PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

484 Practical MATLAB® Applications for Engineers

R.5.73 For example, evaluate f 1 (n) and f 2 (n) by taking the inverse ZT of F 1 (z) and F 2 (z)
given by
a. F 1 (z) = 4z + 3z−^3 − 12z−^8 , using Table 5.2
b. F 2 (z) = z^3 /z − 1, fi rst using long division, then Table 5.2

ANALYTICAL Solutions

a. Using Table 5.2

fn Z Fz 1 ()


^1 [ ()] 1 Z^138 [ 4 z 3 z 12 z] 4 

 
(n 1 ) 3 (n 3 ) 12 (n 8 8)

b. Using long division

fn Z Fz Z z

z

Zz z z

(^2) z
1
2
1 3 12
11
()

[ ()]




 









Then from Table 5.2 fn Z Fz 2 ()

^1 [ ()] 2

(n^21 ) (n ) un()
R.5.74 As an additional example, let F(z) be given. Determine then f(n) and its ROC for
Fz
z
zz

()





2

(^2732)
ANALYTICAL Solutions
By partial fraction expansion
Fz z
z
z
z
()
()





2
312
1
33




, the resulting poles are z 1
1 1/2 and z 2
3
Then the following three ROC are defi ned:
a. Region number 1, |z| > 3, outside the two poles then
f(n) = (2/3)
(n) − (1/2)nu(n) + (1/3)3nu(n)
b. Region number 2, (1/2) < |z| < 3, bounded by the two poles, then
f(n) = (2/3)
(n) − (1/2)nu(n) − (1/3)3nu(−n − 1)
c. Region number 3, |z| < ( 1 / 2 ), region inside the smallest pole, then
f(n) = (2/3)
(n) + (1/2)nu(−n − 1) − (1/3)3nu(−n − 1)
R.5.75 The MATLAB symbolic toolbox command ztrans(f) returns the ZT of the scalar
symbolic object f with the default independent variable n.
R.5.76 For example, let
fn n un
n
()


() nn()()



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0
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