2

2.1

, , . , , .

2.1.1

. , , , . , : . . , .

2.1.2

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

2.2

, , :

1.

2.

3.

4.

5.

2.2.1

t , t = nT , n T , . , ,

22

t

t

t

t

0

0

0

0

() ()

() ()

2.1: : () , , () , , () , , () ,

1 . , , .

2.2.2

. . . , , . M , M - . (M = 2) M - , . (). (). , , . 2.1 . .

2.2.3

x(t) T0

x(t) = x(t+ T0), t (2.1)

T0 . . , T0. , t = ., . 2.2. .

1 Digea ,

2. 23

t0

x(t)

... ...

T0

2.2: T0.

2.2.4

.

2.2.4.1

x(t) , . . ; , ( , ). x2(t), . , Ex,

Ex =

|x(t)|2dt (2.2)

, ,

Ex =

x2(t)dt (2.3)

, |x(t)|. .

2.2.4.2

. 0 |t| . , (2.3) .

, 0 |t| , . , , . . x(t), , Px,

Px = limT

1

2T

TT

x2(t)dt (2.4)

Px = limT

1

2T

TT|x(t)|2dt (2.5)

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

24

. , ( ) ( ), .

( ), . , , x(t). x(t) , V 2s (Volts seconds), V 2 (Volt ). x(t) , A2s (Ampere seconds) A2 (Ampere ), . , ... . .

2.1:

2.3.

: 1

0.5

0 t

0

x(t)

t

x(t)

1

-1

... ...

e2t

1 3 5-1-3

()

()

x(t)

1

-1

... ...

1 3 5-1-3

()

t

-

0

2.3: () (-) .

2.3(), , |t| . , .

Ex =

x2(t)dt = +

0

e4tdt+

12

0

12dt =3

4(2.6)

2.3)(), , |t| . , . (2.4) , T 2.3)(). .

Px = limT

1

2T

TT

x2(t)dt (2.7)

= limT

1

2T

( 1T

(1)2dt+ 11

12dt+

T1

(1)2dt)

(2.8)

= limT

1

2T

(t]1T

+ t]11

+ t]T

1

)(2.9)

= limT

1

2T(1 + T + 2 + T 1) (2.10)

= limT

1

2T(2T + 2) = 1 (2.11)

( 2.3)() ), ( 4 , ). , x2(t) . ,

Px =1

T0

T0

x2(t)dt =1

4

31x2(t)dt =

1

4

( 11

12dt+

31

(1)2dt)

(2.12)

=1

4

(t]11

+ t]3

1

)=

1

4(2 + 3 1) = 1 (2.13)

.

2. 25

2.2:

:

() x(t) = A cos(2f0t+ )

() x(t) = A1 cos(2f1t+ 1) +A2 cos(2f2t+ 2), f1 6= f2

() x(t) = A2 ej2f0t, A C

:

() T0 = 1/f0. . , T0. , , .

Px = limT

1

2T

TT

A2 cos2(2f0t+ )dt = limT

A2

4T

TT

[1 + cos(4f0t+ 2)]dt

= limT

A2

4T

TT

dt+ limT

A2

4T

TT

cos(4f0t+ 2)dt (2.14)

A2/2. , , 2T , T . , . A2/4T , T . ,

Px =A2

2(2.15)

A A2/2, f0( f0 6= 0) . f0 = 0, ! , A2.

()

Px = limT

1

2T

TT

[A1 cos(2f1t+ 1) +A2 cos(2f2t+ 2)]2dt

= limT

1

2T

TT

A21 cos2(2f1t+ 1)dt+ lim

T

1

2T

TT

A22 cos2(2f2t+ 2)dt

+ limT

A1A2T

TT

cos(2f1t+ 1) cos(2f2t+ 2)dt (2.16)

, A21/2 A22/2,

. , , 2

Px =A212

+A222

(2.17)

x(t) =

k=1

Ak cos(2fkt+ k) (2.18)

Px =1

2

k=1

A2k (2.19)

2 f1 6= f2. f1 = f2;

26

() , ,

Px = limT

1

T

T/2T/2

A2ej2f0t

2dt (2.20) |ej2f0t| = 1,

A2 ejf0t2 = A2 2 Px =

|A|2

4(2.21)

- , A A2/2. - Euler

A cos(2f0t+ ) =A

2ejej2f0t +

A

2ejej2f0t (2.22)

, A/2, |A|2/4, .

, x(t) :

:

.

