PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

DTFT, DFT, ZT, and FFT 481

ANALYTICAL Solution

Zf n F z unz z

z

n

n

n

n

[()] 33 () ()

0

1

1

1



 
 







 







∑∑

then

fn un

z

z

3 zz

1

()
 ()



⇔ for |^1 |1 or | | 1

R.5.63 To gain some insight and experience in the evaluation of the ZT, additional trans-

formation examples of various functions and their ROC are presented as follows.

For example, evaluate ZT and its ROC of the discrete function f 4 (n) given by

fn u n

n

(^4) n

1

10

00

()
  
( )











ANALYTICAL Solution

Fz u n zn

n

4 ()
(^1 )


 



∑

Fz zn z

n

n

n

4

1

1

()

( ) 1 


 





∑∑

Fz

z

z

z

4 z

1

11

()
 1








for

R.5.64 Let us evaluate the ZT and ROC of the discrete function f 5 (n) given by

fn un u n

nn

5 ()
23 1() ( )

ANALYTICAL Solution

Zf n[ ( )] 5 
Z un[ 231 nn( )] Z u n[ ( )]

Zun

z

z

[()]2n z

2


 2



for 

and

Zun

z

z

[(n )] |||31z

3

 
 3



for  |

Then ROC is given by |2| < |z| < |3|.