Digital Signal Processing Reference
In-Depth Information
The importance of the impulse response of an LTI system cannot be overempha-
sized. It is also shown that the impulse response of an LTI system can be derived
from its step response. Hence, the input-output description of a system is also con-
tained in its step response.
Next, the general system properties of an LTI system are investigated. These
include memory, invertibility, causality, and stability.
A general procedure for solving linear differential equations with constant co-
efficients is reviewed. This procedure leads to a test to determine the BIBO stability
for a causal LTI system.
The most common method of modeling LTI systems is by ordinary linear dif-
ferential equations with constant coefficients; many physical systems can be mod-
eled accurately by these equations. The concept of representing system models by
simulation diagrams is developed. Two simulation diagrams, direct forms I and II,
are given. However, an unbounded number of simulation diagrams exist for a given
LTI system. This topic is considered further in Chapter 8.
A procedure for finding the response of differential-equation models of LTI
systems is given for the case that the input signal is a complex-exponential function.
Although this signal cannot appear in a physical system, the procedure has wide ap-
plication in models of physical systems.
3.1.
Consider the integrator in Figure P3.1.This system is described in Example 3.1 and has
the impulse response
h(t) = u(t).
(a)
Using the convolution integral, find the system response when the input
x(t)
is
e
5t
u(t)
(i)
u(t - 2)
(ii)
(iii)
u(t)
(iv)
(t + 1)u(t + 1)
(b)
Use the convolution integral to find the system's response when the input
x(t)
is
e
-5t
u(t)
(i) (ii)
(iii) (iv)
(c)
Verify the results of Part (a) and (b), using the system equation
-tu (t)
(t - 1) u(t - 1)
u(t)-u(t - 2)
t
y(t) =
L
x(t)dt.
- q
t
y
(
t
)
x
( )
d
x
(
t
)
Figure P3.1
3.2.
Suppose that the system of Figure P3.2(a) has the input
x(t)
given in Figure P3.2(b).
The impulse response is the unit step function
h(t) = u(t).
Find and sketch the system
output y(t).
