Romanyuk YuA Osnovy Tsifrovoi Obrabotki Signalov Chast 1 Svoistva i Preobrazovaniya Diskretnyh Signalov

7/21/2019 Romanyuk YuA Osnovy Tsifrovoi Obrabotki Signalov Chast 1 Svoistva i Preobrazovaniya Diskretnyh Signalov

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ISBN 5-7417-0144-2 (. 1)

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( 3- )

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.. . .. 16.12.2005. 60 X 84 Vie- .

. . . . 20,75. . -. . 20,0. 700 . -458.

- ( ) - 141700, ., . , ., 9

ISBN 5-7417-0144-2 (. 1) .., 2005ISBN 5-7417-0145-0 - ( ), 2005


3/331

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. [6]-[10].

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4/331

. - - .

. 1. . 2.

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. , (1.10.15) 15-

10- 1. , , .

. . .

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4


5/331

A 1.

1.1.

, . , .

, *(/),

/(, ), , . .

x{t).

- .*(/),

, (. 1.1.1).

( \ ^ *('>/

/ \/ t ~ \f k \ i t()=A-( -v(/) (0 = / - )

/

* 1

y'i/

/

r i 1 V /

)

V \ t

.

5


6/331

(/)

*(/) y{t\

, At (. 1.1.1). - (. 1.1.16).

:(:)

(. 1.1.2), . . , - mq, q -

, a w - ,

-0, 1, 2,..., -1, -2", - .

4

*1-

\ ''

*- kV . 1.1.2

, .

,

.

( ) X ( f )

, -

, /].

, , . . , .

, .

x(t) , *(/) = 0 /< 0.

. .

6


7/331

1.2.

, , S.

, . . , . , ,

d(\ , ) , .

, , .

. d(\, )

, :

) d(, ) > 0 d(, ) = 0, = ;

) d{x ,y ) = d ( , ) ();

) d(, z ) < d(, y) + d(y , z ) ( ).

S d

. , , . .

, [,],

( 1.2.1)

( 1.2.2)


8/331

-

* = ( * - 1 * 1 *) = ( -, )) :

4 x . y ) = Z [ * . ^ 3 ( L2-7)/= 0

2: 1+ 0 = 0 +1 = 1, 0 + 0 = 1+1 =

. (1.2.7) .

, , . [13].

.

,, 2, , ,

, = 0

= ! , = 0.

N

S,

:

=0

S N-

. {}

S , S

. N { }

. .

d 3(x, ) = max |x( k) - y (k) |. (1.2.6)

8


9/331

.1. .

2.

(, ), ,

, . :

(X. ) = ( , ) ;

(, + )= (, )+ (, z); (1.2.9)

(, ) = (, ), (, ) = (, );

(, ) > ,

(, ) = 0 , = 0.

.3. .

||||= V (X, ) > 0. (1.2.10)

0 , :

(*> ) - (*> ) .2.11)

V *) (> ) II! IIII . ,

, . . (, ) = 0.

|cos0| < 1,

|(, )|


10/331

4 .= % * . (1.2.14)

(>)= V 11 (1-2.15)[, / * .

|| 1 = 7 ^ ) = 1 , 0-2.16)

|| |2= (, > = - -2-17)

N 1 = J I h r (i-2^ )V =

(1.2.17) . (1.2.13)

{ }, -

. ,

{} {} {} ,

(X, ) = ( . * * ) = (1.2.19) = 0 = 0 = 0

( pw ),

:

S > A = Z X ( 1-2-20) = 0 = 0

.

:

I w h J l k - b . r (1-2-21)V = 0

, , ,

. . -


11/331

, , , , , . 1.2.1. N

, /7= 1, 2, N.

N

{ },

.V

= Z = z = z (*> >. . . )-

=0 = 0 =0 =0

1.3.

L2

S

*(/), , .

, jt(f)

J|*(0|2dt< 0 0 .

, ,

*(/) - 1.

11


12/331

L 2 .

.

L2 ,

d2(xt )

y{t) (/).


13/331

X + = W O) + (0), 41 ) + (1), * (* ) + (\ ...],

= [(0), ax (l) , a(\ ...].

,

12,

(1.2.13) - (1.2.16). (;)

. * (0 L 2,

( )

12. L 2 12

,

N

x = [jc(0), (1), jc(2), (\ x (N -\ ) ]

12,

12. , 12 N

\|/. N

.

12

{\|/}

00

(1.3.8)

(1.3.10)

(1.3.9)

L 2 12 (1.2.18).

(1.3.11)

13


14/331

(Y. V ) =

0,

2

m = n,

(1.3.12)

(1.3.13)

m^n.

1.3.1. 12

() (. . 1.7):

,

. , ,

(1.3.16) (1.3.17) (), 3. ,

N. () {)-

( N) . N- . , :

N ( , N , ). N.

, = 0,1, N \,. .

12.

() :

(1.3.15)

(1.3.16)

(1.3.17)

14


15/331

1.4. .

*(/), ,

,

* (') = ('), (1.4.1)

{


16/331

.

, (1.4.5).

|(/)}

. 1.4.1.

. 1.4.1.

. -

{(/)}, .

. 1.4.2.

/ t u T

. 1.4.2.

%(t)

hn(t)= ( -/ ). t = :

jx (T )h n(T -x )d x =|()()^ = , =1, 2,

16


17/331

1.5. .

(1.4.1) , , . . .

1)

. ,

2) , 2(N )= 0. ?

(1.5.1) .

, .

, ,

, (1.5.2),

{ (0}- ,

N- (1.4.5)

* (/) N-

{ (0}- ,

(1.5.1) , , (1.5.2),

2{N). , N

.

,

2

(1.5.1)

(1.5.2)


18/331

.

= ^ -5-3)

a = \x{t)V(t)dt. (1.5.4)

. (1.5.1)

( ) = 1 (/ )] [* ( - I * ; : (') ]}


19/331

1.5.1.

. -N

= ^ , * I

II* - II2=

. , ,

, ,2-, , . 1.5.1.

, I , , 2-.

X , 2 . ,

. ,

:

-

1.6. L2

: .

. .

19


20/331

J *( , ( =

,

/ sin 2/1 (/ :/) 1* ( = / ; - , , (1.6.1)

2nfe(t -k A t) 2/

At, (~,):

, * 1 ,| || 2 ^ 1 , , (1.6.2) ! =/ = 2 ^ = /

(. . 2.2). , ,

,

t = kAt:

(1.6.3,2nfe( t -k b t )

/ V" . s m 2 n f (t -k A t )

x ( t )= X * (* ')~ . ,- , (1-6.4)* = _ 2nfe(t -k A t)

. , .

- 00

J k(t) e -J2' f , dt= 2/ (f)e x p (- j2 n fk A t) (1.6.5)

- kAt.

2/ (/ )

(. 1.6.1). t - kAt

, 1.

20


21/331

/= (//)/, /= 1, 2,

.

sin 2 f ( t ~ A t )


22/331

.


23/331

00

cwal (w,0 ) , (1.6.12)n=0

.n | ,t(Q)wal(w, 0 ) c/0. (1.6.13)0

x(0)

N= 2V v - ,

[25].

, wal(, 0 ) [0 ,1) .

wal(, 0).

, wal (, 0) ( -

), ( - ) ,

0 = 1/2. , -,

0 = 1/2 , -,

. - . :

ni= 0, .

, , wal(5, 0). . . - ,

0 = 1/2.

= (101). 2- 1,

0 = 1/8, 3/8, 5/8, 7/8. wal(5, 0) . 1.6.2.

wal(0, 0 ) = 1,

wal(>7, 0) = 1 .

V1

-^ rij2', nj =0, ni=1, =(wv_j... , 0)./= 0

. = 1,

23


24/331

1/8 1/4 3/8 1/2 5/8/ 3/4 7/8 1

+1 1 1 1 1


25/331

p = (p v_\

0)

Pi= , 0 ,

2. -

[2V_I, 2V-1).

. 1.6.4. - , .

)

. 1.6.4. :

- ; - [0, 1).

. , , 2:

wal(w, 0 )wal(w , 0 ) = wal(/7 /, 0) . (1.6.16 ) had( /7,0)

.

(

). 1.6.1.

25


26/331

1.6.1 h h

. . . . . .

0 000 0 000 01 001 001 1 100 42 010 011 3 110 63 011 010 2 010 24 100 110 6 011 3

5 101 111 7 111 7

6 110 101 5 101 57 111 100 4 001 1

0

1.

-:

^ (9 )= cwal(, 0), (1.6.17)*

1

= J x(9)wai(, eye.

N =2V, v - , SN(Q) -

1/N, ,

(0):

) = 1 ^ ( 9 ) , (1.6.18)*=

3ct = NJ (0)^0 ,

fl, k / N < Q < ( k + l )/ N , ) = \

{0, 0.-

:

26


27/331

e \ N ) = J[x (0)-5 W(0)]2^ -

[7]

N - 2 V\

'(0 ) - .

:wal w(0, ) = wal(, 0)wal(/w, ). (1.6.20)

, 0 ;

0 [0,1), [0,1), . 1.6.5. ,

. .

. 1 . 1

. 1.6.5.

27


28/331

1.7. 1,

, N.

