Laplace & Ztransform
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Transcript of Laplace & Ztransform
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8/11/2019 Laplace & Ztransform
1/2
Table of Laplace and Z-transforms
X(s) x(t) x(kT) orx(k) X(z)
1. Kronecker delta0(k)
1 k= 0
0 k01
2.
0(n-k)
1 n= k
0 nkz
-k
3.s1 1(t) 1(k)
111
z
4.as +
1 e
-ate
-akT11
1 ze aT
5.2
1
s t kT
( )211
1
z
Tz
6.3
2
s t
2 (kT)2 ( )
( )31112
1
1
+
z
zzT
7.4
6
s t3 (kT)3
( )( )41
2113
1
41
++
z
zzzT
8. ( )assa
+ 1
e
-at
1
e
-akT ( )( )( )11
1
11
1
zez
zeaT
aT
9.( )( )bsas
ab
++
e
-at e
-bte
-akT e
-bkT ( )( )( )11
1
11
zeze
zeebTaT
bTaT
10.( )2
1
as + te
-at kTe-akT
( )211
1
ze
zTe
aT
aT
11.( )2as
s
+ (1 at)e-at (1 akT)e-akT
( )
( )211
1
11
+
ze
zeaT
aT
aT
12.( )3
2
as + t
2e
-at (kT)2e-akT ( )
( )31112
1
1
+
ze
zzeeT
aT
aTaT
13.
( )ass
a
+2
2
at 1 + e-at akT 1 + e-akT ( ) ( )[ ]
( ) ( )12
1
11
11
11
++
zez
zzaTeeeaT
aT
aTaTaT
14.22
+s sin t sin kT
21
1
cos21
sin
+ zTz
Tz
15.22
+s
s cos t cos kT
21
1
cos21
cos1
+
zTz
Tz
16.( ) 22
++ as e-atsin t e-akTsin kT
221
1
cos21
sin
+ zeTze
TzeaTaT
aT
17.( ) 22 ++
+
as
as e
-atcos t e-akTcos kT
221
1
cos21
cos1
+
zeTze
TzeaTaT
aT
18. ak11
1 az
19. ak
k= 1, 2, 3, 1
1
1
az
z
20. kak-1
( )211
1
az
z
21. k2ak-1 ( )
( )3111
1
1
+
az
azz
22. k3ak-1 ( )
( )412211
1
41
++
az
zaazz
23. k4ak-1 ( )
( )51332211
1
11111
+++
az
zazaazz
24. akcos k11
1+ az
x(t) = 0 for t< 0x(kT) =x(k) = 0 for k< 0Unless otherwise noted, k= 0, 1, 2, 3,
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8/11/2019 Laplace & Ztransform
2/2
Definition of the Z-transform
Z{x(k)}
=
==0
)()(k
kzkxzX
Important properties and theorems of the Z-transform
x(t
) orx
(k
) {x
(t
)} or {x
(k
)}1. )(tax )(zaX
2. )t(bx)t(ax 21 + )()( 21 zbXzaX +
3. )Tt(x + or )k(x 1+ )(zx)z(zX 0
4. )Tt(x 2+ )T(zx)(xz)z(Xz 022
5. )k(x 2+ )(zx)(xz)z(Xz 1022
6. )kTt(x + )TkT(zx)T(xz)(xz)z(Xz kkk K10
7. )kTt(x )z(Xz k
8. )kn(x + )k(zx)(xz)(xz)z(Xz kkk 1110 1 K
9. )kn(x )z(Xz k
10. )t(tx )z(Xdz
dTz
11. )k(kx )z(Xdz
dz
12. )t(xe at )ze(X aT
13. )k(xe ak )ze(X a
14. )k(xak
a
zX
15. )k(xkak
a
zX
dz
dz
16. )(x 0 )(lim zXz
if the limit exists
17. )(x )(1lim1
1zXz
z
if ( ) )z(Xz 11 is analytic on and outside the unit circle
18. )k(x)k(x)k(x 1= ( ) )z(Xz 11
19. )k(x)k(x)k(x += 1 ( ) )(zx)z(Xz 01
20. =
n
k
)k(x0
)z(Xz
11
1
21. )a,t(xa
)a,z(Xa
22. )k(xkm )z(X
dz
dz
m
23. =
n
k
)kTnT(y)kT(x0
)z(Y)z(X
24.
=0k
)k(x )(X 1
