470.динамика цифровых колебательных систем учебное пособие

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  1. 1. . .. .. , 2005 Copyright & A K-C
  2. 2. 2 621.37 (075.5) 3841-0173 89 : ; , .. .. : . / .. ; . . -. 2- ., . . : , 2005. 153 . ISBN 5-8397-0405-9 . , . . 2- : . , - . , 01380 ( , ). , 654200 , 654400 653700 . . 100. .:21 . 621.37 (075.5) 3841-0173 , 2005 ISBN 5-8397-0405-9 .., 2005 Copyright & A K-C
  3. 3. 3 . , . , . . ( ) , . . .2.8, . : . : . , - , , . . . , . . . , 2005 .. Copyright & A K-C
  4. 4. 4 Copyright & A K-C
  5. 5. 5 . . , , , , - . : ; , . , . .. , .. .. , .. , .. , .. , .. .. . . , . , . . (), , , ( , , , ), , ( ), , . Copyright & A K-C
  6. 6. 6 . . , .. , .. , . , .. , . . . . . , , , . . , , ( ) . , . . ( ) , . . , , . , , , (). , , . . .. , .. , .. , .. .. , , . , , ... .. , , .. , .. .. , . Copyright & A K-C
  7. 7. 7 1. () , - . , . . , .. . . , .. , , . . ( )ba, x x . ( )ba, . ( )xfx = , (1.1) , , , , . x x , - -. F ( )ba, (1.1), , ( )bax , ( ) ( )baxf , . ( )ba, 0x ( )01 xfx = . 1x F , ( ) ( )[ ]012 xffxfx == . , 0x 2x , F F F (.. 2 FFF = ). F 2x , ( )23 xfx = , 3 F 0x , ( )[ ]{ }03 xfffx = . Copyright & A K-C
  8. 8. 8 : ( )1= nn xfx . (1.2) nx n- 0x . , ( )xfx = xx = (. 1.1). , F , . x* x x x=f x( ) x= x x0 0 . 1.1 x ( )ba, , ( ) 0= xxf , (1.3) .. ( ( ) = xfx ), ( ) F . F Copyright & A K-C
  9. 9. 9 ( )xf xx = . . 1.1, x (1.2). x F , , (1.2), , x . x . . 1.11.4 , . x x x=f x( ) x=x 0 . 1.2 Copyright & A K-C
  10. 10. 10 x x x=f x( ) * x x=x 0 x0 . 1.3 x x x=f x( ) * x x=x 0 . 1.4 Copyright & A K-C
  11. 11. 11 : x F ( )xf , 1< =xxdx df , (1.3) , 1> =xxdx df . (1.3) , , 1=dxdf . . , , . . 1.5 , 1a 1b , ( ) ( ) 1111 , abfbaf == . , 2, 1a 1b F . : ( )[ ] 0= xxff . (1.4) Copyright & A K-C
  12. 12. 12 x x x=f x( ) ca bx0 0 x=x . 1.5 . ( )xf , ( )ba, , ( )ba, , ( )bax , ( ) ( )baxf , , : ( ) ( ) ( ) ( ) ( ) ( ) ( ){ } ( ) ( ) 0 0 1 1 2 2 3 3 1 , , , , ..., .n n n x f x x x f x f x x f x f f x x f x f f f x x f x f x = = = = = = = = = = (1.5) ( ) ( )1= nn xfxf n- ( )xf . : ( )[ ] ( )xfxff mnmn += . (1.6) N , 0x Copyright & A K-C
