ELEC264
Final Exam
Signals and Systems I
Concordia University
1
Winter 2009
Question 1:
Consider a continuoustime system which has input of signal
)
(
t
x
and output of
)
(
)
(
)
(
t
u
t
x
t
y
=
.
a)
Is this system time invariant? Justify your answer.
b)
Is this system linear? Justify your answer.
Part a:
To prove that the system is time invariant, we should show that for any input
)
(
1
t
x
and
any time shift
0
t
, we have
)
(
)
(
0
1
2
t
t
y
t
y
−
=
, where
)
(
)
(
1
1
t
y
t
x
→
,
)
(
)
(
2
2
t
y
t
x
→
and
)
(
)
(
0
1
2
t
t
x
t
x
−
=
. Otherwise, the system is time variant.
Proof is as follows:
)
(
)
(
)
(
)
(
)
(
)
(
0
0
1
0
1
1
1
t
t
u
t
t
x
t
t
y
t
u
t
x
t
y
−
−
=
−
⇒
=
)
(
)
(
)
(
)
(
)
(
0
1
2
2
t
u
t
t
x
t
u
t
x
t
y
−
=
=
Answer
: Therefore,
)
(
)
(
0
1
2
t
t
y
t
y
−
≠
and the system is time variant.
Part b:
To prove that the system is linear, we should show that for any input
)
(
1
t
x
and
)
(
2
t
x
and
any scalar
a
and
b
, we have
)
(
)
(
)
(
2
1
3
t
by
t
ay
t
y
+
=
, where
)
(
)
(
1
1
t
y
t
x
→
,
)
(
)
(
2
2
t
y
t
x
→
,
)
(
)
(
3
3
t
y
t
x
→
and
)
(
)
(
)
(
2
1
3
t
bx
t
ax
t
x
+
=
.
Otherwise, the system is nonlinear.
Proof is as follows:
{}
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
2
1
2
1
2
1
3
3
t
by
t
ay
t
u
t
bx
t
u
t
ax
t
u
t
bx
t
ax
t
u
t
x
t
y
+
=
+
=
+
=
=
Answer
: Therefore,
)
(
)
(
)
(
2
1
3
t
by
t
ay
t
y
+
=
and the system is linear.
Question 2:
Consider a discretetime system which has input of signal
]
[
n
x
and output of
⎥
⎦
⎤
⎢
⎣
⎡
=
]
[
4
cos
]
[
n
x
n
y
π
.
a)
Evaluate and draw the impulse response of the above system.
b)
If the input to the system is
2
]
[
2
n
n
x
=
, determine whether the output of the system
]
[
n
y
is periodic.
If
]
[
n
y
is
periodic, find its fundamental period and fundamental
frequency.
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View Full DocumentELEC264
Final Exam
Signals and Systems I
Concordia University
2
Winter 2009
Part a:
By definition, the impulse response is
⎥
⎦
⎤
⎢
⎣
⎡
=
=
=
]
[
4
cos
]
[
]
[
]
[
]
[
n
n
y
n
h
n
n
x
δ
π
and therefore,
Answer
:
⎪
⎪
⎩
⎪
⎪
⎨
⎧
≠
=
⎥
⎦
⎤
⎢
⎣
⎡
×
=
=
⎥
⎦
⎤
⎢
⎣
⎡
×
=
0
1
0
4
cos
0
2
2
1
4
cos
]
[
n
n
n
h
Part b:
To prove that
]
[
n
y
is periodic, we should find a positive integer number
N
such that for
any
n
,
]
[
]
[
N
n
y
n
y
+
=
.
⎥
⎦
⎤
⎢
⎣
⎡
=
⎥
⎦
⎤
⎢
⎣
⎡
×
=
8
cos
2
4
cos
]
[
2
2
n
n
n
y
and
( )
⎥
⎦
⎤
⎢
⎣
⎡
+
+
8
cos
]
[
2
N
n
N
n
y
If
k
and
l
are integer numbers,
()
( )
( )
(
)
k
n
N
n
k
n
N
n
n
N
n
16
2
8
8
8
cos
8
cos
2
2
2
2
2
2
±
=
+
⇒
±
=
+
⇒
⎥
⎦
⎤
⎢
⎣
⎡
=
⎥
⎦
⎤
⎢
⎣
⎡
+
l
N
k
nN
N
k
n
nN
N
n
8
16
2
16
2
2
2
2
2
=
⇒
=
+
⇒
±
=
+
+
⇒
Answer
: Therefore, the output
]
[
n
y
is periodic with period of
l
N
8
=
and the
fundamental period and frequency are
8
=
N
and
4
8
2
2
0
ω
=
=
=
N
, respectively.
Question 3:
Consider a continuoustime LTI system which has impulse response of
{}
)
1
(
)
(
)
(
−
−
=
t
u
t
u
t
h
.
If
{ }
)
3
(
)
(
)
(
2
−
−
=
t
u
t
u
t
t
x
is applied at the input of the system,
evaluate the output
)
(
t
y
of the system using convolution integral
∫
∞
∞
−
−
=
τ
d
t
h
x
t
y
)
(
)
(
)
(
as follows:
a)
Draw
)
(
x
and
)
(
−
t
h
for different intervals of “
t
”.
b)
Evaluate the output
)
(
t
y
for the intervals of “
t
” indicated in part (a).
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