, x(t) 0 |t| .

:

T0 , .

|x(t)| < Mx, t Mx < (2.23)

, .

, . .

2.3:

, , .

() x(t) = eat, a > 0, t 0

() x(t) = eat, a < 0, t 0

() x(t) = 2 sin(2t), limx+

sin(x)

x= 0.

() x(t) =

1, t < 0

0, t = 0

1, t > 0

() x(t) = e2t, t [0, 1]

2. 27

:

) , . , , .

) , |x(t)| 0 t +.

Ex =

+0

(eat)2dt =

+0

e2atdt =1

2a( limt+

e2at 1) = 12a

(0 1) = 12a, a < 0 (2.24)

) , . .

Px = limT+

1

2T

TT

x2(t)dt = limT+

1

2T

TT

4 sin2(2t)dt (2.25)

= limT+

2

T

TT

sin2(2t)dt = limT+

2

T

TT

(12 1

2cos 4t

)dt (2.26)

= limT+

( 1T

TT

dt 1T

TT

cos 4tdt)

(2.27)

= limT+

(1

T2T 1

T 1

4sin 4t

]TT

)(2.28)

= limT+

(2 1

4t(sin 4T + sin 4T )

)(2.29)

= limT+

(2 2 sin 4T

4T

)= 2 lim

T+2

sin 4T

4T= 2 0 = 2 (2.30)

Px = 2. , .

) , .

Px = limT+

1

2T

TT

x2(t)dt = limT+

1

2T

0T

(1)2dt+ limT+

1

2T

T0

12dt (2.31)

= limT+

1

2Tt]0T

+ limT+

1

2Tt]T

0= limT+

1

2TT + lim

T+

1

2TT (2.32)

=1

2+

1

2= 1 (2.33)

) , .

Ex =

10

e4tdt =1

4(e4 1) (2.34)

2.2.5

, , . , , , , . , .

2.3

. , (audio transformations) , . ,

28

(concatenated speech synthesis), () , , , . , (aircraft detection), .

, . : , , . , t.

2.3.1

x(t) t0 , y(t). , x(t), y(t) T .

y(t+ T ) = x(t) (2.35)

y(t) = x(t T ) (2.36)

, t0, t t t0. , x(t t0) x(t), t0 . t0 , (), (). , x(t 2) x(t) 2 , x(t+ 2) x(t) 2 . :

2.4:

x(t) = et 2.4 t0 = 1 .

t

1

0

x(t)

e-t

2.4: : x(t)

. x(t) t0 = 1 .

x(t) =

et, t 00, t < 0 (2.37) xd(t) = x(t 1) ( ) t0 = 1 -, 2.5(). t t 1.

x(t) =

e(t1), t 1 0 t 10, t 1 < 0 t < 1 (2.38) xa(t) = x(t + 1) ( ) t0 = 1, 2.5(). t t+ 1.

2. 29

x(t) =

e(t+1), t+ 1 0 t 10, t+ 1 < 0 t < 1 (2.39)

t

1

0

e-(t-1)

x(t-1)

1

-1

1

0

x(t+1)

e-(t+1)

t

()

()

2.5: : () x(t) t0 = 1 -, () x(t) t0 = 1 .

2.3.2

. x(t) y(t) x(t) 0 < a < 1. , , x(t) t, y(t) t/a,

y(t/a) = x(t) (2.40)

y(t) = x(at) (2.41)

, x(t) a > 1

y(t) = x( ta

)(2.42)

, a, t at. a > 1, , 0 < a < 1, .

2.5:

2.6 x(t).

30

t

x(t)

e-t/21

0

-1

-12

2.6: : x(t).

2 . 2 .

x(t)

x(t) =

1, 1 t < 0

et/2, 0 t < 2

0,

(2.43)

2.7() xe(t), x(t) a = 2. , x(t/2), t t/2. :

xe(t) = x(t/2) =

1, 1 t/2 < 0 2 t < 0

et/4, 0 t/2 < 2 0 t < 4

0,

(2.44)

t = 1 t = 2 x(t) t = 2 t = 4 x(t/2).

2.7() xc(t), a = 2. , x(2t), t 2t, :

xc(t) =

1, 1 2t < 0 0.5 t < 0

et/2, 0 2t < 2 0 t < 1

0,

(2.45)

t = 1 t = 2 x(t) t = 0.5 t = 1 x(2t).