  13. 13. 13 ( ) 00 xxfN = . (1.7) 0x N - ( ) F . (1.7) f , , 0x N - ( ) ( ) ( )011022011 ...,,, xfxxfxxfx NN === . (1.8) (1.8) N - F , N , N , . , (. 1.6). x x x=f x( ) x= x 0 . 1.6 , (1.7) ( ) 00 xxfkN = , (1.9) Copyright & A K-C
  14. 14. 14 k . N - ( ) 0= xxfN , (1.10) N- . : N - ( N ) , C 1< =Cx N dx df , (1.11) , 1> =Cx N dx df . (1.12) , . . , , , . 6 . , , . 1 1 (. . 1.1 1.2). 2 1 , 1> dx df . (1.13) Copyright & A K-C
  15. 15. 15 (. . 1.1 1.3). 3 1 , 1< dx dfN , (1.14) (. . 1.1 . 1.3). 2 ( . 1.5). , , . : , , . . . n . 1. = xx , 0> )( , )(nxx = , x , , n < )),(( xnx , )),(( xnx )(nx x . 2. , 1 . Copyright & A K-C
  16. 16. 16 . 3. x - , Nn xnx = 0)),(( . 4. , . 5. x - , ( ) )(nx x . . 6. )(nxx = , 0> )( , )(nxx = , 0 0( ( ), ( ))x n x n < , 0nn > < ))(),(( nxnx . 7. )(nxx = , 0> 0)( > , < )),(( 0nx < )),(( nx 0nn > . , )(nxx = , , ));0[()( = nnxx . 6 7 . (. 1.7), , , , . (. 1.8): , . , , , , . Copyright & A K-C
  17. 17. 17 x n( ) x n( ) 0 x n( )* . 1.7 x n( ) x n( ) 0 x n( )* 0 x n( )* . 1.8 Copyright & A K-C
  18. 18. 18 2. () . () () , ( ) . . 2.1. ( )1+nx , ( )11 +ny , 1 z , 1b 1b , ( )f . ( ) . y n+( 1)x n+( 1) y n( ) 1 1 b1 z-1 f( ) . 2.1 ( ) =f . . , ( ) ( ) 1, sign 1. f < = (2.1) . 2.2. Copyright & A K-C
  19. 19. 19 f( ) 1 -1 0 1 2-1-2 . 2.2 ( ) ( ) 1)2(mod1 +=f , (2.2) . 2.3. f( ) 1 -1 0 1 2 3-1-2-3 . 2.3 , (2.1) (2.2) , . Copyright & A K-C
  20. 20. 20 ( ) ( ) ( )( )11 111 ++=+ nxnybfny . (2.3) . , , . , , , ... , () . . 2.1. , , . , , .. ( ) =f . , . ( ) ( )nybny 111 1 =+ (2.4) ( ( ) 001 y ). ( ) ( )112 += nyny , ( ) ( )nybny 112 = . (2.5) . 112 yby = . (2.6) Copyright & A K-C
  21. 21. 21 ( )ny1 ( )11 +ny , - -. ( )ny1 n- ( )01y . 1y , 0111 = yyb , .. ( = 111 yby ), . . . 2.4, (2.4). y y y =b y 2 1 1 1 y (0)1 2 0 y=y2 1 . 2.4 1 . 1 1 ( ) (2.6) 01 >b . (. . 2.4). Copyright & A K-C
  22. 22. 22 2 1 ( , (1.13), ) (2.6) 11 >b . . 3 1 ( , (1.14), ) (2.6) 11
  23. 23. 23 y y y=-y 2 1 1y (0)1 2 y =y 2 1 . 2.6 , 11 =b (2.6) ( ) (. . 2.7). y y y =y 2 1 1 y (0)1 2 y (0)1 y (0)1 y (0)1 0 . 2.7 Copyright & A K-C