TV-

.

iV- N,. .

N x N .

, :

-1 0 |

1

2 I1(, *)=

0 1 2

1 0 0 0

0 1 0 0

0 0 1 0

1

1 (/?, ) = (1.7.1)1 - ,

[ * .

, 1:

g 1


29/331

,

1". (1.6.8) (1.7.3),

t - A t , t -

. .1. .

2. || *|| .

3. N .4. :

(1.7.5) (1.7.6) (), 3.

5. :

I= ( + w )modJV, . . N.

6. :

7. - WNnk WNnk

N , ,

( N - n ) N N ,

N >() - 2nrtk /N .

- ,

.(1.7.4)

(1.7.5)=

(1.7.6)

(1.7.7)

(1.7.8)

w~nk w {N~n)kr v N - VV N

29


30/331

.8. N

, . N

N ( - nn = N 12) (^" 0 "

- . N

( = 0 )

(N -\ )/2 - .

9. (. 1.7.1).

, ,

2nn/N .

N W k

2 , . . N .

-

* - 1

1 1**0k - N - l

(N-n)Jt

N - .

. 1.7.1

n/N W k

( N - n ) / N - WfjN~n)k

(1.6.8) .

10. :

An k+ mq)N (1.7.9)

-

. 1.6 0,

30


31/331

[0,1). -

* + \ =0, 1, 2, N - I ; N = 2\

_N N

0

, . . wal(w, ) pal (/?, ).

N = 2V

W^. /-

/- ( ) . N= 8

. 1.7.2.

1-0-1.

-

-

-

= = = 1:-

-

d = G

_______T b R e

W8=

1 1 1-1-1-1-1

-1-1-1 1 1

1 _1_1 1 1-1-1

-1-111 -1-1 1

~1 -1 1-1 1 1-1

1-11 -1-1 1-1 1

1-1 1-1 1-1 1-1

-waico,*)

-*wal(l*)

-*wal(2, k)

-waK3. k)

4*wal(4.)

-wal^A)

-wal(6yk)

-*wal7,A:)

J L

1/4 3/4- L .11/2

a) 6)

. 1.7.2. , , N= 8: -; -

\ , . .

w ww ; = i , (1.7.10)

I - ,

. 1, ,

. -

-.

31


32/331

, .

N =

2:

,1 1

1 -1, HL HL

HL -H L(1.7.11)

, (. 1.7.3).

hadft)

had&O)

had(2,0)

had(3.G)

bad(4,0)

had&O)

had(6,0)

had(7,G)

" U b i F b F

= t = F = b F = b

I___L.1/4

___I___t. l. ,.


33/331

W- :

had(A, ) =had( h N , ) = had (, N).

{har(r, m, 0)} , Q = t/T -

t [, ), 1910

L 2[0, 1). . \.1

.

1 0

.)

1II-1

S'T1t 1111 } 1II

1U \'ir

=

1 1

1

S

1

V2

1

-V2

1

-7 2

-1

0

-1

0

-1

0

-1

0

0 0 0 0 V2 V2 - -V2

2 - 2 0 0 0 0 0 0

0 0 2 - 2 0 0 0 0

0 0 0 0 2 - 2 0 0

0 0 0 0 0 0 2 -2

)

. 1.7.4. N = $:

- ; -

, [29]:

har(0,0,0) = 1, 0 [0 ,1);

har(r, w, 0) =

m- 1 m -1/2 < 0 < -

v 2r

m -1 /2 m---------- < 0 < ;2 r 2r

0, 0 e [0 ,1),

'(1.7.14)

0 < r< log2N 1< m < 2r

. 1.7.46.

3 .

33


34/331

,,, /?= log2N.

har(0,1, 0)

. - . , , . - (, ) -

, -. - . 1.14.

, . 1.1-1.7

1. , .

2. , .

3. .4. (1.14), (1.16).5. (1.17).

6. L 2(T). .

^ ). .7

. ?

8. { > t s T I

.

9. - |(, )| < ||||-||||,

, :

34


35/331

10. :

= 1 | (/ ) (/ - ) .'

- |(, )| < ||||*||||,

,

|/( )|


36/331

18. {; /?= 1 ,2 , . . . } -

2( ). 2( ) -

:

19. . .

( - 0 0 , ).

20. .21. . .

.22. -.23. .

- .24. ().

.25. 8x8

.26. ,

.

27. jc(0) = 0, 0 [0,1) . -.

28. jc(/)

2. y(t)

. y(t),

. (1.1), (1.2), (1.3).

29. : ) ; ) , ) .

30. 16 .31. , 16

.

36


37/331

1.8.

1.8.1. .

:

'2d t< . (1.8.1)J M 2, x(t)- ( ),

, , , .

jc(f) - .

[16], :

) x(t) /(,-);

) jc(r) 0,- );

) jc(/) ,

.

: (/)-

, X ( f )

(/)

X ( f ) = l.i.m. \x(t)e~J2nfldl, (1.8.2)->

l.i.m. (limit in the mean):

2

lim J X ( f ) - j x ( t ) e - J2nf'dl d f = 0. (1.8.3)* 00 - T

, X ( f) -

, x ( t )

X ( f )

w

*( 0 = l.i.m. \ X ( f ) e i2' f 'd f (1.8.4)

:

37


38/331

f|* (/)|2A = \\X( f )\2df (1.8.5)- -


39/331

^(co) = R e [ A r ( c o ) ] = J* x(/) cos /dt^-00

00

( co ) = I m [ ^ ( c o ) ] = J x ( / ) s i n c o / < # ,-00

\X(a)\= ^ 2() + 2()

- - (),

, 5(co)() = -arctg -

()

- () .

2) A"(co). (

)

Z ( co ) = * * ( - cd ) .

, () |^()|-

, i?(co) () - .

*(/) - ,

Jf(co) = (-) = () =|()|,

. .

.3) . x{t)

() = ] '()ejal da= | JT(cd) | da=2n 2

1 ^= - j \ X (co)| COS [ f + ( )] rfco.

= 0 ^ (0 ) = J (/ ) - .-

1 00 /= 0 jc(0) = JX.() .

2 4) .

eJ{* ,

.


40/331

\

- :

x(t) = [ " '] = cos / = + eia' ).

, - (. 1.8.1).

Rc , - -

, . -cos- ,

"+() > ,

^() )= 2\\X{ f )\2df. (1.8.13)

-a> 71 * (/ )- ( ),

,

V , (/ ) = | * (/ ) f = X { f ) - X \ f ) (1.8.14)

40


41/331

. . /.

4)

] , (0 2(0 1 dt= ] x2(t) i - ] ejvl dv d t:

-y(co-v)rdt dv = = W V)

= - ) x i (v)Z 2(- v)rfv = Jf, () Jf2(),

(1.8.15)

. e. .5) . :

= ( 0 2(0 = J* l (T) * 2 ( f - T) rfT -00

:00 00

- ia,dt =

(1.8.16)

( )= | J, (x )x 2( f -T ) ^ T

00 00

1*1 ( ) | 2 dx.

$ = t - ,

( )= Jjc,(x)e^ Vx |jc2 (9) ~1,V S = X, (>)2(), (1.8.17)00 00

. . . .

, , * (/ )- ,

h ( t ) - (

-) jc(f) (), h(t)).

:

y ( t )= j x (T )h ( t - T )d t = x ( t )h ( t )


42/331

, .

6) .

(/) :00

)= Jx(0 x{t + x)dt. (1.8.19)00

, .

. ( )- ,

.

(1.8.12), ,

jc (f ) Rx(t)

\|/() = | ()|2 :

Rx() ,(). (1.8.20)

1.8.2.

().

y(t) , . , -

,

. , .

1. ( ).2. ,

.3. ,

. 8-.

:

^)

*(/) ;

42


43/331

# ( )

;

() = ( ) ()

;

{1)

() .

, , , ( ) ,

, t - q o t= . ,

, , , . . ().

, , ( ). :

- ;

- .

, , , .

.

1.8.3.

(. 1,8.2).

Jf(a>) = [ 1= -* \ % = -/2 '

t /2

tor2

( .8 .2 )

43


44/331

(. 1.8.26).

)

- / 2 /2 t

)

. 1.8.2. - , - , -

|"()|

4/ . ,

-

2

. 1.8.3

(2 + 1)

-1, 2, 3,

. (. 1.8.2) .

. 1.8.3 ( ) = 1 = 1 .

. 1.8.3, . 1.8.36. .

44


45/331

, /2,

X ( f ) = ExSm e~Jnf'/

e~jnfx ,

. 1.8.3.

1

) 21

45


46/331

(. 1.8.5)

(0

, -( /)2,

(2/)5(/ + /2) (2/)8(/-/2).

,

, . . 1.8.56.

(1.8.22)

(. 1.8.4). .

- 6) t l A X C O ) '

2F.T/7

/ 5/ w- / 2 0 /2 t

. 1.8.5

-(/)2(> )+ ( /)('2 + e 'J2) = (/)2(),

(1.8.23)

46


47/331

. )

fexp(-aO, t >, 1

[, t<

- - .

mc(t )

. 1.8.6

, a > 0 (. 1.8.6).

