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
z a
ϭ Ϫ
for| |
z a
Ͼ
Use MATLAB to obtain an expression for f(n).
ANALYTICAL Solution
f(n) = Z
−1 [F(z)] = a
n u(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 following functions by using MATLAB:
a. F 1 (z) = z/z − 2
b. F 2 (z) = 2z/(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) = z
k
+ 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 transfer 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 commands are revisited as follows:
Let H(z) = G(z)/F(z) then
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7/23/2008 1:47:25 PM
7/23/2008 1:47:25 PM