24. 24 11 >b , . (1.3) (. . 2.8 2.9). y y 1 2 y (0)1 y=by 2 1 1 0 y=y2 1 . 2.8 y y y=by 2 11 1 2 y=y2 1 y (0)1 y =-y 2 1 . 2.9 Copyright & A K-C 25. 25 . . 2.10. b y 1 1 0 1-1 1 -1 ) b T 1 0 1-1 1 2 ) . 2.10 Copyright & A K-C 26. 26 : , , . ( 01 =y ) 11 b . 11 =b , 1y . (. 2.10) ( ) () 1b . 1b ( 0=T ), 1=T 2=T . , (2.4), . (2.4) ( ) n bCny 111 = , ( )011 yC = . 111 yby = . : 1. 11 b . ( )01y ( ). 2. 11 =b . , . 3. 11 =b . 2=T . , . 2.2. ( ) ( )( )nybfny 111 1 =+ (2.7) ( ( ) 001 y ). ( )112 ybfy = , ( )f . , Copyright & A K-C 27. 27 (2.1) (2.2) . . 1 d d 1 01 1 1 : 01 =Y { }1;11 Y (. . 2.11). , 1 1. 1 -1 0-1-2 1 2 y=y2 1 y =f b y( )2 1 1 y (0)1 y2 y1 . 2.11 Copyright & A K-C 28. 28 1b < -1 : ( ) { }1;11 Y , 2=T (. . 2.12). 1 -1 -1-2 1 2 y=y2 1 y =f b y( )2 1 1 y2 y1 y =-y 2 1 . 2.12 1b = 1, ( )1;11 y - { }1;11 Y . 1b = -1 ( )1;11 y - { }1;11 Y , - 2=T . . 2.13. : . (. 2.13) , ( ), 1b . 1b ( 0=T ); 11 b , -, 1T = ; 11 b Copyright & A K-C 29. 29 , -, 2=T . b y 1 1 0 1-1 1 -1 ) b T 1 0 1-1 1 2 ) . 2.13 Copyright & A K-C 30. 30 . 1 d d 1 01 1 >= = b y f y . 11 >b . 2.14. a , a3 .., 11 ba = . [ )aay ;1 112 yby = , [ )aay 3;1 2112 = yby , [ )aay ;31 2112 += yby . 2-2 a-a y=y2 1y2 y1 y=by 2 1 1 y=by-2 2 1 1 y=by+2 2 1 1 . 2.14 , , , . , . . 2.14. 1b >1 ( 2). (. . 2.15). Copyright & A K-C 31. 31 2-2 a-a Y ( )01 Y ( )11 1 -1 y=y2 1 y2 y1 y=by 2 1 1 y=by-2 2 1 1 y=by+2 2 1 1 . 2.15 ( ) [ )1;01 ay , ( ) ( ) 201 111 = yby . [ )aay ;31 , , ( ) ( ) 212 111 += yby . 2=T ( ) ( )02 11 yy = . ( ) ( ) ( ) =+== 2100 1111 ybYy ( )( ) ( ) 220220 11 2 1111 +=+ bybybb . , ( ) 1 1 1 2 0 b Y + = , ( ) ( ) ( )0 1 2 11 1 1 11 Y b Yy = + == . ((1.11), (1.12)) () ( )01Y . == 2ffN 22 11 2 1 + byb , 1 d d d d 2 1 1 2 >== b y f x fN . Copyright & A K-C 32. 32 , ( )01Y , ( )11Y ( 2) . 1b < -1, , , , 2112 = yby (. . 2.16). 2-2 a-a 1-1 1 -1 y=y2 1 y2 y1 y=by-2 2 11 y=by+2 2 11 . 2.16 2 1 1 2 1 2 , y b y y y = = 1 11 1 2 b Yy == . 1b = 1, [ )1;11 y . 1b = -1 ( ]1;11 y , 2=T . Copyright & A K-C 33. 33 b y 1 1 0 1-1 1 -1 ) b T 1 0 1-1 1 2 ) . 2.17 . . 2.17 . 1b ( 0=T ). 11 =b , Copyright & A K-C 34. 34 1T = , 1 1b > 2=T . 11 =b 2=T , 11 35. 35 ( ) ( ) 1 1 1 11 11 0 b A b b A yny n + = . (2.10) 11 0 ( ), 0>A . Nn = , ( ) 111 1,0 YNyY = . 1,01 = N b . N 1ln 3,2 b N . 1b A. . ( ) ( )112 += nyny . (2.9) Ayby += 112 . . 2.18 01 >b , 0>A . 1Y (2.11). { }4,0;2,0A . 2.19. . Copyright & A K-C 36. 