
4.6.1 DISCRETE-TIME IMPULSE FUNCTION
We now introduce a discrete-time function, which plays a prominent role in analyzing the behavior
of linear discrete-time systems. The unit strength discrete-time impulse occurring at discrete-time
k ¼0isdefined by
d
k
¼
1, k ¼ 0
0, k ¼ 1, 2, 3, ...
(4:369)
Delaying the discr ete-time impulse by n units of discrete-time produces
d
kn
¼
1, k ¼ n
0, k ¼ 0, 1, 2, ..., n 1, n þ 1, ...
(4:370)
It follows directly from the definition of the z-transform that
z{d
k
} ¼ 1 and z{d
kn
} ¼ z
n
(4:371)
An arbitrary discrete-time signal f
k
, k ¼0, 1, 2, . . . can be expressed as a weighted sum of unit
discrete-time impulses, that is,
f
k
¼
X
1