X ( f ) = \~'~ i2*f'dt=-----------------------1---------- |+/2' ' 1 - ( + /2/) +j l n f (1.8.24)

X { f ) = |(/)|[(/)],

( )

\(\ = -

y]a2+(2nf)2

( )

(/) = -arctg(2n//a),

. 1.8.66.

(1.8.25)

(1.8.26)

47


48/331

X ( f )

Re [.Y(/ )] Im [Z (/ )] ,

. 1.8.6.

. | ()|

Re[J f(/)] , (/ )

1[(/ )] - .

x(t)= ',

. 1.8.7.

:,(/) :

x(t) = x ,(t) + x](-t) .

() = * , ( ) +* , () =

2= -------------1--------------;-----( + j) ( -j) +

(1.8.27) . 1.8.76.

. 1.8.7

- :

x {t ) = e m l (1.8.28)

(. 1.8.8). , ,

[~2(/2)2] = 0,1.

2 ^ - In 0,1 3,035

48


49/331

X(a>)= \ e ^ ' 1+jM)dt .

. 1.8.8. a - , -

,

.

pV +ja t= V +j(at = (/+-=^)2+ -^ 7 ,4p 4p 2p 4p

A'(co) = e /4p!) j e dt.

4 = / dt = d^/

(>) = -

- )2 /42

i (1.8.29)

. . (. 1.8.8). .

*(/),

, = ,

49


50/331

/2 /2

^() = J (/) dt = J x(t) dt= const = .-/2 -/2

, eJb}xl2 = 1.

, , , , .

, 7/2

/2


51/331

x(t )

. , .

1.8.1 .

/,

1.8.2 0,9 5,1/ 0,81 / 0,73

1.8.4 0,541 5,3/ 0,84/ 0,46

1.8.5 0,596 4,57/ 0,73/ 0,43

1.8.6 1,15/ 6,16 0,98 1,13

1.8.8 0,83/(3 1,64(3 0,26 0,22

:

-/ > , (1.8.30)

- , /.

/,

.

00\ { t - Q 2x \t )d t

J x\t)dt-

00 / 00

t0= j t x 2(t)dt/ j x 2(t)dt - (- / -

), ^ - .

,

()2= - ^ J^-cO o)2!^ )!2^ ^ ^ J|A"(to)|2dc,

0= -!- | | ^ .2 J

51


52/331

(1.8.31)

. . .

=1/2 .

. .

x(t ) t , 1//,

^ ( )- , 1/, . .

. , .

, , , . - , .

- . [5-7]. .

- , , - , . ,

, - , .

,

:

1.8.4. -

(1.8.32)

/0+

| 8 (t - t0)dt = \ > 0. (1.8.33)

5(/-/0) = 8(/0-0 , (1.8.34)

52


53/331

} 5( t - t a ) d t = j 5( t - t 0)dt = , e> 0. (1.8.35)/0 -e

, -

(-, 0-

j 5 ( T - / 0 ) d T = y ( / - f 0 ), (1.8.36)

(t-/0) - :

0 t < tQ,= = ^0j (1.8.37)

1 t > tQ.

, -. :

= (1.8.38)

) )

^

~ 0

' / ()] ' '

. 1.8.9. --, -

(. 1.8.10), , .

I )

/-/2 2

. 1.8.10. , -,

:

(/) = 5(/);-+0

53


54/331

1 t2lim exp(~ ) = 5(?);

/2T 2x

. n tsin

lim-----^ = 5(0-*->0 Tit

() . , - , ,

.

, ,

/0 x(t)

J x (t )5 ( t - t0)dt =x(t0)9 a < t0 b.

Jt(/) t = t0 ( ),

j x ( t )5( t - t 0)dt = ( l/ 2)[(/0+) + x(t0.) ], a < t 0 < b , (1.8.40)

a

(/0+) x(t0_) - (/) .

(1.8.39) ,

8(/ -/0)

(. 1.8.10), . - ,

:

\x {t )b{t -d )d t = x{a\ (1.8.41)

(/) 8(/-)= x ( t - a ) , (1.8.42)

(0 *8(^ -) = ()-8 (/-), (1.8.43)

S [( f- f0)/a] = |a|8(f-f0) 5(at- /0) = 7 8(/ - ). (1.8.44)

54


55/331

(1.8.41) ,

00

J 5(t)e- j2nfdt = e=1, (1.8.45)

. . - . ( ):

00

6(0= \e n%f'd f (1.8.46)-

, , 1


56/331

+ (/ = 0_) -

(/ = 0+). , t= 0

5 '(0 -1 It .

- :

J x ( t )V n)( t - t 0)dt = ( - l ) " x M ( t0). (1.8.52) () (! ) ( )

t = t,

j x ( t ) P \ t - t 0)d t = ^ [ jS '> (t0. ) + X**\t0_) ]. (1-8-53)-

- (1.8.52) |

I

= ()" ~/' (1.8.54)

, t0 =0, 8()(0 ^ 0 ' )" (1-8-55)

1.8.5.

|

(1.8.49), -!

(1.8.56)

A sm ln fy = - e ' W ) [ 5 ( / - / 0) - 5 (/ + /0)].

2 2j ,

/0

J 8(n)( t - t 0)e- ja dt =


57/331

- !2, / 0.

= 2/ (1.8.56)

Acos 0/= ( ^ 0' + (0' ) [8( - 0) + 8( + 0)],

(1.8.57)

Asin 0/= ( '/- [8( - 0) - 8( + 0)].V j

(/) (1.8.36) - (1.8.38).

y(t)= e~at-(/), > 0

, , . .00

|| -/ -(0|


58/331

. 1.8.10.

+1, t >0,sgn/ = ,6 1 -1, /


59/331

1.8.2.

u(t) = J () - . : /, -

/:

( = J x(x)dx = ~ y(t -)].* t- L

(1.8.60) ,

(/) = - f x(x) dx U(v>) = (1 - eJmT) = ( ) ^ "/2 77 2

(1.8.61)


60/331

sin(co - co0) /2 sin(co + co0) /2 T- (1.8.62)

( - co0)t/2 ( + co0)t/2

= 4/0 ( ) . 1.8.12

(-0) -( + 0)

|AXco)|.

. 1.8.12.

, , , ,

\()\ > 0 :

. , , .

1.8.4. . N (. 1.8.13).

] . (N-1)T

t

. 1.8.13

60


61/331

^ ( ) .

N

* ( ) = ()[1 ++ e~J"2T+.. . +e iNT/2 - jcaNT12 -i

=x ,() - j - p s r =x , =

= X - " -] ),2)S*n /21 sin 77 2

, = 2/ NT, m -, () = 0.

,( (. 1.9.2).

.

61


62/331

2/32/ 2/ i!X(u>6)

2/4 2/ 2 -/. 1.8.14. :

- , -

. .

______________________________________________ 1.8. 2

*(/) X ( f )

x{at )- -X\- \

X ( f + f 0)

62


63/331

3 df 2 / ^ ( / )a t

4 1 < /)

^ ~ j2 n J f

5M - U o * - ^

i j2 n f 2

6/rt f l . ' > 0 1 5 ( / )ct(/) =


64/331

13 * * 2 1 2 + ( 2 / )

14 - y'Sgn/nt

15 t e~al a (t ) 0

( + j2 n f)

162n

17 i [ 8 (/ + /0) - 5 ( / - / 0)] 7 s in 2 / / 0

18 x( / )cos2 /0/ o i [ l ( / + / 0) + I ( / - / 0)]

19 1 * . 8 ( / - ]k--co * n = o \ * ) V * /

20 Z * ( / - 7 ) = Z * ( W > y2" mk= -co 1 =oo \ * J k=-00

64


65/331

1.9.

x(t ) = e Jaiot -


66/331

- , .

x ( t )= J 7'' 2 { ( -). (1.9.5) = -

, -

-. 8-

, = 2/.

, .

,

^() ,

.

. (1.9.6)

. (0:

/2

- / 2

- :

= - J | * ( 0 | = - J x ( t )- x * W t = * - / - > * _/->-/2 1 -/2

X I ,- J e ^ ' - e - ^ d t = X 2 (1.9.6)00 00 1 /2 00

(0, -77 2 /772.

() = J xT(t )e- ja 'dt

= ).

66

(1.9.7)


67/331

-

= - ) .= -

(1.9.8)

. - -

00 . 1 00

= - * -

(1.9.9)

1 - nt 1 = J 5(/ - /) 7 dt= .

-/2 ^

-

1 _ 00

- (0 = 8(/-) = 8(-) =- - * = 00* = -

= 8((- to) = ().

D^ (0 ^ () 00 00

^ 5(f - &)


68/331

, - *(/)

= z 5( t - k T ) = -

jcT(f) -

- D T(t).

-

^(^ ^ ^ () 5(-) =

. = (1.9.11)

= ( ) 5( - ),= -00

= 2/, a . () - ,

.

, , . (1.9.11)

sin*- () = -^ -5 ( - ) (1.9.12) = - 2

. 1.9.2 = 3.


69/331

. 1.81.9

-20. (1-20) 1.8.2. :

I. /= at .

- / .

4. / .

5. y(t) x(t ) a(t )

.6. (0 = lim e~at t> 0,

>0

0

. / = 0.

