Digital Signal Processing Reference
In-Depth Information
C
i
e
s
i
t
+ (C
i
e
s
i
t
)
*
= ƒC
i
ƒ e
ju
i
e
s
i
t
e
jv
i
t
+ ƒC
i
ƒ e
-ju
i
e
s
i
t
e
-jv
i
t
= ƒC
i
ƒ e
s
i
t
e
j(v
i
t +u
i
)
+ ƒC
i
ƒ e
s
i
t
e
-j(v
i
t +u
i
)
= 2 ƒC
i
ƒ e
s
i
t
cos (v
i
t + u
i
)
(3.59)
by Euler's relation. If is zero, this function is an undamped sinusoid. If is negative,
the function in (3.59) is a damped sinusoid and approaches zero as
t
approaches infinity;
the envelope of the function is If is positive, the function becomes
unbounded as
t
approaches infinity, with the envelope
Real roots of the characteristic equation then give real exponential terms in the
natural response, while complex roots give sinusoidal terms. These relationships are
illustrated in Figure 3.19, in which the symbols
s
i
s
i
2 ƒC
i
ƒ e
s
i
.
s
i
2 ƒC
i
ƒ e
s
i
t
.
*
denote characteristic-equation root
locations.
We see, then, that the terms that were discussed in Section 2.3 appear in the
natural response of an LTI system. These terms are independent of the type of exci-
tation applied to the system.
We now consider the stability of a
causal
continuous-time LTI system. As stated
earlier, the general term in the natural response is of the form where is a
root of the system characteristic equation. The magnitude of this term is given by
from (3.59). If is negative, the magnitude of the term approaches zero as
t
approaches infinity. However, if is positive, the magnitude of the term becomes
unbounded as
t
approaches infinity. Hence, denotes instability.
Recall that the total solution of a constant-coefficient linear differential equa-
tion is given by
C
i
e
s
i
t
,
s
i
ƒC
i
ƒ ƒ e
s
i
t
ƒ ,
s
i
s
i
s
i
7 0
[eq(3.51)]
y(t) = y
c
(t) + y
p
(t).
Recall also that, for stable systems, the forced response is of the same mathe-
matical form as the input Hence, if is bounded, is also bounded. If the
real parts of all roots of the characteristic equation satisfy the relation each
term of the natural response is also bounded. Consequently, the
necessary and suffi-
cient condition
for a causal continuous-time LTI system to be BIBO stable is that
y
p
(t)
x(t).
x(t)
y
p
(t)
s
i
6 0,
s
Figure 3.19
Characteristic equation root
locations.
