Digital Signal Processing Reference
In-Depth Information
4
y
(
t
)
3
y
c
(
t
)
2
y
p
(
t
)
1
Figure 3.18
Response of a first-order
system.
0
1
2
3
t
3.6
TERMS IN THE NATURAL RESPONSE
We now relate the terms of the natural response (complementary function) of a
continuous-time LTI system to the signals that were studied in Section 2.3. The
mathematical forms of the terms in the natural response are determined by the
roots of the characteristic equation:
+
Á
+ a
1
s + a
0
a
n
s
n
+ a
n- 1
s
n- 1
[eq(3.56)]
= a
n
(s - s
1
)(s - s
2
)
Á
(s - s
n
) = 0.
With the roots distinct, the natural response is given by
+
Á
+ C
n
e
s
n
t
.
y
c
(t) = C
1
e
s
1
t
+ C
2
e
s
2
t
[eq(3.57)]
C
i
e
s
i
t
,
e
s
i
t
Hence, the general term is given by where is called a
system mode
. The
root of the characteristic equation may be real or complex. However, since the
coefficients of the characteristic equation are real, complex roots must occur in
complex conjugate pairs. We now consider some of the different forms of the modes
that can appear in the natural response.
Real
s
i
s
i
If
s
i
is real, the resulting term in the natural response is exponential in form.
s
i
Complex
If
s
i
is complex, we let
s
i
= s
i
+ jv
i
,
and the mode is given by
C
i
e
s
i
t
= C
i
e
(s
i
+ jv
i
)t
= C
i
e
s
i
t
e
jv
i
t
.
(3.58)
Because the natural response
y
c
(t)
must be real, two of the terms of
y
c
(t)
can be
expressed as, with C
i
= ƒC
i
ƒ e
ju
i
,
