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. s1 1(t) 1(k) 11

1−− z

4. as +

1 e-at e-akT11

1−−− ze aT

5. 2

1s

t kT ( )21

1

1 −

−

− zTz

6. 3

2s

t2 (kT)2 ( )( )31

112

11

−

−−

−

+

zzzT

7. 4

6s

t3 (kT)3 ( )( )41

2113

141

−

−−−

−

++

zzzzT

8. ( )assa+

1 – e-at 1 – e-akT ( )( )( )11

1

111

−−−

−−

−−−

zezze

aT

aT

9. ( )( )bsasab

++−

e-at – e-bt e-akT – e-bkT ( )( )( )11

1

11 −−−−

−−−

−−−

zezezee

bTaT

bTaT

10. ( )2

1as +

te-at kTe-akT

( )21

1

1 −−

−−

− zezTe

aT

aT

11. ( )2ass

+ (1 – at)e-at (1 – akT)e-akT ( )

( )21

1

111

−−

−−

−

+−

zezeaT

aT

aT

12. ( )3

2as +

t2e-at (kT)2e-akT ( )( )31

112

11

−−

−−−−

−

+

zezzeeT

aT

aTaT

13. ( )assa

+2

2

at – 1 + e-at akT – 1 + e-akT ( ) ( )[ ]( ) ( )121

11

1111

−−−

−−−−−

−−

−−++−

zezzzaTeeeaT

aT

aTaTaT

14. 22 ωω+s

sin ωt sin ωkT 21

1

cos21sin

−−

−

+− zTzTz

ωω

15. 22 ω+ss cos ωt cos ωkT 21

1

cos21cos1

−−

−

+−−

zTzTz

ωω

16. ( ) 22 ωω

++ as e-at sin ωt e-akT sin ωkT 221

1

cos21sin

−−−−

−−

+− zeTzeTze

aTaT

aT

ωω

17. ( ) 22 ω+++

asas e-at cos ωt e-akT cos ωkT 221

1

cos21cos1

−−−−

−−

+−−

zeTzeTze

aTaT

aT

ωω

18. – – ak11

1−− az

19. – – ak

k = 1, 2, 3, … 1

1

1 −

−

− azz

20. – – kak-1

( )21

1

1 −

−

− azz

21. – – k2ak-1 ( )( )31

11

11

−

−−

−

+

azazz

22. – – k3ak-1 ( )( )41

2211

1

41−

−−−

−

++

az

zaazz

23. – – k4ak-1 ( )( )51

332211

111111

−

−−−−

−

+++

azzazaazz

24. – – ak cos kπ 111

−+ az

x(t) = 0 for t < 0 x(kT) = x(k) = 0 for k < 0 Unless otherwise noted, k = 0, 1, 2, 3, …

Definition of the Z-transform

Z{x(k)} ∑∞

=

−==0

)()(k

kzkxzX

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 −−−−− − 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(XdzdTz−

11. )k(kx )z(Xdzdz−

12. )t(xe at− )ze(X aT

13. )k(xe ak− )ze(X a

14. )k(xak ⎟⎠⎞

⎜⎝⎛

azX

15. )k(xkak ⎟⎠⎞

⎜⎝⎛−

azX

dzdz

16. )(x 0 )(lim zXz ∞→

if the limit exists

17. )(x ∞ ( )[ ])(1lim 11

zXzz

−

→− 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(x

0

)z(Xz 11

1−−

21. )a,t(xa∂∂ )a,z(X

a∂∂

22. )k(xk m )z(Xdzdz

m

⎟⎠⎞

⎜⎝⎛−

23. ∑=

−n

k

)kTnT(y)kT(x0

)z(Y)z(X

24. ∑∞

=0k

)k(x )(X 1