PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

482 Practical MATLAB® Applications for Engineers

R.5.65 The ROC for a fi nite length sequence is the entire z-plane. Only two points in the
z-plane may be excluded from the ROC. They are z = 0 and z = ∞.

R.5.66 Let us turn our attention now and explore the effects on the z-domain of the pro-
cess of differentiation in the time domain of the causal sequence given by ƒ 6 (n) =
anu(n).

ANALYTICAL Solution

Since

fn aun z

za

6 ()
 n() ||za



⇔ for 

d

da

fn d

da

z

za

[] 6 ()⇔ 





 

Then

na u n z

za

n za



1

()⇔() 2 for||

In general

d

da

fn d

da

Fz

n

n

n

[ ( )]⇔ n[ ( )]

or

n

m

aun z

za

 nm m za







 ⇔ 



()

()

1 for||

R.5.67 The result obtained for the causal sequence analyzed in R.5.66 can be used to evalu-
ate ZT of the noncausal sequence f 7 (n) = anu(−n − 1).
The resulting time–frequency relation is given by

n

m

aun

z

za

 nm m za







  ⇔ 





()

()

1 1 for||

R.5.68 Recall that the zero and pole plot of the system transfer function H(z) = G(z)/F(z) is
referred to as the pole/zero plot. Recall that the values of z that make H(z) go to zero
or infi nity are the system zeros and poles, respectively.
It is customary to indicate the zeros by the character “o,” whereas the poles are
indicated by the character “x,” when represented by a plot on the complex plane.
The location of the poles defi ne the ROC of H(z) in the following way:

a. The ROC never includes a pole since a pole makes the denominator (or F(z)) of

H(z) go to zero, therefore H(z) becomes infi nity.

b. The ROC may be either the exterior of a circle, a bounded ring defi ned by two cir-

cles, or the interior of a circle, where poles are allowed only on its boundaries.