Principles of Mathematics in Operations Research

200 14 Special Transformations

Table 14.3. A Brief Table for Z transforms

y{k)

(2)

(3)

1

k

k^2

(4)

(5) k{m

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

fc^3

, m = 0,1,

ak

kak

k\

e-ak

sin bk

cos bk

e-ak sm bk

e~akcosbk

Valid

Z transform \z\ > r

r](z) r

-l

u-1)^4

(*-«)^2

it

z

z - e - °

z sin b

22 —2z2(2 —cos cos fc+e" 6) 2ft

22 -2z cosb + fi~^2 "

ze""sinb

z^ — 2ze~a cos 6 + e —2"

z(z — ti'~" cos fc)

z^2 — 2^e~a cos b-f-e~2fl

Table 14.4. Properties of Z transforms

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

0)

(10)

(11)

(12)

Inverse

y(k)

ayi(k) + by 2 (k)

y(k + \)

y(k + n)

y(k-c), c> 0

aky(k)

ky{k)

k^2 y(k)

kmy(k), m = 0,1,2,...

Y.ku=oV^k ~ u)y 2 {u)

yi(fc)ya(fe)

£« = 0 2/(U)

Z transform

V(z)

arii(z) + brj 2 (z)

zri(z) - zy(0)

znn{z) - zny(0)

~zn~]y(l) zy(n-l)

«~c77(a)

^) ,d7j(z)

(^4) rfz
-^[-^'W]
(-£)%()
771(2)772(2)
^JrP'^1 1l(/')')2(P^2)<i/'
^(*)
We know -0(2)
z — a ' 2 — a 2
Then, by superposition,
-, and rj 2 {z) V2(k) = f(k
y(k) = a
ky
0 + J2 f(
k - (^1) - ")«"•
u=0