Digital Signal Processing Reference
In-Depth Information
We rewrite this equation as
q
x(t) =
L
x(t)d(t - t)
dt.
(3.8)
-
q
This equation is the desired result, in which a general signal
x(t)
is expressed as a
function of an impulse function. We use this expression for
x(t)
in the next section.
3.2
CONVOLUTION FOR CONTINUOUS-TIME LTI SYSTEMS
An equation relating the output of a continuous-time LTI system to its input is
developed in this section. We begin the development by considering the system
shown in Figure 3.1, for which
x(t) : y(t).
A unit impulse function is applied to the system input. Recall the description
(3.5) of this input signal; the input signal is zero at all values of time other than
at which time the signal is unbounded.
With the input an impulse function, we denote the LTI system response in
Figure 3.1 as
d(t)
t = 0,
h(t)
; that is,
d(t) : h(t).
(3.9)
Because the system is time invariant, the response to a time-shifted impulse func-
tion,
d(t - t
0
),
is given by
d(t - t
0
) : h(t - t
0
).
h(
#
)
The notation will
always
denote the
unit impulse response
.
We now derive an expression for the output of an LTI system in terms of its
unit impulse response
h(t)
of (3.9).
x
(
t
)
(
t
)
y
(
t
)
h
(
t
)
LT I
System
x
(
t
)
(
t
)
y
(
t
)
h
(
t
)
LT I
System
x
(
t
)
(
t
k
)
y
(
t
)
h
(
t
k
)
LT I
System
Figure 3.1
Impulse response of
an LTI system.
