Яковлев А.Н. Основы Вейвлет-преобразования Сигналов

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Text of Яковлев А.Н. Основы Вейвлет-преобразования Сигналов

  • ..

    -

    -

    2003

  • 2

    621.372(075.8)+004.92(075.8) 474

    :

    -

    ()

    - . , . .. (), . . , . .. ()

    ()

    ..

    474 -: . . : - , 2003. 104 .

    ISBN 5-7782-0405-1

    - ( 1), - ( 2) , - ( 3) . - , -. - Wavelet Toolbox MATLAB 6.

    , , , , , .

    621.372(075.8)+004.92(075.8)

    ISBN 5-7782-0405-1 , 2003

    . . , 2003

  • 3

    50- ()

    - -, , - (), .

    (wavelet) (Grossmann) (Morlet) 80- XX -. , (Dobechies), (Meyer), (Mallat), (Farge), (Chui) .

    - () (t x), - (-/). . - () () - , - . - , ( ) - .

    , , , , - , - , -

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    , , , , , , , .

    . - - , .

    , - (), Mathcad, MATLAB, Mathematica. JPEC-200, MPEG-4 Corel DRAW 9/10 , , , , . -, Analog Devices - ADV6xx (ADV601, ADV601LC, ADV611, ADV612), - .

    . - [15, 19, 20, 26].

    . [13], [4], . [5] . [6], - [1014]. ( 100 300 ), .

    - , - .

    [7], .. .. 2002 . ( 5000 ., 38 .., 160 .), . - - , MATLAB

  • 5

    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 .

    - - , Wavelet Seminar () . - - - , -, .

    , - [22, 25].

    - , . - , , .

    - . , - (, , ) -.

    , - , , -, , - , , .

    - . . .. . . . . .. , .. MATLAB.

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    1

    -

    1.1. [13, 14], ( )S t , -

    2

    1

    2[ ( )]t

    tS t dt < , (1.1)

    { ( )}n t , .. 0 0

    0( ) ( ) .... ( ) ... ( )n n n n

    nS t C t C t C t

    == + + + = , (1.2)

    2

    12

    1 ( ) ( )t

    n ntn

    C S t t dt= , (1.3)

    2

    1

    2 2|| || ( )t

    n nt

    t dt = (1.4) , ( )n t . ,

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    1 2[ , ]t t -

    2

    1

    2|| ||( ) ( )0

    tn

    k nt

    t t dt =

    , ,, .k nk n= (1.5)

    ( )n t , ( 2|| || 1n = ), (), { }( )n t -. , - .

    (1.2), nC (1.3), .

    ( )n nC t , (1.2), - ( )S t , - 0{ ,.., ,..}nC C . 0{ ,.., ,..}nC C - , , - (. 1.1).

    ( )S t -

    nC ( -) (1.3).

    (1.2) () - N

    0 00

    ( ) ( ) ... ( ) ( )N

    N N n nn

    S t C t C t C t=

    = + + = . (1.2) : { }( )n t N (-), , =

    2

    1

    2( ) ( )

    t

    tS t S t dt =

    2

    1

    2

    0( ) ( )

    t N

    n nnt

    S t C t dt=

    . (1.6)

    nC

    0C1C 2C iC

    0 1 2 i n

    . 1.1

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    , - N . (1.2) - .

    / = , (1.7) ( 1 ), - ( )S t , ..

    2

    1

    2 2 [ ( )]t

    tS S t dt= = . (1.8)

    (1.8) (1.2)

    [ ]21

    2 ( )t

    tS t dt= 22

    0n n

    nC

    == , (1.9)

    { }( )n t 2

    0n

    nC

    == .

    , 2 1T t t=

    [ ]21

    2 2p

    0

    1 1( )t

    n nnt

    P S t dt C

    == = = . (1.10)

    (1.9) (1.10) .

    -. .

    , - , . -, ; . -, - . - , [13, 14].

  • 9

    , . - , , , . - , , [14]. - [13, 14].

    , .

    1.2. . wavelet ( ondelette)

    () . - - : , , - , .

    - ()

    1( )abt bt

    aa = , (1.11)

    () ( )t , ( b ) ( a ) (. 1.2). 1/ a - a .

    , a b ( )ab t ,

    ( )t . . 1.2 -

    () (). ( )ab t

    ( ~ 1/ a ), a

  • 10

    ( )ab t , .. ( )t .

    , - () 0 , .. ; 0 a .

    , , .

    1

    2

    3

    b0b

    . 1.3 -

    ( ) ( ) ( )ab t ( . 1.3) . , - - ( / constb a = = ) - t.

    . 1.2

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    , () . , ( - ) . , - . , - ( )S t ( ( )S x ).

    1.3. , -

    , . , - , . .

    . -:

    2 2( )t dt

    = < . (1.12)

    . - , . , :

    1( ) (1 )t C t + 1( ) (1 )S C + , 0 > . (1.13) , - ( )t

    - .

