Digital Signal Processing Reference
In-Depth Information
x
(
t
)
y
(
t
)
h
(
t
)
y
(
t
)
h
(
t
)
x
(
t
)
Figure 3.12
Commutative property.
x
(
t
)
y
(
t
)
x
(
t
)
y
(
t
)
h
1
(
t
)
h
2
(
t
)
h
2
(
t
)
h
1
(
t
)
(a)
x
(
t
)
y
(
t
)
x
(
t
)
y
(
t
)
h
1
(
t
)
h
2
(
t
)
h
1
(
t
)
h
2
(
t
)
(b)
Figure 3.13
Associative property.
connection may be changed with no effect on the system impulse response (the
input-output characteristics). Note that the two cascaded systems in Figure 3.13(b)
may be replaced with a single system with the impulse response
h(t),
given by,
from (3.22) and (3.23),
h(t) = h
1
(t)*h
2
(t) = h
2
(t)*h
1
(t).
(3.24)
It follows that for
m
cascaded systems, the impulse response of the total system is
given by
h(t) = h
1
(t)*h
2
(t)*
Á
*h
m
(t).
(3.25)
3.
Distributive property
. The convolution integral satisfies the following
relationship:
x(t)*h
1
(t) + x(t)*h
2
(t) = x(t)*[h
1
(t) + h
2
(t)].
(3.26)
This property is developed directly from the convolution integral, (3.14),
q
q
x(t)*h
1
(t) + x(t)*h
2
(t) =
L
x(t)h
1
(t - t) dt +
L
x(t)h
2
(t - t) dt
-
q
-
q
q
=
L
x(t)[h
1
(t - t) + h
2
(t - t)] dt
-
q
= x(t)*[h
1
(t) + h
2
(t)].
(3.27)
The two systems in parallel in Figure 3.14 illustrate this property, with the output
given by
y(t) = x(t)*h
1
(t) + x(t)*h
2
(t) = x(t)*[h
1
(t) + h
2
(t)].
(3.28)
