Digital Signal Processing Reference
In-Depth Information
3.8.
Show that the convolution of three signals can be performed in any order by showing that
[f(t)*g(t)]*h(t) = f(t)*[g(t)*h(t)].
(
Hint
: Form the required integrals, and use a change of variables. In one approach to
this problem, the function
q
q
g(t)
B
h(t - t-s)f(s)ds
R
dt
L
L
-
q
-
q
appears in an intermediate step.)
3.9.
Find
x
1
(t)*x
2
(t),
where
x
1
(t) = 2u(t + 2) - 2u(t - 2)
and
0,
t 6-4
e
-ƒtƒ
,
x
2
(t) =
c
-4 … t … 4
0,
t 7 4.
3.10.
Find and sketch
u(t)*u(t - 5).
3.11.
For the system of Figure P3.2(a), the input signal is
x(t)
in Figure P3.11 (note that the
h(t) = e
-ƒtƒ
u(-t).
signal is not symmetric) and
Find the system output y(t).
x
(
t
)
3
2
1
t
3
2
1
0
1
23
Figure P3.11
3.12.
(a)
Consider the two-LTI system cascaded in Figure P3.12. The impulse responses of
the two systems are identical, with
h
1
(t) = h
2
(t) = e
-t
u(t).
Find the impulse re-
sponse of the total system.
(b)
Repeat Part (a) for the case that
(c)
Repeat Part (a) for the case that
(d)
Repeat Part (a) for the case that
h
1
(t) = h
2
(t) = d(t).
h
1
(t) = h
2
(t) = d(t - 2).
h
1
(t) = h
2
(t) = u(t - 1) - u(t - 5).
3.13.
(a)
We define a new signal,
z(t)
, a function of two signals
x(t)
and
h(t)
, as
q
z(t) =
L
x(-t + a)h(t + t)dt.
-
q
Express
z(t)
in terms of
y(t) = x(t)*h(t),
the convolution of
x(t)
and h(t).
x
(
t
)
y
(
t
)
h
1
(
t
)
h
2
(
t
)
Figure P3.12