36 -1 1 1 -1 y 2 y 1 y =b y +A 2 1 1 A Y 1 y=y2 1 y (0)1 0 . 2.18 b0 1 0,2 0,4 0,6 0,8-0,2-0,4-0,6-0,8-1 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 Y1 A=0,4 0,2 . 2.19 Copyright & A K-C 37. 37 2.3.2. : , (2.1) (2.2) . , , ( )Aybfy += 112 . 1b A . 1b >0 ( ). , (2.11) 1Y 11 Y (. . 2.18), .. 1 1 1 b A , 1 11 1b A b + . (2.12) . 2.20. -1 1 1 -1 1 b A 0 . 2.20 Copyright & A K-C 38. 38 1b A . 2.21. -1 1 1 -1 y 2 y 1 y =b y +A 2 1 1 A y=y2 1 y =-y 2 1 Y 1y (0)1 . 2.21 1. 1 1 1 + b A , .. 1 11 1b A b + . (2.13) . 2.20. 1b A . 2.20 . . Copyright & A K-C 39. 39 1b >0. 0>A . 2.22. 11 =Y . -1 1 1 -1 y=f b y +A ( ) y 2 y 1 y (0)1 0 y=y2 1 1 1 . 2.22 ( ) ( ) 1122 == Nyny . ( )ny2 (2.10) : ( ) ( ) 1 1 1 1 12 11 0 b A b b A yny n + = + . ( ) 001 =y 1N 1 ln 1 ln 1 1 1 = b Y A N , Copyright & A K-C 40. 40 [ ] , 1Y (2.11). A ( ) 001 =y 1 ln sign 1ln 1 1 1 = b Y A N . , 2 , 1N . 1b A . 2.23. 1Y , (2.11). ( )01y , 1n , (2.9). ( ) ( ) 11 10 bAy < . . -1 1 1 -1 y 2 y 1 Y 1 y=f b y +A ( ) y (0)1 y=y2 1 y =-y 2 1 1 1 . 2.23 Copyright & A K-C 41. 41 0 , ( 1n ). , , . 2.24. b0 1 0,2 0,4 0,6 0,8-0,2-0,4-0,6-0,8-1 0,2 0,4 0,6 0,8 Y1 A=0,4 0,2 . 2.24 1b . 1b >0. , , . T , 1b A. 2=T . 1 0,6, 0,9b A= = . 2.25. ( )1 11 1/6y A b= =. ( ) ( )00 11 Yy = . ( ) ( ) ( )1 1 1 11 1 0y Y bY A= = + . ( ) ( ) 212 112 += Ayby . Copyright & A K-C 42. 42 -1 1 1 -1 y 2 y 1 Y (1) 1 Y ( )0 1 y =b y +A 2 1 1 y =b y +A-2 2 1 1 y=y2 1 0 0,4 0,8- ,40- ,80 0,4 0,8 - ,80 - ,40 . 2.25 2=T ( )01y ( )11y ( ) ( )02 11 yy = , .. ( ) ( ) 200 11 2 11 ++= AAbyby . ( ) ( ) ( ) 2 1 1 11 1 21 00 b bA Yy + == . (2.14) ( ) ( ) 2 1 11 1 1 21 1 b bbA Y + = . (2.15) , 6,01 =b , 9,0=A ( ) 875,001 =Y , ( ) 375,011 =Y (. . 2.25). 2=T . ((1.11), (1.12)) () ( )01Y . Copyright & A K-C 43. 43 211 2 12 ++== AAbybffN , 1 d d d d 2 1 1 2 A ( ) 101 Y , (2.16) ( ) 111 44. 44 0 0.2 0.4 0.6 0.6 0.7 0.8 0.9 A 0.8 b A= -1 b 1 1 . 2.26 3=T . 1 0,6, 0,7b A= = . 2.27. -1 1 1 -1 y 2 y 1 Y ( )2 1 Y ( )0 1 y =b y +A 2 1 1 y =b y +A-2 2 1 1 Y (1) 1 y=y2 1 0 0,4 0,8- ,40 0,4 - ,80 0,8 - ,80 - ,40 . 2.27 Copyright & A K-C 45. 45 ( ) ( )00 11 Yy = . ( ) ( ) ( ) AYbYy +== 011 1111 , ( ) ( )== 22 11 Yy ( )1 1 1b y A+ . ( ) ( ) 223 111 += Ayby . 