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• Pablo González Balcázar

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TABLA DE TRANSFORMADAS z X(s)

x(t)

x(kT) ó x(k)

X(z) 1

1 s 1 s+a 1 s2

1(t)

Delta Kronecker 1, k = 0 δ 0 (k ) =  î 0,k ≠ 0 1, n = k δ0(n-k)=  î 0,n ≠ k 1(k)

e-at

e-akT

t

kT

6

2 s3

t2

(kT)2

7

6 s4

t3

(kT)3

8

a s( s + a)

1 - e-at

1-e-akT

1

2 3 4 5

9

e-at - e-bt b−a ( s + a)( s + b)

e-akT - e-bkT

1 ( s + a) 2

te-at

s ( s + a) 2

(1-at)e-at

2 ( s + a) 3

t2e-at

13

a2 s2 ( s + a )

at-1+e-at

akT - 1 + e-akT

14

w 2 s + w2

sin wt

sin wkT

10

11

12

kTe-akT

z-k

1 1 − z −1 1 1 − e − aT z −1 Tz −1

(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 ) Te − aTz−1

(1 − e−aTz−1)2

(1-akT)e-akT

1 − (1 + aT) e−aT z−1

(1 − e−aTz−1) 2

Control Digital - © JAM 1998

(kT)2e-akT

1 − (1 + aT) e −aT z −1

(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

TABLA DE TRANSFORMADAS z x(t)

x(kT) ó x(k)

15

X(s) s s2 + w 2

cos wt

cos wkT

16

w

e-at sin wt

e-akT sin wkT

)2

(s + a + w 17

2

s+ a

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 e-at cos wt

1 − e − aT z −1 cos wT

e-akT cos wkT

( s + a) 2 + w 2

1 − 2e− aT z −1 cos wT + e −2aT z −2

18

ak

19

k-1

1 1 − az −1 z −1

a k=1,2,3...

1 − az −1 z−1

k-1

20

ka

(1 − az−1 ) 2

(

z −1 1 + az −1

2 k-1

21

ka

(

z −1 1 + 4az−1 + a 2 z −2

k3ak-1

22

k4ak-1

23

ak cos kπ

24

(1 − az−1) 3

(

(1 − az−1)4 (1 − az−1)5 1

25

k (k − 1) 2!

26

k (k − 1)...(k − m + 2) (m − 1)!

z − m +1 (1 − z −1 ) m

k (k − 1) k-2 a 2!

28

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

x(t) = 0, para t < 0 x(kT) = x(k) = 0, para k < 0 A menos que se indique otra cosa, k = 0, 1, 2, 3, ...

Control Digital - © JAM 1998

)

z −1 1 + 11az−1 + 11a 2 z −2 + a 3z −3

1 + az −1 z −2 (1 − z −1 ) 3

27

)

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

)

TEOREMAS Y PROPIEDADES TRANSFORMADA z 1 2 3 4 5 6 7 8 9 10

x(t) ó x(k) a x(t) a x1(t) + b x2(t) x(t+T) ó x(k+1) x(t+2T) x(k+2) x(t+kT) x(t-kT) x(n+k) x(n-k) tx(t)

11

kx(k)

12 13 14

e-atx(t) e-akx(k) akx(k)

15

kakx(k)

16 17

x(0) x(∞)

18 19 20

∇x(k) = x(k)-x(k-1) ∆x(k) = x(k+1)-x(k) n

∑ x( k )

k =0

X(z) a X(z) a X1(z) + b X2(z) zX(z) - zx(0) 2 z X(z) - z2x(0) - zx(T) z2X(z) - z2x(0) - zx(1) zkX(z) - zkx(0) - zk-1x(T) - ... - zx(kT-T) z-kX(z) zkX(z) - zkx(0) - zk-1x(1) - ... - zx(k-1) z-kX(z) -Tz

d X(z) dz

d X(z) dz X(zeaT) X(zea)

-z

z a d z -z X  dz  a 

X 

lim(z→∞) X(z) si el límite existe lim(z→1) [(1-z-1)X(z)] si (1-z-1)X(z) es analítica dentro y fuera del círculo unidad (1-z-1)X(z) (z-1)X(z) - zx(0) 1 1 − z −1

X(z)

∂ x(t,a) ∂a

22

kmx(k)

∂ X(z,a) ∂a  d m  − z  X(z)  dz 

23

∑ x( kT) y(nT-kT)

X(z) Y(z)

21

n

k =0

24

∞

∑ x( k )

k =0

Control Digital - © JAM 1998

X(1)