    . - ( ) (. . 1.2)

    ( ) 0t dt

    = . (1.14)

    - .

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    ( )t , .. -, , - ( )S 0= -. a .

    , , n

    ( ) 0nt t dt

    = . (1.15)

    n - - () , .

    . . ( )ab t ,

    ( )t , ( a ) ( b ).

    1.4. , -

    , . 1.1. -

    ( 20 ( ) exp( / 2))g t t= . , , .

    . 1.4 . 1n = , WAVE- . 2n = MHAT-, - (mexican hat -). . , WAVE-.

    [13]. [30] ,

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    1 4g g -.

    1.1

    ( )t

    ( )

    : ,

    WAVE-, ,

    MHAT- -c mexican hat),

    n- ,

    2exp( / 2)t t

    2 2(1 )exp( / 2)t t

    2( 1) exp( / 2)n

    nn

    d tdt

    2( ) 2 exp( / 2)i

    2 2( ) 2 exp( / 2)i

    2( 1) ( ) 2 exp( / 2)n ni DOG difference of gaussians

    2 22 /80,5t te e 2 2/ 2 22 ( )e e

    LP-Littlewood & Paley 1( ) (sin 2 sin )t t t 1/ 2(2 ) , 2 ,

    0,t

    HAAR- 1, 0 1/ 2,1, 1/ 2 1,0, 0, 0.

    tt

    t t

    < >

    2/ 2 sin / 4

    / 4iie

    FHAT-, (French hat )

    1, 1/3,

    1/ 2, 1/3 1,

    0, 1.

    t

    t

    t

    >

    34 sin /33 / 3

    (Morlet) 20 / 2i t te e 2

    0( ) / 2( ) 2 e (Paul) ( n, - )

    1( 1) (1 )

    n

    ni

    nn +

    + ( ) 2 ( )ne

    HAAR-. t-,

  • 14

    - , 1/ .

    LR , , , --, .

    - , - . 0 . - ( )t -- .

    -, . - , , - . (Daubechies), (db4) Mathcad.

    . Wavelet Toolbox 2.0/2.1 (MATLAB 6) -

    ; . - MATLAB waveinfo (type), . - wavemenu Wavelet Display. , - : , ( Name) .

    4 3 2 1 0 1 2 3 43

    2

    1

    0

    1

    2

    g1t( )

    g2t( )

    g3t( )

    g4t( )

    t

    0 0.5 1 1.5 2 2.5 3 3.5 400.5

    11.5

    22.5

    33.5

    44.5

    55.5

    6

    Sg1( )

    Sg2( )

    Sg3( )

    Sg4( )

    . 1.4

    t

  • 15

    . .1 Wavelet Display db4.

    - [8, . 2.9].

    ( - ) - . - , -. .

    1.5. - () -

    ( WT continuous wavelet transform). ( )ab t -

    ( a ) ( b ) ( )t - a b (1.11).

    () () H (.. ) ( )S t :

    ( ) 1( , ) ( ), ( ) ( )s ab t bW a b S t t S t dtaa

    = = , (1.16)

    21( ) ( , ) ( )s ab

    dadbS t W a b tC a

    = , (1.17)

    C

    2 1( ) d

    = < , ( , ) ,

    ( ) - ( )t . - C = 1.

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    (1.16) , - ( , )sW a b (wavelet spec-trum, time-scale-spectrum - ) - (single spectrum) : ( ) - , .. , b .

    , 0( , )sW b a ( 0a a= ), 0( , )sW a b - (

    0b b= ). ( )S t -

    u , - 0t t= , - - ua = , 0b t= .

    () ( , )sW a b . ( , )sW a b - (. . 1.5). - ab (. 1.6), - () ( b ). , - , - (sceleton), .

    1.6. -

    . , , - . .

    . (1.16):

    1 2[ ( ) ( )]W S t S t + = 1 2( , ) ( , )W a b W a b + .

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    . 0b - 0b :

    0 0[ ( )] [ , ]W S t b W a b b = . . ()

    () ( , )W a b :

    [ ]00 0 0

    1( ) ,a bW S t a Wa a a

    = .

    :

    [ ] ( 1) ( ) [ ( )]m m mt t abW d S S t d t dt

    = ,

    [...]/m m mtd d dt= , 1m . , -, , - ( )S t - , . , , - , .

    - . , - - .

    ( a - - ( )ab t ) - (), . - - , - () , ( ) .

    -- . - -

  • 18

    . b - , - a , , , . - (., , [1]).

    1.7. - -

    . , - () (), -, , , - .

    .

    1.7.1. - Mathcad

    (1.11), (1.16), (1.17). -

    Mathcad . - . - .

    , a b ; , , , - . a b - , - .

  • 19

    1.1. ( ) sin( )s t A t= , 1A = , 2 / 2 / 50T = = , 0 = .

    :