 4DcD</p> <p>c, o)aa (. 1). . 1 -o : (. 1) (Shannon), (. 1). 1 , , o -; ; - - , , - . , - , - , - . -- -, - (11) , , F(oY b) =_ f(t)z(t b) exp(iot) dt , - z. , F(oY b) z(t b)exp(iot), z(t) b o , -- W(aY b) c_(t b)aa_, c(t) b a . - - . , - _z(t)Y z(o)_ , - - a _c(taa)_ - a _</p> <p>c(ao)_. - , ( ) , - . -, - - , - , - ., Dt , Do = Dta2 ( , - - ). - - DtDo5(4p)1. , - , -, o - .3. - "" - , . . [3], , .3.1. R- c L2(R) R- ( ), - c+ L2(R) ( , ) , cjk c+jk, (7) (15), - L2(R). c, , , - f L2(R) (8), - - f c+.bta3a2a1 1aa1k1aa21aa3tk3kk2k1DkDt . 1. -a : () , () , () .1150 .. [ 1996- c+ R-. (cY c+) , c c+. R- c -, c+= c, cjk = c+jk . , c , .. cjk cjkY clm) = 0 j = l, jY kY lY m I.R- , . , R-, , (cY c+) cjk c+jk, cjkY c+lm) = djldkm, jY kY lY m I . -- ( ) , .3.2. , , ; , - .. - . o , . :_c(t) dt = 0 X (19) , , m :_tmc(t) dt = 0 X (20) m- . - , - , - .:_c(t)2dt ` X (21) c(t) `_1 [t[n_1ilic(o) `_1 [k o0[n_1Y o0 , n . . - -. cab(t) , c(t), - . - (., , [7]). - [6], . , , ., d- : t- d- k-; , k- t-. G(t) = exp_iO(t t0) i5exp_(t t0)22s2_ 1s(2p)1a2 : t0, (-) s, O 5. . , -. , - t-, k- --, .HAAR- (. (9)) - , - . t--, (- k1) "" k-, - . - , . - , , - FHAT-, " " (French hat ):c(t) =1 Y [t[ 41a3 Y1a2 Y 1a3 ` [t[ 41 Y0 Y [t[ b 1 Y___</p> <p>c(k) = 3Y(k)_sin kk sin 3k3k_Y Y(k) (Y(k) = 1 k b 0 Y(k) = 0 k 40).FHAT-, o - - , LP- ( (Littlewood &amp; Paley, . [6])), , k- t-, - , .. 166, 11] - 11513.3. - - - , W(aY b) - ( , - ). , , - , . - o , . "", b , a , , c . :cm(t) = (1)mqmt_exp_t22__Y</p> <p>cm(k) = m(ik)mexp_k22_( qmt = qm[F F F[aqtm, m51). - , . 2, , m = 1 m = 2 . - WAVE-, MHAT-, " " (Mexican hat ).MHAT-, ( ), . MHAT- . , - - [6]. - - . (-) , -. DOG- (Difference of Gaussians):c(t) = exp_[t[22_ 0Y5 exp_[t[28_Y</p> <p>c(k) = 1(2p)1a2_exp_[k[22_ exp_2[k[2__X . 2, ( ). k- r- (Morlet) [1]:c(r) = exp(ik0r) exp_r22_Y</p> <p>c(k) = Y(k) exp_(k k0)22_Y , . 2 k0 = 6. k0 - , . (Paul) [8]c(t) = G(m 1) im(1 it)m1 Y</p> <p>cm(k) = Y(k)(k)mexp(k) . 2 m = 4 ( m, ).ctcttctctcctk</p> <p>ck</p> <p>ck</p> <p>ck</p> <p>c</p> <p>ckk</p> <p>c. 2. : () WAVE, ()MHAT, () Morlet, () Paul, () LMB, () Daubechies. ( ) ( ).1152 .. [ 1996 . , - . , -: ., - -- :W(aY b) =W(aY b)exp_iF(aY b)X e 2, , - ( (7)) (Mallat) [9]: LMB-, - , (Lemarie, Meyer, Battle)[10, 11] [5]. - , (), - . .4. - - (-) (). -- - W(aY b). (aY b) = (o , ) -, (-) - - (time-scale spectrum, wavelet spectrum single spectrum - ).4.1. W(aY b) . -. - ab , - , ( - "sceleton"), - . "" "" (. ), . , , - , , (log aY b), . - , . 3 - ( [12] -8; [12]). MHAT-. 3 , -- ab (o , ); ( ), o . 3 -, W(aY b), W(aY b). , - - (a0Y b0) ( ), - t = b0. - , - : W(aY b) - , , .. 3. - : () , () W(aY b), () ,() W(aY b) , (), () - EW(aY b) , ().. 166, 11] - 1153 3 , - ("" "") . , - , de facto . W(aY b), -. , - , , , "" . , - . 3, a -; . 3 . 3 a. - t0 - ( , , . - 4.3). , - t0 - , (. 4)., , , (-, o ). , W(aY b) (a0Y b0) o ( -) b0 (. 4), , a0, .. ( ) , . ( ) , - W(aY b) -, ( ). , - ( ) . ( , ..).- f(o) c(o). , f(o0) - W(aY b), - omin ` ao0 ` omax (. 4); , W(aY b) (a0Y b0) f(o) , omin ` a0o ` omax. . , , - MHAT- (. 2) . , , - W(aY b) , .. (ominY omax) ., . - , , . - (., , [13]), - (FFT). , -, . - ( ) - - ( ) ( ). - - , - . - -. 3 .4.2. - , -- . , - , - . - . - f(t). - W[ f [ = W(aY b).:W_a f1(t) b f2(t) = aW[ f1[ bW[ f2[ == aW1(aY b) bW2(aY b) X (22)at t0tY ba (a0Y b0)1aomax1aominoo0atY b. 4. (, ) ().1154 .. [ 1996, , , -- , - - -. :W_f(t b0) = W(aY b b0) X (23) -, , qtW[ f [ = W[qt f [ ( qt= qaqt). - . ():W_f_ ta0__ = 1a0W_aa0Y ba0_X (24) , , - (. 4.3.1). -- . - , , .-a -o ( - -o ). - -o 2.4.:W_qmt f = (1)m_f(t) qmt__c+ab(t)_dt X (25) , , , f, , - . , , f , -. - - _ f1(t) f+2 (t) dt = C1c__ W1(aY b)W+2(aY b) da dba2 Y (26) , - () -- , :Ef =_ f2(t) dt =_A(o) iB(o)2doX - . . , - - .4.3. - -,...</p>