Digital Signal Processing Reference
In-Depth Information
2.4
SINGULARITY FUNCTIONS
In this section, we consider a class of functions called singularity functions. We de-
fine a
singularity function
as one that is related to the impulse function (to be de-
fined in this section) and associated functions. Two singularity functions are
emphasized in this section: the unit step function and the unit impulse function. We
begin with the unit step function.
The unit step function, denoted as
u
(
t
), is usually employed to switch other signals
on or off. The
unit step function
is defined as
1,
t 7 0
b
u(t) =
t 6 0
,
(2.32)
0,
where the independent variable is denoted as In the study of signals, we choose
the independent variable to be a linear function of time. For example, if
the unit step is expressed as
t.
t = (t - 5),
1,
t - 5 7 0 Q t 7 5
b
u(t - 5) =
t - 5 6 0 Q t 6 5
.
0,
This unit step function has a value of unity for
t 7 5
and a value of zero for
t 6 5.
The general unit step is written as
u(t - t
0
),
with
1,
t 7 t
0
b
u(t - t
0
) =
.
0,
t 6 t
0
A plot of is given in Figure 2.17 for a value of
The unit step function has the property
u(t - t
0
)
t
0
7 0.
u(t - t
0
) = [u(t - t
0
)]
2
= [u(t - t
0
)]
k
,
(2.33)
(0)
k
with
k
any positive integer. This property is based on the relations
= 0
and
(1)
k
= 1, k = 1, 2, Á .
A second property is related to time scaling:
u(at - t
0
) = u(t - t
0
/a), a Z 0.
(2.34)
(See Problem 2.21.)
Note that we have not defined the value of the unit step function at the
point that the step occurs. Unfortunately, no standard definition exists for this
value. As is sometimes done, we leave this value undefined; some authors define
the value as zero, and some define it as one-half, while others define the value as
unity.
