Tabla trasformada z

Tabla trasformada z

1. Table of Laplace and Z-transforms


X(s) x(t) x(kT) or x(k) X(z)
1. – –
Kronecker delta δ
0(k)
1 k = 0
0 k ≠ 0
1
2. – –
δ
0(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
()
2
1
1
1
−
−
−z
Tz

6.
3
2
s
t
2
(kT)
2 ()
()
3
1
112
1
1
−
−−
−
+
z
zzT

7.
4
6
s
t
3
(kT)
3 ()
()
4
1
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
()
2
1
1
1
−−
−−
−ze
zTe
aT
aT

11.
()
2
as
s
+
(1 – at)e
-at
(1 – akT)e
-akT
()
()
2
1
1
1
11
−−
−−
−
+−
ze
zeaT
aT
aT

12.
()
3
2
as+
t
2
e
-at
(kT)
2
e
-akT ()
()
3
1
112
1
1
−−
−−−−
−
+
ze
zzeeT
aT
aTaT

13.
()ass
a
+
2
2
at – 1 + e
-at
akT – 1 + e
-akT ( )( )[ ]
() ( )
1
2
1
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. – – a
k
1
1
1
−
−az

19. – –
a
k
k = 1, 2, 3, …
1
1
1
−
−
−az
z

20. – – ka
k-1
()
2
1
1
1
−
−
−az
z

21. – – k
2
a
k-1 ()
()
3
1
11
1
1
−
−−
−
+
az
azz

22. – – k
3
a
k-1 ()
()
4
1
2211
1
41
−
−−−
−
++
az
zaazz

23. – – k
4
a
k-1 ( )
()
5
1
332211
1
11111
−
−−−−
−
+++
az
zazaazz

24. – – a
k
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(x1+ )(zx)z(zX 0−
4. )Tt(x2+ )T(zx)(xz)z(Xz −− 0
22

5. )k(x2+ )(zx)(xz)z(Xz 10
22
−−
6. )kTt(x+ )TkT(zx)T(xz)(xz)z(Xz
kkk
−−−−−
−
K
1
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(xa
k
⎟
⎠
⎞
⎜
⎝
⎛
a
z
X

15. )k(xka
k
⎟
⎠
⎞
⎜
⎝
⎛
−
a
z
X
dz
d
z

16. )(x0
)(limzX
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(xk
m
)z(X
dz
d
z
m
⎟
⎠
⎞
⎜
⎝
⎛
−
23. ∑
=
−
n
k
)kTnT(y)kT(x
0

)z(Y)z(X
24. ∑
∞
=0k
)k(x

)(X1