7. sgn(0 = o ( t ) -< j ( - t ) .

& .

910. , .

II. ,

=*(/) +*(-/).

15-16. /.

21. N ,

,

NT.

. .

22.

00

12. x(t ) = e ata(t),

69


70/331

a) E

) X

i_ T

J D____ C LT IT

1 I i T 21

23.

x(/) = sin2it/0f; -


71/331

) Re , =1 = ,

ImC = .[ \

26. -2 (

).

_*>

. .:

ReC =2 * /77 sin

= -

2

27.

/ =1

2/.2 / , (/-: .

28.

sm2nf ( { ( t - kAt ) 1

At =-2/.2n f(1(t - kA t) (-,).

. -

* (0 , (/),

.29. 2N+ 1

-, . , .

. (2N + 1)

( 2 ' +] \

IV )S(2 N + 1) = ^(1- )

, - , q- .

71


72/331

:

, ( / ) =sin 7i/(2W+ 1)

/

j

in in

- NT - I t \ > N T

30. x(t ) X(/),

. . X ( f ) = 0 , , [ - / , /]

,

/ = 1/2/,,

* (0 = * (0 D{t ) = :(0 AtZ 8 (- =

= / x(kAt)d(t - kAt).

**-00

X ( f).

. .

31. ,

\ At

.

X h(t -kAt) .* -

32. ,

( - D* - 5 S* = 00 * -00

/ L 2

12


73/331

33. 31 32, :

) { ) x { f ~ .- * = - V/

34* ,

(* ("\ x) = (y ,Z ) , Y ( f ) = ( j l n f ) " - X ( f) .

35t :

* > ) [ * $ > ] ( " ) *36. , ,

1 -2*1/1----- - ,

. 9 . . .

73


74/331

1.10.

, , :

( 0 = - - ] ( < ) ' ,

. . ((). ,

x(t) =


75/331

(1.10.2)p- j c o

, :

( / ) ( ) .

jc(/) , ( ) -

. -, = 0.

(1.10.2)

(p - , + )

. 1,10.1). -

{(3+jI

*(/) = - ! _ 4 ( ) ' dp = Y j Res-J 2k abca * " =A (1.10.3) -( . : .

75


76/331

1.10.1. x ( t ) = exp(p0t) a (t \ 0=(30+j 0 -

. (/)

x(t)= 0 /< 0. (1.10.1)

( ) = je~ip~Po) dt = -r(P~P ) ' t -CO

/= 0P - P o

Re p= P > P0 t >.

ePoa(t) . -

= 0.

1

(10

^ ( / ) p+Pi

= -, (Pj > 0).

1.10.2. x(t)= ~ 2/cosco2/a(/) 2>0.

/

(1 1 >\ 1 &

i

/

1

\ ^ ' ( ej,,h' + ^ ' ) ' dt 0

\

1 , 2= - 2+ j a 2. - 2 ~

( )

. 1.10.3 = 2= - 2 +; 'w 2 = ' = - 2 -;'2.

. 1.10.3.

1.10.1.

( ) ,

. (

) ( ( ) ) ]

,

76


77/331

X(p) = O n P + a ^ P + " + *


78/331

= ) (/ > + 2)]..2 = - 4 ^(+1)

1

~-

= -1,

* = [ ( + 1)3]=., = + 2

= 1.1

1 2 1 1/>+ 1 + 2 ( + 1)2 ( + 1)3 ( + 1)3(/> + 2)

^ | ( + 1)2( + 2 ) - ( + 1) + ^ ( + 1)( + 2 ) + ( + 2)

( + 1)3( + 2)

,

0 } =1.

(22+ 1 2) - 3 2+2 2-0 2=-1.

( ) = 1 1 , 1 ______1 _(/> + 1)3(/> + 2) ( p + l f ( + 1)2 + 1 + 2

,

. x(t) ( ) .

t x ( t ) o - ^ - , (1.10.8dp

(1.10.1). ,

te~al - ---- = ---- ? , (1.10.9dp + ( + )

\( . 0. 0

( + ) " 1'

,

*(/) = ^ t 2e4 - te~l + ' ( - e~2t, t> 0. npi

, . !

/=

78


79/331

-p lt o ii x .

, 1,10.2.

^

(!

(

w

9 iit)

vit):an-

* i'(,)n 1 Lu ir

a rr.

DOM.

*

- V0{t)

. 1.10.4

'('> ^ #0(

:

v(/)-v(l,a t

V(0 = R i ( t) \ i ( t) ~ C - ~ - v(/) - v(0) =

v (t ) = L ~ ^ - / ( / ) - 1(0) = j - J v ( t ) c / t ;

%{#) /(f) ; i0(t) v(t).

V(p ) = RI (p ), (1.10.11)

v(t)V(p) /(/)/(/?) - .

, .

79

1 V


80/331

? - ^ ( ). (1..12)at

:

Ji ^ l e-pdt = x (t )e -p'\*+p\ x (t )e^d t= ( ) - (0). 0

, :

I ( p ) = CpV (p ) -Cv (0 ) (1.10.13)

F (p ) = -U (/ > )+ , (1.10.14)

V ( p) = p L I ( p ) - L i ( 0) (1.10.15)

) = - \ - ( )+ - (1.10.16)pL

(1.10.14) ,

Z (p ) = \l ,

v(0) /,

. , (1.10.16) ,

Z (p ) - pL /(0)/,

. . 1.10.5.

V(p^Rlip))

)

)

vw - ^-) + ^ v ( p ) s r

-------

) ' - 1) + ' L _ 0 _ 1

. 1.10.5

80


81/331

(-)

, |^ . .

{ 1.10.5. , & (. 10.6). v(t)

I:

*{0)p

V'(p)

[fc) -ihM - V a ) 6 ). 1.10.6.

(. 1.10.65) - :

l s V(p)_

RV ( p ) -

v(0).

, /(0 .

[ V(p),

( ) = + .

p ip + d /R C )) '

Sfr* : { -0 -~ \ ! RC. -

It = 1R 2=v (0 ) - IR .

(1*10.6) (1.10.7),

v(/) = IR+ (v(0) - IR)e' t/RC, t >0.

. 1.10..

(1.10.17)

. 1.10.6. , . 1.10.6

81


82/331

1.10.6. ,

/(/) (. 1.10.7). v(/).

/(/) /)L l

) V{p)r\ vL

. 1.10.7. () (

/(/) = /(/), 1()= / /*

:

- + + R pL t

( ) = IL

P\-R+TL+PCJ {-RP + X+ P2LC)R

- ( 1 ~ 0 = , +2/? + 0) LC 2L

V(p ) :

12= - + yja2-(ol = - jto, to = /2- 2

( ) = I *\L

{ P - P S K P - P i )

: . =_ _ _ oil /021

2 2 -27 2 (1.10.6) (1.10.7)

v(/) = .''+ ,= - ~>') = 1 2 2V 7

sin /.

82


83/331

w 0,

v(/) = I*/~sin 0t= 1~sin 0 (1.10.18)

- .

. 1.10.8.

> 0

,

1.10.1

, () + 2(/) ( ) + 2{)

x ( t - T ) u ( t - T ) < * X ( p ) e pT, > 0

w fdX(p )

t x ( t ) < * -----dp

ie e-wx ( t ) ^ X ( p + )

x ( a t ) o - X , > 0 LaJ

* - ^ ( ) - ( 0)at

) 1

Jx, ( ) 2(t -) dzX, ( ) 2() 0

83


84/331

______________________________________ 1.10.!

(0 X ( p )

8(0 1

IItT 1

p

-/ 1

p +a

sin co010

P 2+u>l

C O S CD0t P

2 2p + 0

e~a'cos co01 p + a

{p + a f + ^ l

. 1.10

1. 0 .

x(t), t >0

a) \ - e a' a , Re [p ]>0 ,p (p +a)

6)e+a,sin(o0/ 0 , R e [ p ] > a ,

( p -a) +a>0

) e cos (0/+ / 4 ) P 0 R e [p ]> a,V 2[ (p + a )2+(o2]

r)

- a / (P + a ) 2 - 0 r 1te cosoi0tr --------- :----- 4-, Re [p] > -a,[( p + a ) +co0J

)I 1 1 < f < 2 1< --p--------- =, /?,[0, -

84


85/331

* 0


86/331

3. (1

) h(t)

( )

( )= \h{t)e"'dt.

Re [p] > (3,,

, - , h(t).

. , , ,

, .

)h[t)

~'((t)

()1

+

R e [ P ] > - a

) h{t)= ~2cos 2/, 2 > 0, t >0.

) h(t) = ~2sin 2/, 2 > 0, t >0.

) h(t)= sin 3/, /> 0. ) h(t)= cos 3/, t >0.

4. R, L ,

, t= 0 !

.

.5.

RL-.

6. , , L J

, /(/).

86


87/331

*: RL C t= 0 linn e(t)= sin ,/. i (t )

9 -0 > .taf&^tWpaamJibHO R L

( , R, /= 0

(1) = - *, > 0. / 0)*0. -

Inrtb /7-.

( RLC- t= 0

e(/) = cosco,/. /(/)

II> 0* .

, . - ^ -.

(*\ *

I tx - t= 0 > 0 . (0 )*0 , -

'

(.

