Digital Signal Processing Reference
In-Depth Information
starting point for the design. The LTI model may not be very accurate, but the use of an
LTI model allows us to initiate the design process with standard design procedures.
4.
We can sometimes model a general signal
x(t)
as a sum of functions:
Á
.
x(t) = x
1
(t) + x
2
(t) +
The functions are standard functions for which it is much easier to
find an LTI system response than it is to find the response to
x
1
(t), x
2
(t),
Á
x(t).
The system re-
sponse is then the sum of the responses to the standard functions,
Á
,
y(t) = y
1
(t) + y
2
(t) +
where is the response to This is a key attribute of linear
time-invariant systems that will be used when we consider Fourier analysis in
Chapter 4.
y
i
(t)
x
i
(t), i = 1, 2, Á .
3.1
IMPULSE REPRESENTATION OF CONTINUOUS-TIME
SIGNALS
In this section, a relationship is developed that expresses a general signal as a
function of an impulse function. This relationship is useful in deriving general prop-
erties of continuous-time linear time-invariant (LTI) systems.
Recall that two definitions of the impulse function are given in Section 2.4.
The first definition is, from (2.40),
x(t)
q
d(t - t
0
) dt = 1,
(3.5)
L
-
q
and the second one is, from (2.41),
q
x(t)d(t - t
0
) dt = x(t
0
).
(3.6)
L
-
q
The second definition requires that
x(t)
be continuous at
t = t
0
.
According to (3.6),
if
x
(
t
) is continuous at
t = t
0
,
the sifting property of impulse functions can be stated
as from (2.42),
x(t)d(t - t
0
) = x(t
0
)d(t - t
0
).
(3.7)
We now derive the desired relationship. From (3.7), with
t
0
= t,
x(t)d(t - t) = x(t)d(t - t).
From (3.5), we use the preceding result to express
x(t)
as an integral involving an
impulse function:
q
q
x(t)d(t - t) dt =
L
x(t)d(t - t) dt
L
-
q
-
q
q
= x(t)
L
d(t - t) dt = x(t).
-
q
