Digital Signal Processing Reference
In-Depth Information
−
3
t
u
(
t
);
(v)
x
(
t
)
=
sin(2
π
t
)(
u
(
t
−
2)
−
u
(
t
−
5))
,
h
(
t
)
=
u
(
t
)
−
u
(
t
−
2);
(vi)
x
(
t
)
=
e
(iv)
x
(
t
)
=
e
2
t
u
(
−
t
)
,
h
(
t
)
=
e
x
(
t
)
1
−
2
t
−
5
t
;
(vii)
x
(
t
)
=
sin(
t
)
u
(
t
)
,
h
(
t
)
=
cos(
t
)
u
(
t
)
.
,
h
(
t
)
=
e
t
0
3.6
For the four CT signals shown in Figs. P3.6, determine the following
convolutions:
(i)
y
1
(
t
)
=
x
(
t
)
∗
x
(
t
);
(ii)
y
2
(
t
)
=
x
(
t
)
∗
z
(
t
);
(iii)
y
3
(
t
)
=
x
(
t
)
∗
w
(
t
);
(iv)
y
4
(
t
)
=
x
(
t
)
∗
v
(
t
);
(v)
y
5
(
t
)
=
z
(
t
)
∗
z
(
t
);
−1
(vi)
y
6
(
t
)
=
z
(
t
)
∗
w
(
t
);
(vii)
y
7
(
t
)
=
z
(
t
)
∗
v
(
t
);
(viii)
y
8
(
t
)
=
w
(
t
)
∗
w
(
t
);
(ix)
y
9
(
t
)
=
w
(
t
)
∗
v
(
t
);
(x)
y
10
(
t
)
=
v
(
t
)
∗
v
(
t
).
(i)
z
(
t
)
1
3.7
Show that the convolution integral satisfies the distributive, associative,
and scaling properties as defined in Section 3.6.
−1
t
0
1
3.8
When the unit step function,
u
(
t
), is applied as the input to an LTIC sys-
tem, the output produced by the system is given by
y
(
t
)
=
(1
−
e
−1
−
t
)
u
(
t
).
Determine the impulse response of the system. [Hint: If
x
(
t
)
→
y
(
t
) then
d
x
/
d
t
→
d
y
/
d
t
(see Problem 2.15).]
(ii)
3.9
A CT signal
x
(
t
), which is non-zero only over the time interval,
t
=
[
−
2
,
3], is applied to an LTIC system with impulse response
h
(
t
). The
output
y
(
t
) is observed to be non-zero only over the time interval
t
=
[
−
5
,
6]. Determine the time interval in which the impulse response
h
(
t
)
of the system is possibly non-zero.
w
(
t
)
1
(1 +
t
)
(1 −
t
)
t
−1
(iii)
3.10
An input signal
v
(
t
)
1
−
t
0
≤
t
≤
1
x
(
t
)
=
1
0
otherwise
e
2
t
e
−2
t
is applied to an LTIC system whose impulse response is given by
h
(
t
)
=
e
t
−1
0
1
−
t
u
(
t
). Using the result in Example 3.8 and the properties of
the convolution integral, calculate the output of the system.
(iv)
−
(
t
−
2)
u
(
t
−
2) is applied to an LTIC system whose
impulse response is given by
3.11
An input signal
g
(
t
)
=
e
Fig. P3.6. CT signals for
Problem P3.6.
5
−
t
4
≤
t
≤
5
r
(
t
)
=
0
otherwise
.
Using the result in Example 3.8 and the properties of the convolution
integral, calculate the output of the system.
3.12
Determine whether the LTIC systems characterized by the following
impulse responses are memoryless, causal, and stable. Justify your
answer. For the unstable systems, demonstrate with an example that a
bounded input signal produces an unbounded output signal.
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