; .

* ( / ) -

\

0 / 2/

. 1.11.1

87


88/331

*(/)

At ,

x(t)

(. 1.11.1).

, . . *(/ ) = 0 t x(0)g(/) + (x t -x i4 )g(/-kA/).

At- 0, i

y(t) = x(0) g ( t ) + J x'(x)g(t - x ) d x . (1.11.2)0

88


89/331

t

(0= g(O)x(0 + \g\i )x ( t - t ) dx, (1.11.3)0

y{t)= x(0)g (0 + Jx'[t- x )g (x) dx, (1.11.4)0

A t ) = g ( 0 ) x ( t ) + ) g \ t - x ) x ( x ) dx. (1.11.5)0

(1.11.2) - (1.11.5)

.

(0

(. 1.11.2). At,

(/).

. 1.11.2

x(kAt)[(/ - At) - o(t - At - /)].

:

*(/ )= ^ x{kAt)- [ ( j ( t - k A t ) - o ( t - A t - /)]/.- /

At

x ( t ) At^ (kAt)b(t- At). (1.11.6)

=- (1.11.6)

8- x(kAt)At,

t = kAt (. 1.11.3). (1.11.6) ,

89


90/331

-

At >0.

/>, At->dz ,

.

(1.8.32), :

x( f)= J*(T )8( f- = (0 8(0- (1.11.7)

.

h(t). (1.11.7)

-:

= t :()

h(t) = 8(t). ,

. , .

h(t)

5(/-) (-), At^ x(kAt)h(t-kAt) y(t).* = -00 * = -00

At> 0,

00

y ( t ) = ^x (x )h ( t -x )dx = x ( t )h{t\ ( 1.11.8)

.

. h(t) =0 t


91/331

, ,

Z (p\ t= 0

*(/) ( ) . , ,

i ( t ) o I ( p ) = X ( p )/ Z ( p )

: ( )

l/Z(p) . *(/) = 5(/) 1,

( )

, . ( 1 / , (1.11.9)

. . .(/)

.

// ( /)= | ( ) /- )* / . (1 .11 .10)

1.11.1. R L /= 0

x{t)= Ee~at t> 0.

1 -*/ . (/) = 1 (1.11.10)

L

Hf) = E \ e ' " e L dx =

R v r , ( R \I- 0 -1 _

dZ= L [a - (R/L )\ e

V J, t>0 .

>0,

* > - !( _R t \

1- 1 , /> 0.

&- Ea{t).

1.11.2. , , . 1.11.4 .

91


92/331

, . M t),)

dt[

J

v(0)

X(p) 1 (/7)-

k K n

. 1.11.4

( ) = ~

, , ( ) h(t)= (/),

. 1.11.4 . (1.11.8)

x(t)\I i t

(/) = Jx(t )h{t - )dx -Jx(x)a(/ - x)dx= Jx(x)dx.o o o

y(t ) = Jx(T)rfx+^(0),0

. 1.11.3. - t -0

/(/) = ~

v(f) .

:

V(p ) = 1() Z {p )= 1() R l t ) .R + (l/pC )

(),) \1 .Up) P K U S C )

, (. . 1.10)

92


93/331

a) 6)

. 1.11.5. () ()

h(t), , ,

-' = /( ).

(1.11.8),

v(0= Ji(t)fc(/-T)rfx= )( - ') \ e-{'- ')IRC\d i = vC )

- ( e- "R C t > 0.

RC

[-(1/)]

. 1.11.6.

. 1.11.6. , . 1.11.5

, >0,

Aa(t):

v (t ) = A R ( l -e ' " * c ) , t >0.

AR .

93


94/331

1.11.4. . 1.11.7. t= 0

x(t) ,

. 1.11.7. i(t)

t > tr

. 1.11.7

. 1.1\.1.. . c ( t ) 1/ /?,

{ R i p ) _

pL + R ( + )

= R/L. 0 2= .

-\ 2 = - 1.

(1.10.7) R

g ( t ) = l - e - p = l - e ~ L> (1.11.11)

(1.11.5)

( 0= g (0) (0 + fg ' ( t~ (" dr.

R R

g(0 ) = 0, g \ t ) = j e L g X t -x ) = j e *

94


95/331

/(/) =y ( t ) l R

i (0 = j - e ' L'"* U t dx+ j - e ' ^ U . e ' a,t"') ti

* - 1r - a L

. 1.11

1. R, -

I.

)

0~

, t =

*(/) = kt.

y{t).

. 0 = RC

.2. R

,

,

*(/)

. t > 2 T

3. L,

, t=

,

. t{ < t < t2 t > t2

95

f(t)!

111i t


96/331

4. L,

, /= 0 ~/

r ! L a = r/L.

:. \

r - a L - -te

5. , te~

:

r - a L -te EL

( r - a L f - e

2 L- r e

6. )

g (0 ,

h(t) .

) ,

, h(t) = g'(t).

) - .

7. ](/) - ] (t) .

v

I 2

I 2 3 4 5 6

) .

) 2(t) 2(t).

8. u2(t) z/,(/) = k(t + T)


97/331

1.12.

,

]/) [/0-/ /0+ / J , /0 2/ (. 1.12.1).

x(r) = A(t)cos[2nf01+ (0]> (1.12.1)

A(t) (/) -

. (

/0 0)

. , - .

- .

, :

97


98/331

Acos[2nf0/+ 0] = R e ( ^ V 2n/o'),

As\n[2n/0/+ 0] = Im(Ae j(?0ej2i:fnt).

Aem

. (1.12.1)

,

/0 ,

.

. (1.12.1)

* (0 = (0cos 2n f 0 1- (0sin 2/01. (1.12.2)

(0 = A(t)cos (/) xs(t) = A(t) sin (/) (1.12.3)

x(t), (0 - , /) -

. .

2f ti.

.

cos2;rf j

s inlxf j

. 1.12.2.

,

(0 cos 2nf01= (1 /2)[(/) cos 22f 0t - x s(/) sin 22/0/] +1 /2( t).

98


99/331

2/0

()

xc(t).

xs(/).

,

, 90 .

x{t) . (1.12.3)

Ait) = 4 * ( 0 +*,2(0.*,(/) (7) = arctg .*(0

, cp(f)

.

7(0 = * + > , ( ' ) = (1-12.5)

.

(0 = |(0|-

\ | / (0 = 2 / 0/ + ( / ) ,

:

/. /. 1 dy r 1 d xs 0 = / + = / + - r -a r c tg ^ -

2 dt 2 dt , . ( 12-6)1 X X - X X

-/ +2 I+ 2

, , , (. 1.12.3).

99


100/331

:(/)

(/)

x(t ) = R e[y( t)e J2nfo']. (1.12.7)

*(0

. 1.12.3

,

.

*(/ ) (1.12.1)

,

X ( f ) = } x(t)e~J2nf,dt= \X(f)\ ej *A f ) ,

I*(/)| = \x (-/)| > * (" / ) = (/)>

. e. , - (. 1.12.1).

N

= ) W )e -j2nf'dt =|(/)| ej^ f )

2/.|/>

-

. 1.12.4

f (1.12.7)

* ( = ! / - / ) + ^ - ( / + /0)]. (1-12.8)

~

* +( = / - / ) ,

* ( = | - (/ + / ) ] -

(1.12.9)

100


101/331

(/ ) ((/ )

(/) . 1.12.4.

[ - / , / ].

.

.

. . , . ,

, [Sign/] =j l n t , (1.12.11)

) = [ * ] = [ X ( / ) ] + [X(/)Sign / ] =

= () + 7[(0 (1/ 0].

*(/).

, (1.12.10)

101

xA(t) =Re [*,(/) ] + 7 Im f-].

Re [*, (/ )] = *(/)>

/> ,/ < ,

(. 2. 0)

X( / ) = X( / ) + X(/ ) Sign/, ( 1.12.11)

/ > 0,

/ < 0.

( 1.12.12)


102/331

xA(t) = x{t ) + j x r {t), (1.12.13)

xr {t) = < X r ( f ) = -y ^( /)S ign/ . (1.12.14)

(/) (1.12.12), . .

:

/ - * W

. (1.12.12) (1.12.15) = ty

, . .

,

(1.12.12) (1.12.15). (1.12.12) :

I fnt ( - jSign/)

(. 1.12.5).

90.

(1.12.15)

x(t) = 1/7>

//(/) = - /Sign/

. 1.12.5.

102


103/331

, , x(t) = coslnft

(0 = sin2/7, x(t) = sinln ft xr ( t ) = - cos ln f t .

,

cos ln fnt + sin 2 f n,

( = Z sin 1- cos 2 -

.

, . . .

/0, h(t) t = - t 0

t = t0 h(t) tQ. ,

,

. , t0

.

, -/1

. , , . .

/0 > 2/.

(1.12.2) (1.12.7)

*(0 = (t) cos ln fQt - xs(t ) sin 2 rc/01=Re[y(/)eJ2/' ].

(/) ej2nfo1 (/)

/0. (t )eJ2nfot

.

xr{t) =1[ (/)'2"' ] = xc(/)sin2n/01+ xs(t) co s2 nf at=

= xc(/)cos(2ttf Qt - n / l ) - x s(t )sin (l n f t - n i l ) .

x(t) xr {t\ ,

c o s l n f t sin2rc^/,

xc(t) ,(/) .

