Table of transformation of laplace & z
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Table of transformation of laplace & z
![Table of Laplace and Z-transforms
X(s) x(t) x(kT) or x(k) X(z)
1. – –
Kronecker delta δ0(k)
1 k = 0
0 k ≠ 0
1
2. – –
δ0(n-k)
1 n = k
0 n ≠ k
z-k
3.
s
1
1(t) 1(k) 1
1
1
−
− z
4.
as +
1
e-at
e-akT
1
1
1
−−
− ze aT
5. 2
1
s
t kT
( )21
1
1 −
−
− z
Tz
6. 3
2
s
t2
(kT)2
( )
( )31
112
1
1
−
−−
−
+
z
zzT
7. 4
6
s
t3
(kT)3
( )
( )41
2113
1
41
−
−−−
−
++
z
zzzT
8.
( )ass
a
+
1 – e-at
1 – e-akT ( )
( )( )11
1
11
1
−−−
−−
−−
−
zez
ze
aT
aT
9.
( )( )bsas
ab
++
−
e-at
– e-bt
e-akT
– e-bkT ( )
( )( )11
1
11 −−−−
−−−
−−
−
zeze
zee
bTaT
bTaT
10.
( )2
1
as +
te-at
kTe-akT
( )21
1
1 −−
−−
− ze
zTe
aT
aT
11.
( )2
as
s
+
(1 – at)e-at
(1 – akT)e-akT
( )
( )21
1
1
11
−−
−−
−
+−
ze
zeaT
aT
aT
12.
( )3
2
as +
t2
e-at
(kT)2
e-akT
( )
( )31
112
1
1
−−
−−−−
−
+
ze
zzeeT
aT
aTaT
13.
( )ass
a
+2
2
at – 1 + e-at
akT – 1 + e-akT
( ) ( )[ ]
( ) ( )121
11
11
11
−−−
−−−−−
−−
−−++−
zez
zzaTeeeaT
aT
aTaTaT
14. 22
ω
ω
+s
sin ωt sin ωkT 21
1
cos21
sin
−−
−
+− zTz
Tz
ω
ω
15. 22
ω+s
s
cos ωt cos ωkT 21
1
cos21
cos1
−−
−
+−
−
zTz
Tz
ω
ω
16.
( ) 22
ω
ω
++ as
e-at
sin ωt e-akT
sin ωkT 221
1
cos21
sin
−−−−
−−
+− zeTze
Tze
aTaT
aT
ω
ω
17.
( ) 22
ω++
+
as
as
e-at
cos ωt e-akT
cos ωkT 221
1
cos21
cos1
−−−−
−−
+−
−
zeTze
Tze
aTaT
aT
ω
ω
18. – – ak
1
1
1
−
− az
19. – –
ak-1
k = 1, 2, 3, … 1
1
1 −
−
− az
z
20. – – kak-1
( )21
1
1 −
−
− az
z
21. – – k2
ak-1
( )
( )31
11
1
1
−
−−
−
+
az
azz
22. – – k3
ak-1
( )
( )41
2211
1
41
−
−−−
−
++
az
zaazz
23. – – k4
ak-1
( )
( )51
332211
1
11111
−
−−−−
−
+++
az
zazaazz
24. – – ak
cos kπ 1
1
1
−
+ az
x(t) = 0 for t < 0
x(kT) = x(k) = 0 for k < 0
Unless otherwise noted, k = 0, 1, 2, 3, …](https://image.slidesharecdn.com/tabellatrasformataz-200812182044/85/table-of-transformation-of-laplace-z-1-320.jpg)
![Definition of the Z-transform
Z{x(k)} ∑
∞
=
−
==
0
)()(
k
k
zkxzX
Important properties and theorems of the Z-transform
x(t) or x(k) Z{x(t)} or Z {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
−−−−− −
K1
0
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(X
dz
d
Tz−
11. )k(kx )z(X
dz
d
z−
12. )t(xe at−
)ze(X aT
13. )k(xe ak−
)ze(X a
14. )k(xak
⎟
⎠
⎞
⎜
⎝
⎛
a
z
X
15. )k(xkak
⎟
⎠
⎞
⎜
⎝
⎛
−
a
z
X
dz
d
z
16. )(x 0 )(lim zX
z ∞→
if the limit exists
17. )(x ∞ ( )[ ])(1lim 1
1
zXz
z
−
→
− if ( ) )z(Xz 1
1 −
− is analytic on and outside the unit circle
18. )k(x)k(x)k(x 1−−=∇ ( ) )z(Xz 1
1 −
−
19. )k(x)k(x)k(x −+=∆ 1 ( ) )(zx)z(Xz 01 −−
20. ∑=
n
k
)k(x
0
)z(X
z 1
1
1
−
−
21. )a,t(x
a∂
∂
)a,z(X
a∂
∂
22. )k(xkm
)z(X
dz
d
z
m
⎟
⎠
⎞
⎜
⎝
⎛
−
23. ∑=
−
n
k
)kTnT(y)kT(x
0
)z(Y)z(X
24. ∑
∞
=0k
