Discrete and Digital Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 77

Computation of the IZT can be quite involved as it requires integration in the
complex z-plane. For this reason, it is advisable to use Tables to determine the
IZT.

2.3.3.1 The Single-Sided ZT

As with the LT, one can define double-sided and single-sided transforms. Mim-
icking the approach used for the LT, this topic will focus only on the single-sided
ZT, which is defined as:

X ð z Þ¼ X

1

x ð n Þ z n : ð 2 : 11 Þ

n ¼ 0

If x(n) = a n Example (1) u(n), then its ZT is given by:

X ð z Þ¼ X x ð n Þ z n ¼ X a n z n ¼ X

1 1 1

ð az 1 Þ n

n ¼ 0 n ¼ 0 n ¼ 0

:

¼ 1 ð az 1 Þ 1

1 az 1 ¼ z

1

1 az 1 ½ from Tables ; Formula 10 ¼ z a ð if j z j [ a Þ:

Figure ( 2.9 ) shows this function for a = 3.

Example (2) If x(n) = u(n), then from Example (1), its ZT is given by z/(z - 1)
for |z| [ 1, i.e., the ROC is |z| [ 1).

If x(n) = d(n), then it ZT is given by X ð z Þ¼ P 1

n ¼ 0 d ð n Þ z n ¼

Example (3)

1 þ 0 þ 0 þ¼ 1 :

Fig. 2.9 A plot of the mag-
nitude of the z-transform
X(z) = z/(z - a) for a = 3

| X ( z ) |

Im ( z )