Digital Signal Processing Reference
In-Depth Information
In this equation, is called the
complementary function
and is a
particular
solution
. For the case that the differential equation models a system, the comple-
mentary function is usually called the
natural response,
and the particular solution,
the
forced response
. We will use this notation. We only outline the method of solu-
tion; this method is presented in greater detail in Appendix E for readers requiring
more review. The solution procedure is given as three steps:
y
c
(t)
y
p
(t)
y
c
(t) = Ce
st
1.
Natural response.
Assume to be the solution of the homoge-
neous equation. Substitute this solution into the homogeneous equation [(3.50) with
the right side set to zero] to determine the required values of
s
.
2.
Forced response.
Assume that
y
p
(t)
is a weighted sum of the mathematical
form of
x(t)
and its derivatives which are different in form from
x(t).
Three exam-
ples are given:
x(t) = 5 Q y
p
(t) = P;
x(t) = 5e
-7t
Q y
p
(t) = Pe
-7t
;
x(t) = 2 cos 3t Q y
p
(t) = P
1
cos 3t + P
2
sin3t.
This solution procedure can be applied only if
y
p
(t)
contains a finite number of
terms.
3.
Coefficient evaluation.
Solve for the unknown coefficients of the forced
response by substituting into the differential equation (3.50). Then use the
general solution (3.51) and the initial conditions to solve for the unknown coeffi-
cients
P
i
y
p
(t)
C
i
of the natural response.
System response for a first-order LTI system
EXAMPLE 3.11
As an example, we consider the differential equation given earlier in the section, but with
constant; that is,
x(t)
dy(t)
dt
+ 2y(t) = 2
y
c
(t) = Ce
st
.
for
t G 0,
with
y(0) = 4.
In Step 1, we assume the natural response
Then we
substitute
y
c
(t)
into the homogeneous equation:
dy
c
(t)
dt
+ 2y
c
(t) = 0 Q (s + 2)Ce
st
= 0 Q s =-2.
y
c
(t) = Ce
-2t
,
The natural response is then where
C
is yet to be determined.
Because the forcing function is constant, and since the derivative of a constant is zero,
the forced response in Step 2 is assumed to be
y
p
(t) = P,
where
P
is an unknown constant. Substitution of the forced response
y
p
(t)
into the differen-
tial equation yields
