Digital Signal Processing Reference
In-Depth Information
v
(
t
)
1
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
at
(a)
i
(
t
)
1
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
at
Figure 2.43
(a) System input signal;
(b) bounded output signal.
(b)
Therefore, the circuit is not BIBO stable. However, if the input to the circuit is the
decaying exponential
v(t) = e
-at
u(t),
a 7 0
,
shown in Figure 2.43(a), then
1
aL
(1 - e
-at
)u(t),
i(t) =
which is shown in Figure 2.43(b) (for the particular case that ) and is bound-
ed for all time. Therefore, the system response is bounded for some inputs, but un-
bounded for others. A system with this characteristic is
not
BIBO stable, but is
sometimes said to be
marginally stable
.
Stability is a basic property required of almost all physical systems. Generally,
a system that is not stable cannot be controlled and is of no value. An example of an
unstable system is a public address system that has broken into oscillation; the out-
put of this system is unrelated to its input. A second example of an unstable system
has been seen several times in television news segments: the first stage of a space
booster or a missile that went out of control (unstable) and had to be destroyed.
aL = 1
Time Invariance
A system is said to be time invariant if a time shift in the input signal results
only
in the
same time shift in the output signal.
For a time-invariant system for which the input
x
(
t
) produces the output
produces
y(t), y(t) = T[x(t)],
x(t - t
0
)
y(t - t
0
).
That is,
y(t - t
0
) = T[x(t - t
0
)]
(2.73)
