1
x1 (t ) → X 1 ( s ), ROC = R1
x 2 (t ) → X 2 ( s ), ROC = R2
ax1 + bx 2 (t ) → aX 1 ( s ) + bX 2 ( s )
&
x(t)
X(s)
%
! "#$
2
-)*+, ' (&
/0 1
x(t ) = x (t ) − x (t ) .
1
2
1
{s} > −1
X 1(s) =
s +1
1
{s} > −1
X 2 ( s) =
( s + 1)(s + 2)
X ( s) =
1
1
1
s +1
−
=
=
s + 1 (s +1)(s + 2) (s + 1)(s + 2) s + 2
R1
-1
R
R2
-2
-1
{s} > −2
-2
3
x (t ) → X ( s ), ROC = R
x (t − t 0 ) → e − st 0 X ( s ), ROC = R
4
s
! " #$
x (t ) → X ( s ), ROC = R
e s0t x (t ) → X ( s − s0 )
R
r1
R1
r2
r1 +
23
(s0)
r2 +
23
(s0)
5
s
s-s0=a
X(s-s0)
4
! " #$
3 6 78s=a X(s) - 45
% 3 6 78s=a+s0
->
s0=j
0
= -9 :; <
e jω 0t x (t ) → X ( s − jω 0 ), ROC = R
2& 4&
??
& < @&; A B C; <
%; 1 D
6
&' (&&) %
x (t ) → X ( s ), ROC = R
x ( at ) →
R
s
1
X ( ), ROC =
a
a
|a|
a = −1
.@ %
& F.&ROC
EE
x(t)
-9 :; <
&= E E
x ( − t ) → X ( − s ), ROC = − R
7
&' (&&) %
a < −1
−1 < a < 0
0 < a <1
1< a
; &"O P N |a|6 I J K H)
; &"O P N 1/|a| 6 I 0 L HM
1/a 6 I 0 L H*
a 6 I J K HG
G;<
R
*;<
R
r1
r2
23
r1/a
R
r2/a
23
r1/a
r2/a
23
8
! (&*+, x (t ) → X ( s ), ROC = R
x * (t ) → X * ( s * ), ROC = R
X ( s ) = X * ( s * ) ->
4
LL x(t) Q #
LL x(t) - F4
3 6 78s=s0 X(s) E
% 3 6 78R s=s*
X(s)
9
/ 0 1 23 &4 .
x1 (t ) → X 1 ( s ), ROC = R1
x 2 (t ) → X 2 ( s ), ROC = R2
x1 (t ) * x 2 (t ) → X 1 ( s ) X 2 ( s )
&
%; A
s
23
/ROC
X1(s)X2(s) = 1
/ 4<
-' (&
10
! " #$ ! (&*6 78 5
x (t ) → X ( s ), ROC = R
dx (t )
→ sX ( s )
dt
&
1
,
s
x ( t ) = u (t )
X (s) =
dx (t )
= δ (t )
dt
X ( s ) = 1, ROC = s − Plane
{s} > 0
-' (&
11
s
! " #$ ! (&*6 78 9
x (t ) → X ( s ), ROC = R
− tx (t ) →
dX ( s )
, ROC = R
ds
12
s
%> E ; A
! " #$ ! (&*6 78 9
=' .A ! "#$
> : &-)G+, ' (&
x (t ) = te − at u (t )
e − at u (t ) →
1
,
s+a
te − at u (t ) → −
{s} > − a
d
1
1
=
,
2
ds s + a
(s + a)
1
t 2 − at
e u (t ) →
,
3
2
(s + a)
{s} > − a
{s} > − a
t n −1
1
e − at u (t ) →
,
n
( n − 1)!
