Principles of Mathematics in Operations Research

14.4 Z Transforms

14.4 Z Transforms

Definition 14.4.1 The Z Transformation y to rj is

oo ,.

/ ^ V^ V(U) , ~,

rj(z) = 2^ ^-~, where z e C.

u=o^2

We call the function rj the Z transform of y if

3r6R9 r](z) converges whenever \z\ > r,

in such cases y is the inverse Z transform of n.

n(z) = Z {y(t)} , y(t) = Z"^1 {V(z)}.

Proposition 14.4.2 If y satisfies

(i) y(k) = 0fork = -1,-2,...,

(ii) y(k) = 0(kn), neZ+,

then y has a Z transform.

If 7]{z) is the Z transform for some function \z\ > r, then that function is

v(k). i^Jcz

k-lv(z)dz,k = 0,1,2,...

yW"\0, fc = -l,-2,.

where C is positively oriented cycle of radius r' > r and center at z — 0.

For Z transform related information, please refer to Tables 14.3 and 14.4.

Remark 14.4.3

Z {y(k + 1)} = j/(l) +'& + y^. + ... = Zri(z) - zy(Q).

z zl

The Laplace transform ofy'(t) is s'ij(s) —y(Q).

Remark 14.4.4 The procedure to follow for using Z transforms to solve an

initial value problem, is as follows:

51. y(k) ^ r,(z).


52. Solve the resulting linear algebraic equation 'q(z) = Z {y(k)}.


53. Find the inverse Z transform y(k) = Z~x {t]{z)}.
    S4- Verify that y(k) is a solution.

Example 14.4.5

y(k + 1) = ay{k) + f(k), k = 0,1,...; y(0) = y 0 , a ^ 0

z 1

zrj(z) - zyQ = arj{z) + <j){z) => r/{z) = y 0 H 4>{z) = 7/1(2) + rj 2 (z).