103


104/331

= , ,

, :

, 1 const , xr ( t ) = -------d x -0.

71-1

: () -

- = t, x r ( t )

. ,

:

( = ( / + / 0). (1.12.16)

( = * ( ' ) ( -/ 2 /0,

*(/) :

,(/) = (/) (2 /0 (1.12.17)

x(t)= Re [( 0 '2/0/) ]. (1.12.18) /0 . ,

(1.12.1) /0

.

\xA(t )\ = jx 1(t) + x2r ( t )

A(t ) = Jx2c( t) + x2(t).

/0 ,

( / ) . /0

|J^(/)|2 /0

104


105/331

- fo) 2\XA(f i [ df . 1.12.6 -

, .

. 1.12.6.

- ; - ;-

1.12.1.

x(t) X ( f \

.

' 2X+( f ) , / > ,

;

* , ( / ) =, / < 0,

/

/ , ( 0 = 2f eJ2nftdt = !L (e J2lzf -1).

oJ Jnt

x(t) =R e[^ (0] = x 02 f sin 2/// 2nfet,

xr ( t ) = lm [^(0] = X 02f, sin2(n fat) /(nfet).

. 1.12.7 ,

. , ,

.

105


106/331

0.4

0.6

. 1.12.7.

. 1.12

1. , /) x,(t ) -

,

x(t),

.

2. x(t)

x(t) = cosoy + 2cosco2/.

, , , , .

4.

x(f) = Asin2nf0t, t> 0.

3. x(t)

^(),

.

0) . .

5.

xA{t\

, aw** -

/ < / < / 2 / > 0 .


107/331

6. ,

;(/)

xA(t) -*(/) +jx r (t ) :

7. x'A{t\ (()

- 5. () ,

() f Q= (/ +/ 2)/2.8. xr( t ) - *(/).

, xr ( t ) * (0 , . .

.9. -

, ,

.11. (0 t 0

. t= 0

.

.12. xA(t)

x'A(t).

13. , (/ )

y(t) /0

X A( f ) xA(t\ . .

(/) = (0 cos 2nf0t+ (0 sin 2 /0/,

xs(t) = (/) cos 2nf0t- x(t)sin 2nf0t.

00

^x{t)x'r (t)dt= 0,

x ) = ------5-.sin (oet

.

10. x(t)= Asin 2nf0tSignt,

r ( f ) = X A( f + f 0).

107


108/331

14. , ,

( )= \xA{t) xA{t + -z)dt,

. ?

15. .

16. .

17. - .

18. , .

19. :

) :() = cos ln fy ; - < /< , ) (/) = - < t


109/331

p j */4() =-------- e~J(*'

cljay

x ( t ) , ,

0 .

23. ,

*(/) (/) * (/ ),

_ (t )e > = jy (t).24. :

)/ ( * ) = --. )/ ( , ) - ^ .1+ X

25. ,

() Q()

*( ) = () + ()

.

1.13.

.

, . , . . [33].

, j ,

. cas (cosine-and-sine).

109


110/331

. x(f ) - ,

. :

(/ )= jx(f)ca s2n ffdtr (1.13.1)-tr

cas2nft= cos2rc/Y + sin 2nft. (1.13.2)

x(t)

: x ( t )= J K(/)cas2it/ tdf, (1.13.3)

-

(/ )

X(f )\

(/) = Re X ( f ) - ImX ( f ) . (1.13.4)

(/ ) = (/ ) +


111/331

X ( f ) = Ax sin / __infz = ^ sin /

/ /

sinit/t .cosn/ x- --------- SinH/x =

n f*

_ ^ _ s i n 2 n / T ,4_sin2n ft - ' J A T

2 / /

(1.13.4)

^ sin2rc/ . sin2/ ( / ) = -f -- J

2 / /

(1.13.8)

(/ ) .

1.13.1, .

) .it )

. 1.13.1

(/ )

Q ( f).

[(-, t > 0 , a > 0 , *(/) = )( ~ c v i i

? = J| (co)|2|


123/331

. (1.14.13):

^ o * m n ( a^ aWA o 9b) + W y(a9b). (1.14.16)

. 0

- 0:

Wx }(a,b) = Wx(a,b -b 0). (1.14.17)

. () () :

; ( - . - ) (1-14.18)0 0 0

:

W1.

,

x(t)

, . .

-

, - . . , , ,

( ) - . , , ( )

123


124/331

. ( , ..) -

, . , , .

- ?

L2(R) /.

:

(1.14.13)

x(t)

.

() - - Y|/(r).

, - ()

. ,,. ,

- | , ^.

(, \|/) = (X , ) /2,

(1.14.23) (1.14.24) (1.14.13)

(1.14.20)

(1.14.21)

(1.14.22)

|() = ( + .), (1.14.24)

124


125/331

W,(a,b) = - j = } * (/ )V

- ) d t = - J 2nJ\a\

jX {o > )e Ja% J,(o>)da =

1 = f ()' {aa)d(a= ^ = f X {(o)e ^ r\ [a(.) = 0* ) -

, - /.

2

eJM , (1.14.25)

()

]=

(1.14.26)

. 1.14.7. - ,


126/331


127/331

(1.14.30)

Wx(a, b):

=wx(2\ k 2 ") . (1.14.31)

(1.14.30) , x(t)

.

(1.14.30) ,

, . , ,

v (0 = M,oo(0 (* 1.14.8).

/oi(01 2. *

'V07(

1

0.5

7 V i b ( 0 V n ( 0 V i2 W p i n

\ Ji l .\ I

!____!

1 J 4 4 1n

v 2o(0 v 2i (0

0 1 2 3 4 * 6 7 * 9

0.5 * (0

* 6 1 2 J 4 5 6 7 9 9

. 1.14.8.

127


128/331

, , . .

- , - . ,

- .

, , . ( ), . 2.3.

N =2V ( v - ),

:

, = 0 (. . = 1)

2V-1 = N -1 .

/(1, 2,3 ,. .. ) \|fmk(t ) , . / = v

= 0, . . \\tm 0(t )

(. 1.14.8; N=8).

. 1.14.9 .

- (. 1.14.2). -

Wx(, ).

(, )

1, - - q\

, .

-

w = 0,1, 2 ,... , v; : = 0,1, 2 , 2V~W-1.

I 2

128


129/331

-. Wx(a,b )

(. 1.14.9 ). . 1.14.10

Wx(a,b) (, ) ;

, - ( ).

. 1.14.9.

129


130/331

0

10 20 30 40

. 1.14.10.

, -

Wx(ayb);

Wx (, ).

(

) . (

Wx(, ))

, .

130


131/331

A 2.

, - *(/) - - . () . ( ), . 1.7

, .

. 2.1.1.

. 2.1.1. -

2.1. .

t

* ( 0 = * ( 0 \ 0

1111kAt t-------- ^

131


132/331

.

At.

y(t) (. 2.1.1)

(. . 1.9):

71S i n w - - - . 2

At ,

, , ( A r - A t ) T 0,

lim [M ')] = A v (0 = S ^ ' = 30

-, /, . .

D^ {t) = At S( t -kAt ) . (2.1.1)=-

.

Da,{()

x(t ) :

(/) = (0-> =

^ (2.1.2)= / x(k/\t)S(! - At),

. 2.1.2.

(2.1.2) . .

(/)

-:

k\i t

-'(') = A r ^ x ( h \ r ) S ( r - k M )

/

- -, N

1 \

/ I \I

\ / V

. 2.1.2

At I* ' > = * I

132


133/331

(/) = J *(/) 5(/ = ^x(t)b(kAt-t)dt. (2.1.3)

. , ,

x(kAt), ()

. ,

x(kAt) ,

. (2.1.2) ; .,(

. (. 1.11):

= /

( = 0, 1, 2, ...), (2.1.4)

, a dr /,

, , . .

(/) xA(t) '(/)

:

-, At, .

(2.1.2) (2.1.5) At-x(kAt).

tk = kAt, = 0, 1, 2, 3 ......

. 2.1.2.

(2.1.4)

'(/) = / ^ x ( k A t ) - S { t - k A t ) k=-

= *(/)! ( t - k A t ) = x ( t ) D f (t ) .(2.1.5)

133


134/331

x\(t)

(2.1.2) . .

(/),

, . - .

2.2.

(2,1.1).

- :

>*(/) = / 5(/ - At ) = At - - - -

/

2

b { t - k b ) e - j ^ d t = -

2

.

2 /

>/ ( /) = / X 8 (/- * /)= X * 'it = = -

(2.2.1)

(1.9.10)

(2.2.2)

/ = / = 1// . ,

-, .

/) ( (/)

f i t .

/-1

) ) (1)

111-\/ / / V V. 2.2.1.

134


135/331

(0 = At 8(/ - / ) > ( / ) = X 5 ( / - n fA) (2.2.3)= - = -

. 2.2.1.

*(/).

X(/ ) - x(t).

(2.1.5) - ,

( = ( X 5 ( / - / )= J X ( J - v ) b ( y - n f J d v .= -0 0 = _*

-

* , (/ ) = Z X ( f- n f > ) = X ( f ) + ' X ( f nf i ). (2.2.4) - = I

, At

, /= A f = 1/ At.