2.3.3

x(t). x(t), 180 . y(t) = x(t).

y(t) = x(t) (2.46)

y(t) = x(t) (2.47)

, , t t. , x(t) x(t).

2. 31

t

x(t/2)

e-t/4 ()1

0

-1

-24

t

x(2t)

e-t ()1

0

-1

-0.51

2.7: : () x(t) a = 2,() x(t) a = 2.

2.6:

2.8,

0 1

1

x(t)

t3

1/3

2.8: : x(t)

x(t), x(t).

1 3 x(t) 1 3 x(t). x(t) = t/3, x(t) = t/3.

x(t) =

t/3, 1 t 30, (2.48) x(t), . x(t) t t x(t)

x(t) =

t/3, 1 t 3 3 t 10, (2.49) x(t) 2.9.

32

0

1

x(-t)

-1 t-3

1/3

2.9: : x(t)

2.4

(. , ) . : () (step function), () (rectangular pulse), () (triangular pulse), () (Delta function).

2.4.1 u(t)

, t < 0, . t = 0. . u(t), :

u(t) =

1, t > 00, t < 0 (2.50) -, t < 0 () t > 0 ( ). t = 0 (. t < 0), u(t). 2.10. ,

t

1

0

u(t)

2.10: u(t).

t = t0 > 0, u(t t0). , eat, a > 0 t = . , , 2.11 eatu(t). . . . , t!

2.12(). - (2.12)() , , 2.12(). u(t) - T u(tT ). 2.12(), [2, 4]

x(t) = u(t 2) u(t 4) (2.51)

2. 33

t

1

0

e-tu(t), > 0

2.11: eatu(t), a > 0.

t2

1

4

x(t)

0 t2

1

4

x(t)

0

2.12: .

2.4.2

, . rect, rectangular3.

( T

2,T

2

)= Arect

( tT

)=

A, t (T/2, T/2)0, (2.52) 2.13.

t-T/2

A

T/2

x(t)

0 t-T/2

A

T/2

x(t)

0

-A

2.13: .

rect(). t. ,

3, .

34

Arect( tT

)= A(u(t (T/2)) u(t T/2)) = A

(u(t+

T

2) u(t T

2))

(2.53)

2.12,

rect( t 3

2

)= u(t 2) u(t 4) (2.54)

2.4.3

, tri, triangular4.

( T, T

)= Atri

( tT

)=

A(

1 |t|T), t (T, T )

0, (2.55)

2.14.

0 T

A

t

Atri(t/T)

-T

2.14: .

t. tri() - , rect. .

2.4.4 (t)

( , , (t), ), . , . 5.

(t) = 0, t 6= 0 (2.56)

(t)dt = 1 (2.57)

. , , ( (2.57) ). 0. , , 1 ( 1 = 1). , , , ! , ; , 2.15 , , . ; , . - - , .

2.4.4.1

x(t), t = 0. t = 0, x(t) t = 0 x(0),

x(t)(t) = x(0)(t) (2.58)

4, .5 t.

2. 35

t-/2

1/

/2

p(t)

0t0

(t) 0

2.15: : .

, x(t) (t), (t T ) ( t = T ),

x(t)(t T ) = x(T )(t T ) (2.59)

; x(t), x(t) ! , , ( ) ! 2.16.

x(t)

0 0

(t)

t t 0

x(0) (t)

t

x(0)

X =

2.16: .

, , ,

x(t) =

1, t = 2

1, t = 0

2, t = 3

0,

(2.60)

x(t) = 1(t+ 2) 1(t) + 2(t 3) (2.61)

, (2.58), +

x(t)(t)dt = x(0)

+

(t)dt = x(0) (2.62)

(;;), x(t) t = 0. : x(t)

36

t = 0. +

x(t)(t T )dt = x(T )

+

(t T )dt = x(T ) (2.63)

, .

. . , , . , t , .

, .

(at) =(t)

|a|, a < {0} (2.64)

(t) = (t) (2.65)

2.4.4.2

,

d

dtu(t) = (t) (2.66)

. . ( 0 1, t = 0), !6

, - - . : u(t) ddtu(t), ; : , t = 0.

d

dtu(t) = 0, t 6= 0 (2.67)

: t1 < 0 < t2, t2

t1

d

dtu(t)dt = u(t)

]t2t1

= u(t2) u(t1) = 1 0 = 1 (2.68)

t2t1

d

dtu(t)dt = 1, t1 < 0 < t2 (2.69)

- - ! (;;). t1 t2 +, ! .