(s + a)
{s} > − a
13
s
! " #$ ! (&*6 78
D
2 s 2 + 5s + 5
X (s) =
,
2
( s + 1) ( s + 2 )
X (s) =
=! "#$
{s} > − 1
2
1
3
−
+
,
2
( s + 1)
s +1 s + 2
-)S+, ' (&
>
{s} > − 1
x (t ) = 2te − t u (t ) − e − t u (t ) + 3e −3t u (t )
14
! " #$ ! (&*: (;7
x (t ) → X ( s ), ROC = R
t
−∞
x (τ ) d τ →
1
X (s)
s
&
TU
15
= $ ' , &1, $ ' ! >? <
- / .@&V
E x(t)=0 > 4 t<0 H)
4 $ & WEX
I X Y & x(t) HM
= 34A
&
L&E x(0+) <E L&! "#$
% E ; A t ∞ = x(t)
.@
x ( 0 + ) = lim sX ( s )
s→∞
lim x (t ) = lim sX ( s )
t→∞
s→0
16
= $ ' , &1, $ ' ! >? <
4A
24&
&
%
/
L&E <E L& Z8
' .A ! "#$
A2&
-)[+, ' (&
x (t ) = e −2 t u (t ) + e − t (cos 3t )u (t )
2 s 2 + 5 s + 12
X (s) = 3
s + 4 s 2 + 14 s + 20
2 s 3 + 5 s 2 + 12 s
lim sX ( s ) = lim 3
=2
s→∞
s → ∞ s + 4 s 2 + 14 s + 20
x (0 + ) = 2
17
+, @A .
%
\ ? ] 4/
= "
=; _O ^
Ha> E`@&'E ?
18
LTI !
BC7D&E F &4 , &GH 5
19
! $,I
-> ! "#$
>b ; : ?
Y (s) = H (s) X (s)
?E :! "#$
%; A
LTI
>4KA
EE ! "#$
I cA#! "#$
=
H(s) ! "#
$
s=j
H(s)
/>4
_/ C
% .& &
20
JGK
%; A 3
t<0
e
I cA# d_O LTI >4KA
%; A R.S. ' .A e F4
d_O>4KA e
X
%; A
%; K ; A $
!-$
R Q 5O
f.4&ROC
R Q 5O - ?
;A J
21
JGK
-)g+, ' (&
h (t ) = e − t u (t )
H (s) =
1
,
s +1
{s} > − 1
%; A 23 > ; A
h (t ) = e
H (s) =
ROC
%; A d
_O>4KA
-)h+, ' (&
−t
−2
,−1 <
2
s −1
{s} < 1
%; K 23 > ; A
ROC
%; K
d_O>4KA
22
JGK
H (s) =
s
e
,
s +1
-),+, ' (&
{s} > − 1
I cA#Q #%
86 78
1
e u (t ) →
,
s +1
{s} > − 1
−t
s
e
e − ( t +1) u (t + 1) →
,
s +1
; A ; A ; A ROC
%; A R.S.
{s} > − 1
%; K d_O>4KA -1<t<0
h (t ) = e − ( t +1) u (t + 1)
I cA#
3
<
23
JGK
d_O I
U%
3 t>0 i I cA# /; A d_O I 4KA
%; A 4A j Uh(t)
%; A 23 > +j U 4KA .U>4KA X
86 78
j Uj U; A ROC E
%; A "#$ % >4KA
4KA X
41
24
!$
&
H(s)
>4KA X
.E ; A
#LTI >4KA e
% j 2&
25
!$
-Mk+, ' (&
s −1
H (s) =
( s + 1)( s − 2 )
d_O I
-1
# l>4KA
1
#>4KA
2
-1
-1
h (t ) = −
_O>4KA
d
2 −t 1 2t
e + e u (−t )
3
3
1
2
2
h (t ) =
h (t ) =
1
2 −t
1
e u (t ) − e 2 t u ( − t )
3
3
2 −t 1 2 t
e + e u (t )
3
3
26
!$
U%
D
>4KA X
d_O LTI >4KA e
6 78 ; A ; A ; A ROC ; A d_O>4KA
;A
>4KA
#
F4 %
- .@ % j 2&j U; A 6 78
. E ; A # H(s) >4KA X
_O>4KA e
23 j U 23 >
6 78m
.@
s
%
3.& 6 78m LL n o
27
M+g+, n o
L &0
#
7D
28