Digital Signal Processing Reference
In-Depth Information
To calculate the system output, we perform a quasi-graphical convolution technique.
First, sketch the time-reversed, time-shifted impulse response function,
h(t - t),
as shown in
Figure 3.11(c). From this sketch, it is seen that the signal
h(t - t)
will be nonzero over the
-
q
6t … (t - 1)
interval
and for other values of
t, h(t - t)
will be zero. The input signal,
x(t),
is zero for
t 7-1,
so both functions in the product
x(t)h(t - t)
will be nonzero only
for
t … 0.
Next, evaluate the convolution integral for the time interval
During this interval, both exponential functions are nonzero and
-
q
6t … (t - 1), t … 0.
the output is given by
t - 1
t - 1
e
-2
e
t
2
e
t
e
-(t -t)
dt =
3
e
-t
e
2t
dt =
, -
q
6 t … 0.
y(t) =
3
-
q
-
q
Because
x(t)
is zero for
t 7-1,
for
t 7 0,
the output signal is given by
-1
-1
e
-2
e
-t
2
e
t
e
-(t -t)
dt = e
-t
e
2t
dt =
y(t) =
3
, t 7 0.
3
-
q
-
q
The output signal,
y(t),
is plotted in Figure 3.11(d). It is an even function of time, because in
this example,
x(t) = h(-t).
■
In this section, the convolution integral for continuous-time LTI systems is de-
veloped. This integral is fundamental to the analysis and design of LTI systems and
is also used to develop general properties of these systems. To illustrate convolu-
tion, we evaluate the integral first by a strictly mathematical approach and then by a
quasi-graphical approach.
3.3
PROPERTIES OF CONVOLUTION
The convolution integral of (3.14) has three important properties:
1.
Commutative property
. As stated in (3.15), the convolution integral is
symmetrical with respect to
x(t)
and
h(t):
x(t)*h(t) = h(t)*x(t).
(3.22)
An illustration of this property is given in Figure 3.12. This figure includes a com-
mon representation of an LTI system as a block containing the impulse response.
The outputs for the two systems in Figure 3.12 are equal, from (3.22).
2.
Associative property
. The result of the convolution of three or more func-
tions is independent of the order in which the convolution is performed. For example,
[x(t)*h
1
(t)]*h
2
(t) = x(t)*[h
1
(t)*h
2
(t)] = x(t)*[h
2
(t)*h
1
(t)].
(3.23)
The proof of this property involves forming integrals and a change of variables
and is not given here. (See Problem 3.8.) This property is illustrated with the two
cascaded systems in Figure 3.13(a). For cascaded LTI systems, the order of the
