Digital Signal Processing Reference
In-Depth Information
u
[
n
]
u
[
n
n
0
]
1
1
•••
•••
•••
•••
•••
2
10
1
2
3
n
10 1
n
0
n
0
n
1
n
0
1
Figure 9.3
Discrete-time unit step functions.
[
n
]
[
n
n
0
]
1
1
•••
•••
•••
•••
•••
2
10
1
2
n
10 1
n
0
1
n
0
n
0
1
n
Figure 9.4
Discrete-time unit impulse functions.
Recall that this definition applies for
n
an integer only. The unit step function is il-
lustrated in Figure 9.3. Dots at the end of a vertical line are used to denote the val-
ues of a discrete signal, as shown in Figure 9.3. The time-shifted unit step function is
denoted as
u[n - n
0
],
where
n
0
is an integer and
1,
n G n
0
b
u[n - n
0
] =
(9.10)
0,
n 6 n
0
.
This function is also plotted in Figure 9.3 for positive.
The second signal to be defined is the
discrete-time unit impulse function
also called the
unit sample function
. By definition, the discrete-time unit impulse
function is given by
n
0
d[n],
1,
n = 0
b
d[n] =
(9.11)
0,
n Z 0.
This function is plotted in Figure 9.4. Note that the discrete-time impulse function is
well behaved mathematically and presents none of the problems of the continuous-
time impulse function. In fact, the discrete-time unit impulse function can be ex-
pressed as the difference of two step functions:
d[n] = u[n] - u[n - 1].
(9.12)
This result is seen by plotting
u
[
n
] and
-u[n - 1].
The shifted unit impulse function
is defined by
1,
n = n
0
b
d[n - n
0
] =
(9.13)
0,
n Z n
0
and is also plotted in Figure 9.4, for n
0
7 0.