. 2.2.2 ,

, .

J.IVJI

/

Df m* (U


136/331

, , (, /7 = 0),

. .

1933 .. , [18].

*(/) , . .

, ,

1 (/)1 (- , ),

[ - / , f e] (. 2.3.1). *(/)

At= 1/2/.

- kAt.

2 (/ )

.

2.3.

X ( f ) -

* (1.6.1):

. 2.3.1

(2.3.1)

( * , < 1 "f , .m s in 2 n f ' 0 - )

* (*,*) A / i 2 - )z(2.3.2)

J


137/331

(2.3.3)

* = ] ( 2/ ( f ) e J2*fkAldf.

( 2/(/) = = X ( f) M ,/

= x(kAt). (2.3.4)

: ,

[- / , /] / = 1/2/, /

* (/)

x{kAt) *(/)

:

(2.3.5) :

X ( * (0 = Yj : ( * |(? -)5(- :^ =

(2.3.6)

(2,3.6)

(2.3.5)


138/331

Atx (kAt ) . (2.3.6)

-

2/.2 ( )* . (/) = 2/.2/# (/ ) , X ( f ~ ) = (- = - *

, , (2.3.5). ,

x(t)

.

. , (2.3.5) .

. , ,

(/) ,

:

(/), = , 1, 2 , 3,

;

&tx(k&t)\

[-/ , /]

, .

, . . (. 2.3,2).

/

. At = \l2 fe

(2.2.2) (/), . 2.3.26.

, , , (aliasing)

" -

138


139/331

At ( A t).


140/331

(/). ,

(2.3.7) (2.3.3),

* = ?) (/ ) {/) /


141/331

*| *

;

;

;

5-;

. .

2.4.

A ] j( f)

00 00x{kAt) = Jx(t)5(f- kAt)dt- Jx(/)-8(kAt - t)dt,

-OO -0

. e. x(t)

8(/)

, h(t)

0< /(. 2.4.1). . 2.4.1

kAt x(kAt)= J x(t)h(kAt - t)dt= J x(t)h(kAt- t)dt= [jt(f) h( t)]t_ ,

kAt- -o o

. e. / = kAt.

141


142/331

(2.1.2)

(/) = At^ x(/:Af) 5(~ )= = -

= [(/ ) (/ )]- * 5(/-;/)

(2.4.1)

(2.4.1) , ( f)

(/) = [(/) (/ )] h(t).

(2.4.1) ,

( / ) = / ) - ( / ) ] ( / ) (2.4.2)

D f (/ ) = 5 (/ ~ n fa) - , - -

X (/ ) = [(0 ] , (/ ) = [/*(/)] -

. (2.4.2) . , (2.4.2)

. 2.4.2

\(


143/331

|#(/)| const, .

0(/ ) = 2/, 2 (/ ) = 1

(/).

,

0( {( =

/ ). 2.4.3

, 1(/ )

,

h(t) (. 2.4.3).

.

h( t )=0 ^


144/331

.

. 2.4.46

= ( ; - 0 / 2 ) / ) = ^ . ^ 8 (2.4.3)n jv

1 */ 1

(&/) = J x ( t )dt = f x ( t )h (kAt - t )d t =9 - 9 -

, ,

(/) = At x(A:Af)5(/-/:A/) = - -

= i [ x ( f ) (/)] /J 8 ( / - ) .

-,

Xa = [ X ( f ) - H ( f ) ] D /t( f ) =

^ 1( / ) = ^ ( / ) 7 ^ e - jn/e7l/0

(2.4.4)

(2-4.5)

(2.4.6)

, 0

sin 71/ 0it0

(/) = -7/0.

0 = /; < 1; At = \/2 f e. ,

1% /,

/

S i n T i a -2/.

7ia:

'2/.

> 0.99. a < 0,15, . .

15% At. / =f a

= -13,5.

144


145/331

2.4.2. ,

. 2.4.5.

*(/ ).

;

At = \!2fe. . 2.4.5

- ().

. 2.4.5.

At x(t)

!.

At = /t.

< 1. , X ( f ) |/|^ /-

,

x(t).

145


146/331

.

At x(kAt)b(t -k A t) -k= ->

h( t) ,

= (0 =

(/) = / (*/) ':

y(t) = At x(kAt)IIz(t-kAt),

fl 0 < /< I,

t.

j2nfkAt ^ Tl f g-jnf _

/

X i f - n f J - x sm nfx e-Jnfi

/

(2.4.7)

(2.4.8)

(2.4.9)

. 2.4.6 X ( f ) Y( / ).

; '( .

/, i t ,

. 2.4.6.

,

X { f - n f a) -

~ ^ sin / ---------- .

/

Y { f ) = ( / At ) = 1/4. ,

sin / / / -.

146


147/331

, 00

(/) =/ ^ x(kAt)b{t - kAt) (2.4.10)= -

.

Z * (/ - / * )= -

.

(2.4.7) .

Y { f ) / > .

(2.4.7) (2.4.10)

(2.4.8), . 2.4.7. , -

*(0

b(t )

. 2.4.7

, (/ ) -

,

sin /

n f x

.

x(t) -00

At ^ ITx(t kAt) = -

147


148/331

(/) =

(1.9.12)

s(t) = x(t)- X I l ^ t - k A t ) ,* = -00

1 - /2 < /< /2,

(2.4.11)

(2.4.12)[

-

sin(rcwc/ At)At fla, ( / At)

6 ( f - n / A t ) . (2.4.13)

(2.4.11)

S ( f ) = X ( f ) P ( f ) =

sin(rcwx/At)= Z

A t . t U

T 00A ( nnz/At)

sin(nnt/ At)

\ X ( f - f , ) b ( f , - n l A t ) d f , = (2.4.14)oo

X ( f - n / A t ) .At a, (nnz/At)

, S ( f )

X ( f ) ,

f A 1/At

sin(rc/7x/A/)(nnz/At)

(. 2.4.8).

. . 2.4.8 S ( f )

= ( /At)= 1/4.

. 2.4.8.

148


149/331

2.5.

(2.3.5) ,

x(t)

{At) -

sin 2// - , . 2.5.1.

2nf,t

sin 2 f J

7\ '" -- "/' - /^ _/0 /t

. 2.5.1.

. 2.3,

x(t)

()

At 2 nfe t

(. 2.5.2).

At ^ x(kAt) 6(t - At) = -

5- Atx(kAt) (. 2.5.2).

(2.3.5)

* (/ ') = ' } X x (kAt )8 ( t-kAt )h0( t ' - l)d t .- =-

-,

149


150/331

)

(*08((-*/)*

VO.0

. 2.5.2. - ; -

* ( 0 = / ^ x(kAt)h0(t ' -k At ) = jT x(kAt) = -0

sin2nfa(tr-kAt)

2nfe(t ' -kAt) '

/0(/) * t< 0, ,

.

. tQ9

hQ( t - t 0) t< ,

H ML/=1/2/ '\

y n m r v

.

*/] :

150


151/331

, , () . - , . -,

, h(t)

,

*f ln|tf(/)|

J .~7^


152/331

/ ,= (2+5)/ . .

const

|(/)|='

!/|s/e,

/, / , - /

X(/ ), x(Y),

. -

(2.5.2) , . . |#(/)|^0

\f\> f R~/- f a - f e

. ,

|#(/)|

g # (/ ) [- / e, /]

(. 2.5.5). ,

NX 2

/

, . . (/ ) # (/ ) * X(/ ),

x{t). . 2.5.5

1, 2.

2.5.1. (. 2.5.6).

. 2.5.6.

152


153/331

h { t ) hQ(t)

2

/ < , ,

, . 2.5.6.

,

//(/) h ( t ) ,

,

. 2.5.66.

/ = + f e .

, . 2.5.7,

, 0 < t < 2 9 (2.5.3)

, t.

00

( / ) = (2.5.4)

H0(f ] )

"(>

(2.5.5)

.

h(t), ,

,

. . 2.5.7.

153


154/331

2.5.2.

* ( 0 =

\ \ ~ \

sin 2nfe ( t - ) 0< t


155/331

.

/ 11

/ /. 2.5.10

155


156/331

f c .

(. 2.5.11).

1 1

1+ 21+

/

\2J2

1+

X2

= 2

2

J 'S

J e j

= 2-1.

(2.5.9) , 2f \2

J \

\ f l J

,

lg

2- 1

-

(2.5.9)

(2.5.10)

(2.5.11)

V 2-1

l8( f (2.5.9)

/ = - ^ ~ - f ~ a

, |i/(/)|2

1 1----- "~1+ 2 2

[/[, /2].

156


157/331

:

f = f c , ,

10 lg 0,5 = - \

|#(/)|2 /

, , / = 0;

(2-\ ) |#(/)|2

/ = 0;

;

9

(. 2.5.10);/

v = 1 fc

|(|2 * v -2"

, , 10 lg( v '2w) = -20lg v .

, -2 00,3 = -6 ;

,

---------------------- = 6 /7.

= 10 / -60 /.

