Digital Signal Processing Reference
In-Depth Information
h
1
(
t
)
x
(
t
)
y
(
t
)
x
(
t
)
y
(
t
)
h
1
(
t
)
h
2
(
t
)
h
2
(
t
)
Figure 3.14
Distributive property.
Therefore, the total system impulse response is the sum of the impulse responses:
h(t) = h
1
(t) + h
2
(t).
(3.29)
In summary,
1.
the impulse response completely describes the input-output characteristics
of a continuous-time LTI system, and
2.
the commutative, associative, and distributive properties give the rules for
determining the impulse response of an interconnection of LTI systems.
Note that all results in this section were proved for LTI systems only. An ex-
ample of the use of these properties will now be given.
Impulse response for an interconnection of systems
EXAMPLE 3.7
We wish to determine the impulse response of the system of Figure 3.15(a) in terms of the im-
pulse responses of the four subsystems. First, from (3.29), the impulse response of the paral-
lel systems 1 and 2 is given by
h
a
(t) = h
1
(t) + h
2
(t),
as shown in Figure 3.15(b). From (3.24), the effect of the cascaded connection of system
a
and
system 3 is given by
h
b
(t) = h
a
(t)*h
3
(t) = [h
1
(t) + h
2
(t)]*h
3
(t)
,
as shown in Figure 3.15(c). We add the effect of the parallel system 4 to give the total-system
impulse response, as shown in Figure 3.15(d):
h(t) = h
b
(t) + h
4
(t) = [h
1
(t) + h
2
(t)]*h
3
(t) + h
4
(t).
■
This section gives three properties of convolution. Based on these properties,
a procedure is developed for calculating the impulse response of an LTI system
composed of subsystems, where the impulse responses of the subsystems are
known. This procedure applies only for linear time-invariant systems. An equiva-
lent and simpler procedure for finding the impulse response of the total system in
terms of its subsystem impulse responses is the transfer-function approach. The
