Digital Signal Processing Reference
In-Depth Information
systems only. The bilateral Laplace transform, introduced in Section 7.8, must be
employed for noncausal systems.
We now relate bounded-input bounded-output (BIBO) stability to transfer func-
tions. Recall the definition of BIBO stability:
BIBO Stability
A system is stable if the output remains bounded for all time for any bounded input.
We express the transfer function of an
n
th-order system as
Á
b
n
s
n
+ b
n- 1
s
n- 1
Y(s)
X(s)
=
+
+ b
1
s + b
0
[eq(7.49)]
H(s) =
,
+
Á
+ a
1
s + a
0
a
n
s
n
+ a
n- 1
s
n- 1
where
a
n
Z 0.
The denominator of this transfer function can be factored as
Á
+ a
1
s + a
0
= a
n
(s - p
1
)(s - p
2
)
Á
(s - p
n
).
a
n
s
n
+ a
n- 1
s
n- 1
+
(7.60)
The zeros of this polynomial are the
poles
of the transfer function, where, by defin-
ition, the poles of a function
H(s)
are the values of
s
at which
H(s)
is unbounded.
We can express the output
Y(s)
in (7.49) as
+
Á
+ b
1
s + b
0
a
n
(s - p
1
)(s - p
2
)
Á
(s - p
n
)
X(s),
b
n
s
n
+ b
n- 1
s
n- 1
Y(s) =
k
1
s - p
1
k
2
s - p
2
k
n
s - p
n
Á
=
+
+
+
+ Y
x
(s),
(7.61)
where is the sum of the terms in this expansion that originate in the poles of
the input Hence, is the forced response. We have assumed in (7.61) that
has no repeated poles. We have also assumed in the partial-fraction expansion
of (7.61) that If a constant term, will appear in the partial-fraction
expansion. (See Appendix F.) As discussed in Section 7.6, in the mathematical models
of many physical systems,
The inverse transform of (7.61) yields
Y
x
(s)
X(s).
Y
x
(s)
H(s)
b
n
= 0.
b
n
Z 0
b
n
/a
n
b
n
= 0.
+
Á
+ k
n
e
p
n
t
y(t) = k
1
e
p
1
t
+ k
2
e
p
2
t
+ y
x
(t)
= y
c
(t) + y
x
(t).
(7.62)
The terms of originate in the poles of the transfer function; is called the
system's natural response. (See Section 3.5.) The natural response is always present
in the system output, independent of the form of the input signal
y
c
(t)
y
c
(t)
x(t).
Each term of