. , .

v = ^~ J

|tf(v)|2 =----- - ---- . (2.5.12)1 1+ (v )

8 < 1- , , (v ) - - ,

157


158/331

f cos( arccos v), Ivl < 1,() = \ (2.5.13)

[cosh( arccos hv), |v|> 1.

r(v) = 2v7;.l(v)-r.2(v), >2, (2.5.14) 0(v ) = 1 7] (v ) = v.

. 2.5.12 -1 < v < 1. v > 1

(v). v l ( v ) 2~]v

,

(v)

-1 < v < 1 . v = l

v = l ( f = f j ( 1) = 1

.

(v )

. 2.5.13

8.

. 2.5.13

158


159/331

-

.

2.5.3. 3-

f c .

, 3fc.

v = 1 (1) = 1 .

|//(V)|2=-----1 _ = ,51 ' 1+22(1)

6 = 1. 3- (2.5.14)

3(v) = 2v2(v ) - 7J(v) = 2v[2v7J (v ) - 0(v )] - 7J (v ) = 4v3- 3v.

, v = 1,

10 lg[l /(1+992) ] -40 .

-, .

. . .

, . , , . . . . .

,

(/) = / x(kAt) 5( t -kAt) . (2.5.15)*=-

. ,

159


160/331

\ (0 = * ]C x(kAt )q( t -kAt )= (2.5.16)Jt= -oo

00 = ^ x(A/)J 8(x kAt)q(t - x)d =


161/331

(2.5.2),

, (/ ) = -0 sin /0

, / 0 1, ,

----------- 1 + \\ 1, - /31 6 11

sin

(. 2.5.15).

'(| g(t) i

i .O f , / /


162/331

, x{t) kAt

( + 1)/

x(t) = x(kAt)+ x'(kAt)(t- kAt)+ X ^ ^ ^ ~ + (2.6.1)

dx(t)x"(kAl)= 'd 2x(0~

dt t= / dt2

* (/ )-

, :

x(kAt)= [jc(ArAr) - jc((At -1 ) /) ];At

x"(kAt)= [x (kAt) - ' (( -1 ) ] =At

- !-r [ x ( k A t ) - 2 x ( ( k - 1)A t )+ x ( (k - 2 ) A t ) ] .At2

, , . , ,

x[n)(kAt), . , , , .

. , , . , . (2.6.1).

(2.6.1). At (. 2.6.1).

162


163/331

, , (. 2.4.5). At

, (. 2.6.1).

. 2.6.1.

(2.6.1):

(t) = x(kAt) , At < t< (k + 1)

h{t)

(). ,

,

( f ) = l t sinnf At - ' = _ L . sin(Jt/ /A ) .

*///

. 2.6.26, . , . -

/ = /

, / = //2.

//(/) .

,, -

RC = At/.

163


164/331

.

*

)

>*

|

1

\t t

|(|

. 2.6.2.

-

164


165/331

, //(/) :

At, (1/ '2/ ) - .

, At,

, . 2.6.3.

(. 2.6.4). (2.6.1):

, ,

. x(kAt)

, . () , 2t (. 2.6.4).

At. h(t) # (/ )

. 2.6.46, .

j2nfdkt-

(0=/ *>-

At

. 2.6.3. -

k(t )= x(kAt)+ x\kAt){t - At) -

= ( ) + [jc(*A/) - (( - l )Af ) ]( f - kAt).

165


166/331

a*co

/ 2/ -1/At 1

. 2.6.4.

, :

. 2.6.5. - ,

. , , , .

[4],

x(t) =sin 2nfQt

/ > / 0( 2,2//7), (2.6.2)

8 =1*(0-*(0|

K o L, , = 0,01, / > 22/0,

10 , . 2.6.1. , ()

, f c -


167/331

f A, . . f c= // 2. ,

,

V= (/ /f c) >1 (. . 2.5)

20 . (2 .6 .2 )

f c f e=10f c

, f e 10^..

2.7.

, , [-, ]. , (0

.

*(/ ) 2( ) :

\()\ * -2

f A> 0, \fe(2 ,2 //). = 0,01, f a >2 ,2 fei. .

X ( f ) = jx( t )e -J2p/,dt.-

Af = \12

= (1/2) |^ )~ ]2^ 1dt= A f X ( n A f ) .-

X ( f ) = | ] / ( / )2,"^ ' e~j2p f'dt =

= / * ( / ) \ e j2n(n&f~f ) , dt. -

167


168/331

1( c i2itlnbf-f)tjt 1 c i2x(nbf-f)tI7' _ 2 sin 2nT{nAf - f )

i j2 n(n bf - f ) l-r 2rr(nAf - f )

X(/ )

* (/ ) = Z Af = l/2T (2.7.1)2 n T ( f - n A f )

( ) . X ( f ) /

,

/ - 2

/ = 1/27

2

, . , , t [- , ],

X ( f \ [(/)].

, ,

, :00 00

J>(0|2


169/331

(/) 2 ,

(/)(-/) 4

\X ( f ) \2 /


170/331

: , , , .

, , (. 3.3).

2.8.

, -

|/| [/0 /0+/], /0 2/ (. 2.8.1).

)( .

V "

j > o

----------------- 1-1-1

- j;>

)

m

0

X

'

-**

)

T 1

f

0

, :

L /

T i

- /i y y

( f. 2.8.1

170


171/331

* (0 = cos[2 /0/+ (/)], (2.8.1)

A(t) (0 -

. . (

/0 0)

. ,

. - .

= 2 (/ + )

( - ).

At =1/2/ , . . +(/)

_(/ ) / = 2/

,

.

2/,

, . , . 1.12.

.

X _ ( f ) .

(/) = jt(/) + yjtr (/), ( / )- ,

* (/ )

171


172/331

( (/ ) = -y sig n/).

. 1.12, ,

[/ - /, / + /]. ,

X (/ ) = 2+( / )

.

(f)

At= 1/ 2fe

2+(/ ) / =2/ (. 2.8.1).

2+(/ nfA) ,

(/) ^ ( kAt).

. 2.8.2

. , (^)

. . (), .

. 2.8.2.

(/) ( kAt)

( 2 .8 .2 )

172


173/331

h (t ) = Sm27lfet e J2nfl>' (2.8.3)' nt

, , . .

^ (/ ) . 2.8.1

. , /0/2/ - , - . /0,

().

(t )= (/) +jxs(/) = A(t)eJ n ,

.

, + (/ ) ,

*(/) = x ( t ) ~ j x r (/).

\ ( t) At= 1/2/

2_(/ )

/ =2/ (. 2.8.1 ). \ (/)

\ [At)

V_8in2ltf et e-j2nf0, } nt

jc(/) = ,4(/)cos[2rc/01+ (0] = (,(08 2// - xs(t)sin2/01 (2.8.4)

(/) = ,4(/)cosq>(0 xs(/) = A(t)sin (/)

,

2/, -

173


174/331

| (&/)} {5(&/)}, At = \/2fe (. . 1.12).

. 2.8.25.

) /0

, . . .

( f )

xs (/) -

. ,

.

f f l= 2 f e (4 /

).

/ = 2(/0 + /), .

Jt(f)

|jcc (/ )} {&/)},

(/) uxs(/), * (/ ),

cos2xfj , -

* |*/| '

,

sin2Jtfj

jA'A/j At= 1/2fe

(. 2.8.26).

(0= cos2nf0t -

sin2 n f 0t.

( 2 . 8 . 5 )

174


175/331

, 100 , ,

.

(/) .

(/)

/= 2 + 1 (2.8.6)

4/0

/0 , ,

, . . = 2, 4, 6,... ( ).

(2.8.6) (/) -

/0.

(2.8.4)

x ( k A t ) = ( kAt)co$2nf0k A t -x s(kAt)sir\2nf0kAt =

=jcc (A/)cosrc(w + l/2)-j ts (A/)simt (w + l/2).

x(2kA t) = (2/)*(1)^ :[(2/ + 1) ] = xs[(2 + l) A /]*(-l)*+

( 2kAt) = jc(2A;A/)-(-1)*

xs [( 2 * +1) /] = [(2 * +1) /] (-1 )* + (2.8.7)

,

:(/)

(. 2.8.3). , /,

, . .

t = kAt

(kAt ) xs[kAt).

175


176/331

( , .2.8.3 ) ().

, , . 2.8.4.

| (~\" \ 1 1

\*,[(2'+1)/j

>(2)

f

| | 1

I , ()(-

. 2.8.4.


177/331

:(/) * -

- () (. 2.4.5). :(/)

- () . (). , .

. ,

, (/) = (2/), -

, ,

(*) + *5 (5/). (4/) = -

' 2

/. .

, . .

* (/ ),

At :

0 1 2 3 4 5 6 7 8 9

( ) 0 , *2 *3 *4 *5 *6 *7 *i *9

>() *0 0 ~* 0 *4 0 - * * 0 1 0

( * ) 0 X, 0 - X j 0 *5 0 *7 0 *9

. . jc(:/)

.

>>,(&) 2{)

= {k )+ jy2 (* ) (2-8-8)

,

177


178/331

y( k) = xc (2k) + jx s (2k+ 1).

y { t) x ( f ) ,

() ( ) ,

y ( t ) = x ( t ) e x p j 2 n ^ - t ,

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