Transformada z

TABLA DE TRANSFORMADAS z
X(s)
1

x(t)

x(kT) ó x(k)
Delta Kronecker 1, k = 0 δ 0 (k ) =  0,k ≠ 0 1, n = k δ0(n-k)=  0,n ≠ k 1(k) e-akT kT

X(z)
1

2 3 4 5

z-k

1 s 1 s+a 1 s2 2s3 6 s4

1(t) e-at t

6

t2

(kT)2

7

t3

(kT)3

8

a s( s + a)

1 - e-at

1-e-akT

9

e-at - e-bt b−a ( s + a)( s + b)

e-akT - e-bkT

(1 − z−1 ) 2 T2 z −1(1 + z −1 ) (1− z −1 ) 3 T3z −1(1 + 4z −1 + z −2 ) (1 − z−1 ) 4 (1 − e −aT ) z −1 (1 − z −1 )(1 − e −aT z −1 ) ( e −aT − e− bT ) z −1 (1 − e−aT z −1 )(1 − e − bT z −1 )
1 − (1 + aT) e−aT z−1
1 − (1 + aT) e −aT z−1

1 1 − z −1 1 1 − e − aT z −1 Tz −1

10

1 ( s + a) 2 s ( s + a) 2 2 ( s + a) 3 a2 s2 ( s + a ) w 2 s + w2

te-at

kTe-akT

(1 − e−aTz−1)2

Te − aTz−1

11

(1-at)e-at(1-akT)e-akT

(1 − e−aTz−1) 2

12

t2e-at

(kT)2e-akT

13

at-1+e-at

akT - 1 + e-akT

(1 − e−aTz−1) 2 [( aT−1+e −aT ) + (1−e −aT −aTe −aT ) z −1] z −1 (1 − z −1 ) 2 (1 − e −aT z −1 )
z−1sinwT 1 − 2 z −1 cos wT + z −2

14

sin wt

sin wkT

Control Digital - © JAM 1998

TABLA DE TRANSFORMADAS z
15 16

X(s) s s2 + w 2
w

x(t)
cos wt e-at sin wt
2

x(kT) ó x(k)
cos wkTe-akT sin wkT

X(z) 1 − z −1 cos wT 1 − 2 z −1 cos wT + z −2
e − aT z −1sinwT 1 − 2e− aT z −1 cos wT + e −2aT z −2

(s + a + w
17 s+ a

)2

e-at cos wt

e-akT cos wkT

( s + a) 2 + w 218 19 20 21 ak a k=1,2,3... ka
k-1 k-1

1 − 2e− aT z −1 cos wT + e −2aT z −2 1 1 − az −1 z −1 1 − az −1 z−1 (1 − az−1 ) 2 ka
2 k-1

1 − e − aT z −1 cos wT

z −1 1 + az −1

22

k3ak-1

z−1 1 + 4az−1 + a 2 z −2

(

(1 − az−1) 3 (1 − az−1)4 (1 − az−1)5
1 1 + az −1 z −2 (1 − z −1 ) 3 z − m +1 (1 − z −1 ) m z −2 (1 − az −1 ) 3 z − m +1 (1 − az −1 ) m

(

) ) )

23

k4ak-1

z−1 1 + 11az−1 + 11a 2 z −2 + a 3z −3

(

24 25 26 27 28

ak cos kπ k (k − 1) 2! k (k − 1)...(k − m + 2) (m − 1)! k (k − 1) k-2 a 2! k (k − 1)...(k − m + 2) k-m+1 a (m − 1)!

x(t) = 0,...