Processing of Signals - Digital Signal Processing: A Practitioner's Approach - page 7

of the powerful tools representing discrete-time systems. The z-transform of a
discrete-time signal h k is denoted as H ð z Þ and is given by

1

h k z k

H ð z Þ , : ð 1 : 8 Þ

k ¼1

As a simple illustration, consider a sequence of numbers

0 : 5 k ; k 0 ;

h k ¼ ð 1 : 9 Þ

0 ; k < 0 :

Using (1.8) for the sequence h k we get

H ð z Þ¼ 1
k ¼ 0

5 k z k

0 :

¼ 1

k

5 z 1

k ¼ 0 ð 0

: Þ

5 z 1 25 z 2 125 z 3 ¼ 1 þ 0 : þ 0 : þ 0 : þ: ð 1 : 10 Þ

Using the simple geometric progression relation

1

1
1 a ;

a k

¼ j a j < 1 ;

k ¼ 0

we get

1
ð 1 0

H ð z Þ¼

Þ ;

ð 1 : 11 Þ

5 z 1

:

<

5z 1

under the condition 0 5. Note that this condition

represents the region of the complex z-plane in which the series (1.10) converges
and (1.11) holds, and is called the region of convergence (ROC). We can also write
h k in the form

: 1, that is, jj> 0 :

h k ¼ 0 : 5 h k 1 þ k ; ð 1 : 12 Þ

where k is the Kronecker delta function, given by

1 ; k ¼ 0 ;

k ¼ ð 1 : 13 Þ

0 ; k 6¼ 0 :