Digital Signal Processing Reference
In-Depth Information
The impulse function is defined by its
properties
rather than by its
values
. The two de-
finitions of the impulse function, (2.40) and (2.41), are not exactly equivalent; use of
the rectangular pulse in defining the impulse function is
not
mathematically rigorous
and must be used with caution [4]. However, (2.40) allows us to derive in a simple,
nonrigorous manner some of the properties of the impulse function (2.41). In addi-
tion, (2.40) is useful when applying the impulse function in signal and system analysis.
We say that the impulse function “occurs” at because this con-
cept is useful. The quotation marks are used because the impulse function (1) is not an
ordinary function and (2) is defined rigorously only under the integral in (2.41). The
operation in (2.41) is often taken one step further; if
f
(
t
) is continuous at
d(t - t
0
)
t = t
0
t = t
0
,
then
f(t)d(t - t
0
) = f(t
0
)d(t - t
0
).
(2.42)
The product of a continuous-time function
f
(
t
) and is an impulse with its
weight equal to
f
(
t
) evaluated at time the time that the impulse occurs. Equations
(2.41) and (2.42) are sometimes called the
sifting
property of the impulse function.
This result can be reasoned by considering the impulse function,
,
to have a
value of zero, except at Therefore, the only value of
f(t)
that is significant in
the product is the value at
One practical use of the impulse function (2.41) is in modeling sampling oper-
ations, since the result of sampling is the selection of a value of the function at a par-
ticular instant of time. The sampling of a time signal by an analog-to-digital
converter (a hardware device described in Section 1.2) such that samples of the sig-
nal can be either processed by a digital computer or stored in the memory of a digi-
tal computer is often modeled as in (2.41). If the model of sampling is based on the
impulse function, the sampling is said to be
ideal,
because an impulse function can-
not appear in a physical system. However, as we will see later, ideal sampling can
accurately
model physical sampling in many applications.
Table 2.3 lists the definition and several properties of the unit impulse func-
tion. See Refs. 2 through 6 for rigorous proofs of these properties. The properties
d(t - t
0
)
t
0
,
d(t - t
0
)
t = t
0
.
f(t)d(t - t
0
)
t = t
0
, f(t
0
).
TABLE 2.3
Properties of the Unit Impulse Function
q
1.
f(t)d(t - t
0
)dt = f(t
0
), f(t)
continuous at
t = t
0
L
-
q
q
2.
f(t - t
0
)d(t)dt = f(-t
0
), f(t)
continuous at
t =-t
0
L
-
q
3.
f(t)d(t - t
0
) = f(t
0
)d(t - t
0
), f(t)
continuous at
t = t
0
d
dt
u(t - t
0
)
4.
d(t - t
0
) =
t
1,
t 7 t
0
b
5.
u(t - t
0
) =
L
d(t-t
0
)dt =
0,
t 6 t
0
- q
q
q
t
0
a
1
ƒ a ƒ
L
¢
≤
6.
d(at - t
0
)dt =
d
t -
dt
L
- q
- q
7.
d(-t) = d(t)
