Digital Signal Processing Reference
In-Depth Information
We wish now to relate the general solution of differential-equation models to the
response of physical systems. We, of course, assume that the differential-equation
model of a physical system is reasonably accurate.
From these developments, we see that the natural-response part of the general
solution of the linear differential equation with constant coefficients is independent
of the forcing function the natural response is dependent only on the structure
of the system [the left side of (3.50)], and hence the term
natural response
. It is also
called the
unforced response,
or the
zero-input response
. This component of the re-
sponse is always present, independent of the manner in which the system is excited.
However, the amplitudes
x(t);
C
i
of the terms depend on both the initial conditions and
e
s
i
t
the excitation. The factors
in the natural response are called the
modes
of the
system.
The forced response of (3.51), is also called the
zero-state response
of the
system. In this application, the term
zero state
means
zero initial conditions
. The
forced response is a function of both the system structure
and
the excitation, but is
independent of the initial conditions.
For almost all LTI models of physical systems, the natural response
approaches zero with increasing time; then, only the forced part of the response
remains. (The requirement for this to occur is that the system be BIBO stable.)
For this reason, we sometimes refer to the natural response as the
transient re-
sponse
and the forced response as the
steady-state response
. When we refer to the
steady-state response of a stable LTI system, we are speaking of the forced re-
sponse of the system differential equation. For a stable system, the steady-state
response is the system response for the case that the input signal has been applied
for a very long time.
y
p
(t),
EXAMPLE 3.13
Time constant for the system of Example 3.11
For the first-order system of Example 3.11, the solution is given by
y(t) = y
c
(t) + y
p
(t) = 3e
-2t
+ 1.
3e
-2t
,
e
-2t
.
The natural response is the term and the system has one mode, The steady-state
response is the term 1. The system response is plotted in Figure 3.18. The time constant in this
response is (See Section 2.3.) Therefore, the natural response can be ignored
after approximately 2.0 units of time
1
2
= 0.5.
t =
(4t),
leaving only the steady-state response. (See
Section 2.3.)
■
In this section, we consider systems modeled by linear differential equations
with constant coefficients. A classical-solution procedure for these equations is
reviewed. The components of the solution are then related to attributes of the
response of a physical system.