(2.66) . , u(t) . +

d

dtu(t)x(t)dt = u(t)x(t)

]+ +

u(t)d

dtx(t)dt (2.70)

= limt+

x(t)u(t) limt

x(t)u(t) +

0

d

dtx(t)dt (2.71)

6 .... , .

2. 37

= limt+

x(t) 0 +

0

d

dtx(t)dt (2.72)

= limt+

x(t) x(t)]+

0(2.73)

= limt+

x(t) limt+

x(t) + limt0

x(t) (2.74)

= x(0) (2.75)

x(t) t = 0. . !

(2.66)

t

()d = u(t) =

1, t > 00, t < 0 (2.76)2.4.4.3

+

d

dt(t)x(t)dt =

+

(t)d

dtx(t)dt = d

dtx(t)

t=0

(2.77)

n +

dn

dtn(t)x(t) = (1)n d

n

dtnx(0)

t=0

(2.78)

2.4.5 ej2f0t

, Euler, :

Aej =

38

2

1

0

-1

-24

3

2

1

-0.5

1

0.5

-1.5

-1

0

1.5

0

2

0

-243

210

-1

-1.5

-0.5

1.5

1

0.5

0

j2 f0 t

j2 f0 t

sin(2 f0 t)

cos(2 f0 t)

cos(2 f0 t)

sin(2 f0 t)

2.17: ej2f0t, , .

0

j2 f0t

1

2

32

0

-2

0

-0.5

1

-1

-1.5

0.5

1.5

0

-j2 f0t

1

2

32

0

-2

0

-0.5

1

-1

-1.5

0.5

1.5

0

j2 f0t + 0.5e

-j2 f0t

1

2

32

0

-2

-1.5

1

1.5

0.5

0

-0.5

-1

2.18: . ( ) ( ).

A = |A|ej (2.82)

,

2. 39

0

j(2 f0t + /4)

1

2

32

0

-2

-1.5

-1

-0.5

0

0.5

1

1.5

0

-j(2 f0t + /4))

1

2

32

0

-2

-1.5

-1

-0.5

0

0.5

1

1.5

0

j(2 f0t + /4)

+ e-j(2 f

0t + /4)

1

2

321

0-1

-2

0.5

0

-0.5

-1

-1.5

1.5

1

)) =

4

2/2

=

4

2.19: = /4 . ( ) ( )

(2.83,2.84):

A, f0, Aej(2f0t+).

A, f0, Aej(2f0t+).

A, f0, A2 e

j(2f0t+).

A, f0, - A2 e

j(2f0t+).

2.5

, - .

: , , . . , Kirchhoff, (, ,.). .

- /, . . , -, () ., . , . Single Input - Single Output (SISO) .

SISO T [] x(t) , y(t).

y(t) = T [x(t)] (2.85)

40

2.5.1

, .

2.5.1.1 -

: .

xi(t), i = 1, 2, yi(t), i = 1, 2,

x(t) = x1(t) + x2(t) (2.86)

y(t) = y1(t) + y2(t) (2.87)

N ., a.

-

x(t) y(t) (2.88)

ax(t) ay(t) (2.89)

a.

, :

x1(t) y1(t) (2.90)x2(t) y2(t) (2.91)

- , a, b,

ax1(t) + bx2(t) ay1(t) + by2(t) (2.92)

- . , , - , . ,

y(t) = 2x(t+ 1) 3x(t 4) (2.93)

, y(t) =

x(t) (2.94)

, y(t) = x2(t) (2.95)

. , .

2.7:

y(t) =1

3x(2 t) + x(t) (2.96)

.

:

2. 41

ax1(t)

y1(t) =1

3ax1(2 t) + ax1(t) = a

(13x1(2 t) + x1(t)

)(2.97)

bx2(t),

y2(t) =1

3bx2(2 t) + bx2(t) = b

(13x2(2 t) + x2(t)

)(2.98)

ax1(t) + bx2(t),

y(t) =1

3(ax1(2 t) + bx2(2 t)) + ax1(t) + bx2(t) (2.99)

=1

3ax1(2 t) +

1

3bx2(2 t) + ax1(t) + bx2(t) (2.100)

= a(1

3x1(2 t) + x1(t)

)+ b(1

3x2(2 t) + x2(t)

)(2.101)

= y1(t) + y2(t) (2.102)

.

2.8:

y(t) =1

x(t+ 1)(2.103)

.

