PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

DTFT, DFT, ZT, and FFT 479

Gz gnzn

n

()
( )

 



∑

Gz() Fz Hz()

. ()

Hz

Gz

Fz

()

()

()

and recall that

gn f kh n k

k

()
()( )



0

∑

R.5.58 Since the ZT of f(n) is given by

Fz f nzn

n

()
( )

 



∑

then ROC defi nes the set of values or regions of z (in the complex plane) that makes

the summation ( ∑+∞n (^) =−∞ f(n)z−n ) converge (fi nite).
The ROC is denoted by R and is given by a ring defi ned by z1 < |z| < z2, whose
inner and outer radii are r1 and r2, respectively, which defi ne the behavior of f(n) as
n approaches plus or minus infi nity.
R.5.59 For example, let us revisit the causal exponential function f 1 (n) given by
f n unrn
n

11 ()
()

 



∑

Then ZT is given by

Zf n F z runzn n rz

n

n

n

[()] 11 () 1 () 1 n

0



 





∑∑

Zf n F z rz n

n

[()] 11 () ( 11 )

0





∑

Recall that the preceding series converges if it is of the form

a

a

n

n






0 

1

1

∑ for| |a ^1

Then

Fz

rz

1

1

1

1

1

()