Digital Signal Processing Reference
In-Depth Information
Recall that a system is bounded-input-bounded-output (BIBO) stable if the output
remains bounded for any bounded input. The boundedness of the input can be
expressed as
ƒ x(t) ƒ 6 M for all t,
where
M
is a real constant. Then, from (3.15), we can write
q
q
-
q
ƒ x(t - t)h(t) ƒ dt
`
`
ƒ y(t) ƒ =
x(t - t)h(t) dt
F
L
L
-
q
q
=
L
ƒ x(t - t) ƒ ƒ h(t) ƒ dt
(3.38)
-
q
q
q
F
L
Mƒ h(t) ƒ dt = M
L
ƒ h(t) ƒ dt,
-
q
-
q
since
q
q
-
q
ƒ x
1
(t)x
2
(t) ƒ dt.
`
`
x
1
(t)x
2
(t) dt
F
L
L
-
q
Thus, because
M
is finite,
y(t)
is bounded if
q
ƒ h(t) ƒ dt 6
q
.
(3.39)
L
-
q
If satisfies this condition, it is said to be
absolutely integrable
. It can be shown
that this requirement is also sufficient. (See Problem 3.15.) Thus, for an LTI system
to be BIBO stable, the impulse response must be absolutely integrable, as in
(3.39). For an LTI
causal
system, this criterion reduces to
h(t)
h(t)
q
ƒ
h(t)
ƒ
dt 6
q
.
(3.40)
L
0
Stability for an LTI system derived
EXAMPLE 3.8
We will determine the stability of the causal LTI system that has the impulse response
given by
h(t) = e
-3t
u(t).
In (3.40),
q
-
q
ƒ
h(t)
ƒ
dt =
L
q
q
e
-3t
-3
1
3
6
q
,
e
-3t
dt =
=
L
0
0
