Digital Signal Processing Reference
In-Depth Information
If (3.41) or (3.42) is differentiated (see Leibnitz's rule, Appendix B), we
obtain
ds(t)
dt
h(t) =
.
(3.43)
Thus, the unit impulse response can be calculated directly from the unit step
response, and we see that the unit step response also completely describes the
input-output characteristics of an LTI system.
Step response from the impulse response
EXAMPLE 3.10
Consider again the system of Example 3.8, which has the impulse response given by
h(t) = e
-3t
u(t).
Note that this system is causal. From (3.42), the unit step response is then
t
t
e
-3t
-3
t
1
3
(1 - e
-3t
)u(t).
e
-3t
dt =
s(t) =
L
h(t) dt =
L
=
0
0
0
We can verify this result by differentiating
s(t)
to obtain the impulse response. From (3.43),
we get
ds(t)
dt
1
3
(1 - e
-3t
)d(t) +
1
3
(-e
-3t
)(-3)u(t)
h(t) =
=
= e
-3t
u(t).
(1 - e
-3t
)d(t) = 0?
Why does
[See (2.42).]
■
In this section, the properties of memory, invertibility, causality, and stability
are considered with respect to LTI systems. Of course, by definition, these systems
are linear and time invariant. An important result is that the BIBO stability can
always be determined from the impulse response of a system, by (3.39). It is then
shown that the impulse response of an LTI system can be determined from its unit
step response.
3.5
DIFFERENTIAL-EQUATION MODELS
Some properties of LTI continuous-time systems were developed in earlier sections
of this chapter, with little reference to the actual equations that are used to model these
systems. We now consider the most common model for LTI systems. Continuous-time
LTI systems are usually modeled by
ordinary linear differential equations with constant
coefficients
. We emphasize that we are considering the
models
of physical systems,
not the physical systems themselves.