3=T ( )01y , ( )11y ( )21y ( ) ( )03 11 yy = , .. ( ) ( ) 200 1 2 11 3 11 +++= AAbAbyby . ( ) ( ) ( ) 3 1 1 2 1 11 1 21 00 b bbA Yy ++ == . (2.18) ( ) ( ) 3 1 11 2 1 1 1 21 1 b bbbA Y ++ = . ( ) ( ) AAbyby ++= 11 2 11 02 , ( ) ( ) 3 1 2 11 2 1 1 1 21 2 b bbbA Y ++ = . (2.19) , 6,01 =b , 7,0=A ( ) 801,001 =Y , ( ) 219,011 =Y , ( ) 832,021 =Y . 2=T () ( )01Y . 21 2 11 3 13 +++== AAbAbybffN , 33 1 1 d d 1 d d Nf f b x y = = < . Copyright & A K-C 46. 46 , , ( )01Y , ( )11Y ( )21Y ( 3=T ) . (2.16) (2.17) ( )Ab ,1 , 3=T . (2.16) (2.18) 1 2 1 3 1 1 1 bb b A ++ + , (2.17) (2.19) 1 2 1 3 1 2 1 1 21 bb bb A ++ + < . . 2.28 , 3=T . 0 0.2 0.4 0.6 0.6 0.7 0.8 0.9 A 0.8 b1 A= -b1 1 . 2.28 T . Copyright & A K-C 47. 47 ( ) ( )011 YTY = , ( ) ( ) 21111 += ATYbTY , ( ) ( ) = += 2 0 11 1 11 01 T i iT bAYbTY . ( ) T T i i b bA Y 1 1 0 1 1 1 2 0 = = , (2.20) ( ) T T i Ti b bbA TY 1 1 0 1 11 1 1 2 1 = = . (2.21) ( )Ab ,1 , T . 0>A (2.16), ( ) 111 A : 1 12 1 11 1 2 , 1 b A Y b A Y = = . 0 49. 49 1 12 1 11 1 2 , 1 b A Y b A Y + = = . , B. . 2.30 , ( )01y . , 01 A ( ) 1 1 1 0 b A y > , 0 50. 50 . 11 51. 51 + = 22 1 2 1 1 sin)cos1( bb X Y . . )e()e()e( 1 jjj XYH = , . - (), .. )()()( 1 = XYH , , .. )e(arg)( = j H . + = 22 1 2 1 2 sin)cos1( 1 )( bb H . ( )2 f . ( ) += cos21 1 2 1 2 bbf . : = sin2 d )(d 1 2 b f . 2, . + = cos21 1 )( 1 2 1 2 bb H [ ] 2;0 [ ] ;0 , { } ,0 . 1b . , Copyright & A K-C 52. 52 01 >b , 01 53. 53 cbb b + = cos21 |)|1( 2 1 1 2 1 2 1 . 1 2 11 2 1||4 cos b bb c = , 1 2 11 2 1||4 arccos b bb c = . , 01 =b ccos . c 1b . 01 >b 1 2 11 2 14 cos b bb c = . (2.29) : 2 1 2 1 1 2 1 d cosd b b b c = . (2.30) 11 b . 11 =b 1cos =c , 0=c . , 1b 1cos =c , (.. =c ). (2.29) 016 1 2 1 =+ bb , Copyright & A K-C 54. 54 , 832,11 =b . 11 b , 01 55. 55 3. . . . . . 3.1, 1b 2b 1b 2b . y n+( 2)x n+( 2) 1 b1 z-1 b2 z-1 y n+( )1 1 y n( ) 1 . 3.1 : ( ) ( ) ( ) ( )212 12111 ++++=+ nxnybnybny . , . . Copyright & A K-C 56. 56 3.1. ( ) ( ) ( )nybnybny 12111 12 ++=+ ( ) ( ) ( ) 012 12111 =++ nybnybny (3.1) ( ) ( )1,0 11 yy . 021 2 = bqbq , (3.2) 2 4 2 2 11 2,1 bbb q + = . (3.3) (3.1) ( ) 22111 nn qCqCny += . (3.4) 1C 2C ( ) ( )1,0 11 yy . (3.4) 1 1 2 1 1 1 2 2 (0) (1) . y C C y C q...