1. The document defines the Laplace transform, Z-transform, and their relationship to continuous-time and discrete-time signals. 2. Tables are provided that list common transforms and their corresponding functions in the s-domain, z-domain, and time/discrete-time domains. 3. Important properties and theorems of the Z-transform are outlined, including linearity, shifting, derivatives, and the inverse z-transform.

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