Digital Signal Processing Reference
In-Depth Information
Fig. P2.16. (a) Input-output
pair for an LTI CT system.
(b) Periodic input to the LTI
system.
x
(
t
)
y
(
t
)
1
1
t
t
−0.5
0.5
−1
1
(a)
x
p
(
t
)
1
t
−2.5
−1.5
−0.5
0.5
1.5
2.5
(b)
2.16
Figure P2.16(a) shows an input-output pair of an LTI CT system. Calcu-
late the output
y
p
(
t
) of the system for the periodic signal
x
p
(
t
) shown in
Fig. P2.16(b).
2.17
The output
h
(
t
) of a CT LTI system in response to a unit impulse function
δ
(
t
) is referred to as the impulse response of the system. Calculate the
impulse response of the CT LTI systems defined by the following input-
output relationships:
(i)
y
(
t
)
=
x
(
t
+
2)
−
2
x
(
t
)
+
2
x
(
t
−
2);
t
+
t
0
(ii)
y
(
t
)
=
x
(
τ
−
4) d
τ
;
t
−
t
0
t
e
−
2(
t
−τ
)
x
(
τ −
4) d
τ
;
(iii)
y
(
t
)
=
−∞
∞
(iv)
y
(
t
)
=
f
(
T
− τ
)
x
(
t
− τ
)d
τ
where
f
(
t
) is a known signal and
−∞
T
is a constant.
2.18
The output
h
[
k
] of a DT LTI system in response to a unit impulse function
δ
[
k
] is shown in Fig. P2.18. Find the output for the following set of inputs:
(i)
x
[
k
]
= δ
[
k
+
1]
+ δ
[
k
]
+ δ
[
k
−
1];
h
[
k
]
1
1
∞
(ii)
x
[
k
]
=
δ
[
k
−
4
m
];
k
m
=−∞
−1
1
(iii)
x
[
k
]
=
u
[
k
]
.
2.19
A DT LTI system is described by the following difference equation:
−2
y
[
k
]
=
x
[
k
]
−
2
x
[
k
−
1]
+
x
[
k
−
2]
.
Fig. P2.18. Output
h
[
k
] for
input
x
[
k
] = δ[
k
] in Problem
2.18.
Determine the output
y
[
k
] of the system if the input
x
[
k
]isgivenby
Search WWH ::
Custom Search