PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

488 Practical MATLAB® Applications for Engineers

R.5.79 The MATLAB symbolic toolbox command iztrans(Fz) returns the inverse ZT of the
symbolic object Fz, as a function of n. For example, let

Fz

z

za



for| |za 

Use MATLAB to obtain an expression for f(n).^

ANALYTICAL Solution

f(n) = Z−^1 [F(z)] = anu(n)

MATLAB Solution

>> syms a z

>> Fz = z/(z-a);

>> fn = iztrans(Fz)

fn =

a^n

R.5.80 Let us gain some additional experience by evaluating the inverse ZT of the follow-
ing functions by using MATLAB:
a. F 1 (z) = z/z − 2
b. F 2 (z) = 2 z/(z − 2)^2

MATLAB Solution

>> syms z n

>> F1z = z/(z-2);

>> f1n = iztrans(F1z)

f1n =

2^n

>> F2z = 2*z/(z-2)^2;

>> f2n = iztrans(F2z)

f2n =

2^n*n

R.5.81 The implicit assumption in the preceding discussion is that the system transfer
function H(z) is a proper rational function of the independent complex variable z.

R.5.82 Recall that a partial fraction expansion of H(z) can be performed only if H(z) is a
proper rational function. If H(z) is not a proper rational function, then H(z) consists
of a fi nite length sequence that can be extracted from H(z) by synthetic division
(dividing the polynomials of the numerator by the polynomial of the denominator
of H(z)), plus a rational function H 1 (z) as illustrated as follows:
Let H(z) = zk + H 1 (z), where k is an integer. Then the remainder H 1 (z) becomes a
rational function and can be decomposed into a partial-fraction expansion.

R.5.83 Recall that a number of useful MATLAB discrete functions that relate the trans-
fer function H(z) to its poles and zeros were presented in Chapter 7 of Practical
MATLAB® Basics for Engineers. For completeness, and as a quick review, some com-
mands are revisited as follows:
Let H(z) = G(z)/F(z) then