Digital Signal Processing Reference
In-Depth Information
Fig. P3.15. Feedback system for
Problem P3.15.
x
(
t
)
y
(
t
)
Σ
h
1
(
t
)
h
2
(
t
)
−
5
t
u
(
t
);
(i)
h
1(
t
)
= δ
(
t
)
+
e
−
2
t
u
(
t
);
(ii)
h
2(
t
)
=
e
−
5
t
sin(2
π
t
)
u
(
t
);
(iii)
h
3(
t
)
=
e
−
2
t
+
u
(
t
+
1)
−
u
(
t
−
1);
(v)
h
5(
t
)
=
t
[
u
(
t
+
4)
−
u
(
t
−
4)];
(vi)
h
6(
t
)
=
sin 10
t
;
(vii)
h
7(
t
)
=
cos(5
t
)
u
(
t
);
(viii)
h
8(
t
)
=
0
.
95
(iv)
h
4(
t
)
=
e
t
;
1
−
1
≤
t
<
0
−
10
≤
t
≤
1
0 otherwise
.
3.13
Consider the systems in Example 3.10. Analyzing the impulse responses,
it was shown that the systems were not memoryless. In this problem,
calculate the input-output relationships of the systems, and from these
relationships determine if the systems are memoryless.
(ix)
h
9(
t
)
=
3.14
Determine whether the LTIC systems characterized by the following
impulse responses are invertible. If yes, derive the impulse response of
the inverse systems.
(i)
h
1(
t
)
=
5
δ
(
t
−
2);
(ii)
h
2(
t
)
= δ
(
t
)
+ δ
(
t
+
2);
(iii)
h
3(
t
)
= δ
(
t
+
1)
+ δ
(
t
−
1);
(iv)
h
4(
t
)
=
u
(
t
);
(v)
h
5(
t
)
=
rect(
t
/
8);
(vi)
h
6(
t
)
=
e
−
2
t
u
(
t
)
.
3.15
Consider the feedback configuration of the two LTIC systems shown in
Fig. P3.15. System 1 is characterized by its impulse response,
h
1
(
t
)
=
u
(
t
). Similarly, system 2 is characterized by its impulse response,
h
2
(
t
)
=
u
(
t
). Determine the expression specifying the relationship between the
input
x
(
t
) and the output
y
(
t
).
3.16
A complex exponential signal
x
(
t
)
=
e
j
ω
0
t
is applied at the input of an
LTIC system with impulse response
h
(
t
). Show that the output signal is
given by
y
(
t
)
=
e
j
ω
0
t
H
(
ω
)
ω=ω
0
,
where
H
(
ω
) is the Fourier transform of the impulse response
h
(
t
)given
by
∞
−
j
ω
t
d
t
.
H
(
ω
)
=
h
(
t
)e
−∞
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