Signals and Systems, Midterm Exam
Solutions
Spring 2008, Edited by bypeng
1.
[10]
Consider a system
H
to be tested as being
memoryless
,
causal
,
linear
,
time invariant
,
and
invertible
. Three signals
1
( )
x t
,
2
( )
x
t
, and
3
( )
x
t
are sent to the system, and the corresponding
output signals
1
( )
y t
,
2
( )
y
t
, and
3
( )
y
t
are obtained as shown in Figure 1.
Based on the three input-output pairs, is it possible to determine each of the five properties for
system
H
? If yes, what is it? If no, why? Justify your answer.
Figure 1
Solution:
i.
NEITHER memoryless NOR causal. Observe that
1
3
( )
( )
x t
x
t
=
for
2
t
<
, but
1
3
( )
( )
y t
y
t
≠
for
0
2
t
<
<
.
ii.
NOT linear. Observe that
3
1
2
( )
( )
( )
x
t
x t
x
t
=
+
, but
3
1
2
( )
( )
( )
y
t
y t
y
t
≠
+
.
iii.
NOT time invariant. Observe that
2
1
( )
(
1)
x
t
x t
=
−
, but
2
1
( )
(
1)
y
t
y t
≠
−
.
iv.
NOT invertible. Observe that
1
2
( )
( )
x t
x
t
≠
, but
1
2
( )
( )
y t
y
t
=
.
2.
Consider a system as shown in Figure 2, where
( )
h t
is the impulse response of the LTI sub-system
in the block, and 2D is the operation of time delay for 2 units.
H
H
H
1 2 3
4 5
1
2
3
0
4
x
1
(
t
)
1 2 3
4 5
1
2
3
0
4
x
2
(
t
)
1 2 3
4 5
1
2
3
0
4
x
3
(
t
)
1
2 3 4 5
1
2
3
0
4
y
1
(
t
)
1
2 3 4 5
1
2
3
0
4
y
2
(
t
)
1
2 3 4 5
1
2
3
0
4
y
3
(
t
)
h
(
t
)
2D
+
x
(
t
)
y
(
t
)

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