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TRANSCRIPT
-
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) , . ,
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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