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