: ax1(t),

y1(t) =1

ax1(t+ 1)(2.104)

bx2(t),

y2(t) =1

bx2(t+ 1)(2.105)

ax1(t) + bx2(t),

y(t) =1

ax1(t+ 1) + bx2(t+ 1)6= y1(t) + y2(t) (2.106)

. ,

ax(t) 6 ay(t) (2.107)

, .

. , . .

2.9:

d

dty(t) + 2y(t) = x(t) (2.108)

42

.

: x1(t),

d

dty1(t) + 2y1(t) = x1(t) (2.109)

x2(t), d

dty2(t) + 2y2(t) = x2(t) (2.110)

a b ,

d

dtay1(t) + 2ay1(t) = ax1(t) (2.111)

d

dtby2(t) + 2by2(t) = bx2(t) (2.112)

d

dtay1(t) + 2ay1(t) +

d

dtby2(t) + 2by2(t) = ax1(t) + bx2(t) (2.113)

(2.108)

x(t) = ax1(t) + bx2(t) (2.114)

y(t) = ay1(t) + by2(t) (2.115)

. N - :

Nk=0

dk

dtkaky(t) =

Nk=0

dk

dtkbkx(t) (2.116)

. ak, bk . , .

-. , . , . , .

2.5.1.2

, . t . ,

x(t) y(t) (2.117)

-, x(t t0) y(t t0) (2.118)

. t0, , - t0.

, y(t) = 3x(t+ 2) 2 cos(x(t 2)) (2.119)

, y(t) = tx(t) (2.120)

. .

2. 43

2.10:

y(t) = 3x(t+ 2) 2 cos(x(t 2)) (2.121)

.

: x(t t0),

y(t) = 3x(t t0 + 2) 2 cos(x(t t0 2)) (2.122)

t = t0,

y(t t0) = 3x(t t0 + 2) 2 cos(x(t t0 2)) (2.123)

, .

2.11:

y(t) = tx(t) (2.124)

.

: x(t t0),

y(t) = tx(t t0) (2.125)

t = t0,

y(t t0) = (t t0)x(t t0) (2.126)

, .

2.12:

d2

dt2y(t) = 4x(t) (2.127)

.

: x(t t0),

d2

dt2z(t) = 4x(t t0) (2.128)

z(t) x(t t 0). t = t0,

d2

dt2y(t t0) = 4x(t t0) =

d2

dt2z(t) (2.129)

.

, .

44

2.5.1.3 -

. ,

y(t) = 2x(t) (2.130)

- ,

y(t) = ex(t1) (2.131)

. - . , . .

2.5.1.4

. ,

y(t) = 2x(t 1) + sin(x(t)) (2.132)

, y(t) = x(t 2)2 + 4x(t+ 4) (2.133)

, y(t) , x(t+ 4).

, - . , , .

, . ; - - :

1. . - (), , . , . , , - ( , , , , , ) .. .

2. . , (MP3, JPEG, MPEG ), ( ) (,, ), ( ).

3. . , (. ) , - ( ) . , .

2.5.1.5

:

|x(t)| < Mx = |y(t)| < My, Mx,My < (2.134)

, , .

, y(t) = x(t 1) + t (2.135)

2. 45

,

y(t) =t

x(t+ 2)(2.136)

y(t) = sin(x(t)) (2.137)

. .

2.13:

y(t) = x(t 1) + t (2.138)

.

: x(t) Mx, .

|x(t)| < Mx (2.139)

y(t)

|y(t)| = |x(t 1) + t| |x(t 1)|+ |t| < Mx + |t| + (2.140)

t . .

2.14:

y(t) = ex(t2) (2.141)

.

: x(t) Mx, .

|x(t)| < Mx (2.142)

y(t) |y(t)| =

ex(t2) |eMx | < + (2.143) t

46

- .

: x(t) y(t) (2.147)

-,

x(t t0) y(t t0) (2.148)

. t0, , t0.

: .

: .

: :

|x(t)| < Mx = |y(t)| < My, Mx,My < (2.149)

. , .

.

2.15:

, , , .