)k(x )(X 1](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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Table of transformation of laplace & z
- 1. Table of Laplace and Z-transforms X(s) x(t) x(kT) or x(k) X(z) 1. – – Kronecker delta δ0(k) 1 k = 0 0 k ≠ 0 1 2. – – δ0(n-k) 1 n = k 0 n ≠ k z-k 3. s 1 1(t) 1(k) 1 1 1 − − z 4. as + 1 e-at e-akT 1 1 1 −− − ze aT 5. 2 1 s t kT ( )21 1 1 − − − z Tz 6. 3 2 s t2 (kT)2 ( ) ( )31 112 1 1 − −− − + z zzT 7. 4 6 s t3 (kT)3 ( ) ( )41 2113 1 41 − −−− − ++ z zzzT 8. ( )ass a + 1 – e-at 1 – e-akT ( ) ( )( )11 1 11 1 −−− −− −− − zez ze aT aT 9. ( )( )bsas ab ++ − e-at – e-bt e-akT – e-bkT ( ) ( )( )11 1 11 −−−− −−− −− − zeze zee bTaT bTaT 10. ( )2 1 as + te-at kTe-akT ( )21 1 1 −− −− − ze zTe aT aT 11. ( )2 as s + (1 – at)e-at (1 – akT)e-akT ( ) ( )21 1 1 11 −− −− − +− ze zeaT aT aT 12. ( )3 2 as + t2 e-at (kT)2 e-akT ( ) ( )31 112 1 1 −− −−−− − + ze zzeeT aT aTaT 13. ( )ass a +2 2 at – 1 + e-at akT – 1 + e-akT ( ) ( )[ ] ( ) ( )121 11 11 11 −−− −−−−− −− −−++− zez zzaTeeeaT aT aTaTaT 14. 22 ω ω +s sin ωt sin ωkT 21 1 cos21 sin −− − +− zTz Tz ω ω 15. 22 ω+s s cos ωt cos ωkT 21 1 cos21 cos1 −− − +− − zTz Tz ω ω 16. ( ) 22 ω ω ++ as e-at sin ωt e-akT sin ωkT 221 1 cos21 sin −−−− −− +− zeTze Tze aTaT aT ω ω 17. ( ) 22 ω++ + as as e-at cos ωt e-akT cos ωkT 221 1 cos21 cos1 −−−− −− +− − zeTze Tze aTaT aT ω ω 18. – – ak 1 1 1 − − az 19. – – ak-1 k = 1, 2, 3, … 1 1 1 − − − az z 20. – – kak-1 ( )21 1 1 − − − az z 21. – – k2 ak-1 ( ) ( )31 11 1 1 − −− − + az azz 22. – – k3 ak-1 ( ) ( )41 2211 1 41 − −−− − ++ az zaazz 23. – – k4 ak-1 ( ) ( )51 332211 1 11111 − −−−− − +++ az zazaazz 24. – – ak cos kπ 1 1 1 − + az x(t) = 0 for t < 0 x(kT) = x(k) = 0 for k < 0 Unless otherwise noted, k = 0, 1, 2, 3, …
- 2. Definition of the Z-transform Z{x(k)} ∑ ∞ = − == 0 )()( k k zkxzX Important properties and theorems of the Z-transform x(t) or x(k) Z{x(t)} or Z {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 −−−−− − K1 0 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(X dz d Tz− 11. )k(kx )z(X dz d z− 12. )t(xe at− )ze(X aT 13. )k(xe ak− )ze(X a 14. )k(xak ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ a z X 15. )k(xkak ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − a z X dz d z 16. )(x 0 )(lim zX z ∞→ if the limit exists 17. )(x ∞ ( )[ ])(1lim 1 1 zXz z − → − if ( ) )z(Xz 1 1 − − is analytic on and outside the unit circle 18. )k(x)k(x)k(x 1−−=∇ ( ) )z(Xz 1 1 − − 19. )k(x)k(x)k(x −+=∆ 1 ( ) )(zx)z(Xz 01 −− 20. ∑= n k )k(x 0 )z(X z 1 1 1 − − 21. )a,t(x a∂ ∂ )a,z(X a∂ ∂ 22. )k(xkm )z(X dz d z m ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − 23. ∑= − n k )kTnT(y)kT(x 0 )z(Y)z(X 24. ∑ ∞ =0k )k(x )(X 1