1. y(t) = 2x(t 1) + 3x(t 3)

2. y(t) = t2x2(t+ 2) x(t)

3. y(t) = x2(t 4),

4. y(t) = log10(|x(t)|),

5. y(t) =1

x(t), x(t) 6= 0

:

1. y(t) = 2x(t 1) + 3x(t 3) . .

y1(t) = 2ax1(t 1) + 3ax1(t 3)

ax1(t).

y2(t) = 2bx2(t 1) + 3bx2(t 3)

bx2(t).

y1(t) + y2(t) = 2ax1(t 1) + 3ax1(t 3) + 2bx2(t 1) + 3bx2(t 3)

y1+2(t) = 2ax1(t 1) + 3ax1(t 3) + 2bx2(t 1) + 3bx2(t 3)

, .

x(t t0),

y(t) = 2x(t t0 1) + 3x(t t0 3)

2. 47

t0

y(t t0) = 2x(t t0 1) + 3x(t t0 3)

.

, (.. y(0)), ( x(1), x(3)).

, , , |x(t)| < Mx, ,

|y(t)| = |2x(t 1) + 3x(t 3)| < 2Mx + 3Mx = 5Mx = My

2. y(t) = t2x2(t+ 2) x(t) .

y1(t) = t2ax21(t+ 2) ax1(t)

ax1(t).

y2(t) = t2bx22(t+ 2) bx2(t)

bx2(t).

y1(t) + y2(t) = t2ax21(t+ 2) ax1(t) + t2bx22(t+ 2) bx2(t)

y1+2(t) = t

2(ax1(t+ 2) + bx2(t+ 2))2 (ax1(t) + bx2(t))

, .

, x(t t0),

y(t) = t2x2(t t0 + 2) x(t t0)

t0

y(t t0) = (t t0)2x2(t t0 + 2) x(t t0)

.

, (.. y(0)), (x(2)).

, , , |x(t)| < Mx, -,

|y(t)| = |t2x2(t+ 2) + (x(t))| < |t2x2(t+ 2)|+ |x(t)| < t2M2x +Mx +

t .

3. , .

, x(t t0),

y(t) = x2(t t0 4)

t0

y(t t0) = x2(t t0 4)

.

, .

, |x(t)| < Mx, |y(t)| = |x2(t 4)| < M2x .

48

4. .

y1(t) = log10(|ax1(t)|)

ax1(t). y2(t) = log10(|bx2(t)|)

bx2(t).

y1(t) + y2(t) = log10(|ax1(t)|) + log10(|bx2(t)|)

y1+2(t) = log10(|ax1(t) + bx2(t)|) 6= y1(t) + y2(t)

.

, x(t t0),

y(t) = log10 |x(t t0)|

t0

y(t t0) = log10 |x(t t0)|

.

.

|x(t)| < Mx, |y(t)| = | log10(|x(t)|)| < log10(Mx) < .

5. ! ,

2.6

1. :

() e2tu(t 2)

() u(t2 4)

() 4rect(t2

5

)() rect

(t+1

2

)+ rect

(2t1

2

)

2.

x(t) =

A, |t| 20, (2.150) x(t), x(t 1), x(t+ 1), - .

3. T

x(t) =

1, 0 t < T/2 2T t+ 2, T/2 t < T (2.151)

4.

x(t) =

t, 0 t < 1

0.5(3 2t), 1 t < 3

0,

(2.152)

x(t) x(2t), x(t/2).

5. x(t) t < 3. t .

x(1 t) (2.153)x(1 t) + x(2 t) (2.154)x(1 t)x(2 t) (2.155)

6.

x(t) =

1, 1 t < 0

2, 0 t < 1

t+ 2, 1 t < 2

0,

(2.156)

2. 49

x(t) x(t 1), x(2 t), x(2t), x(t/2).

7.

x(t) =

t+ 1, 1 t < 0

1, 0 t < 1

2, 1 t < 2

t 3, 2 t < 3

0,

(2.157)

x(t2), x(1 t), x(2t+2),x(2 t/3), (x(t) + x(2 t))u(1 t), x(t)

((t +

3/2) (t 3/2)).

8. - , , , -, .

() y(t) = t/x(t)

() y(t) =x(t 1) sin(x(t))

() y(t) = x(t) 3x(t+ 2)

9. - , , , -, .

() y(t) = x(t) sin(t)

() y(t) = ddtx(t)

() y(t) = x(2t)() y(t) = x(t 1) + x(1 t)

10.

() (t4 + 4)(t) = 4(t)

() e3t(t 4) = e12(t 4)() cos(t2 )(t) = (t)

() t3+1t2+15(t 1) =

18(t 1)

11.

() + (t)e

j2ftdt = 1

() + e

5(xt)(2 t) = e5(x2)

() + (t 8) cos(t)dt = 1