Short Answer Review
In the figure below, x t( ) is the original signal (in the upper left corner)
-4 -2 0 2 4
0 0.5 1 1.5 2
Time (sec)
x(
t)
-4 -2 0 2 4
0 0.5 1 1.5 2
Time (sec)
xa
(t
)
-4 -2 0 2 4
0 1 2
Time (sec)
xb
(t
)
-4 -2 0 2 4
0 1 2
Time (sec)
xc
(t
)
-4 -2 0 2 4
0 1 2
Time (sec)
xd
(t
)
-4 -2 0 2 4
0 1 2
Time (sec)
xe
(t
)
1) Which signal represents 2 t x⎛ ⎞⎜ ⎟
⎝ ⎠ ?
x t
a( )
x t
b( )
x t
c( )
x t
d( )
x t
e( )
2) Which signal represents x t(2 −1)?x t
a( )
x t
b( )
x t
c( )
x t
d( )
x t
e( )
3) Which signal represents x t( +1)?
x t
a( )
x t
b( )
x t
c( )
x t
d( )
x t
e( )
4) Which signal represents x(1.5 )t ?
x t
a( )
x t
b( )
x t
c( )
x t
d( )
x t
e( )
5) Which signal represents x t( −1)?
x t
a( )
x t
b( )
x t
c( )
x t
d( )
x t
e( )
6) Which signal represents a compressed x( )t ?
x t
a( )
x t
b( )
x t
c( )
x t
d( )
x t
e( )
In the following figure, the original signaly t( ) is in the top panel
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-1 -0.5 0 0.5 1
Time (sec)
y(
t)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-1 -0.5 0 0.5 1
Time (sec) y a
(t
)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-1 -0.5 0 0.5 1
Time (sec) y b
(t
)
8) Which signal has the highest frequency? y t( )
y t
a( )
y t
b( )
9) Which signal has the lowest frequency? y t( )
y t
a( )
y t
b( )
10)
y t
( )
=
y ct
a( )
for what value of c ?c
=
0.5
c
=
1.0
c
=
1.5
c
=
2.0
11)
y t
( )
=
y ct
b( )
for what value of c ?c
=
0.5
c
=
1.0
c
=
1.5
c
=
2.0
The original signal z(t) is in the top left panel.
0 2 4 6
0 1 2 3
Time (sec)
z(
t)
-6 -4 -2 0
0 1 2 3
Time (sec)
z d
(t
)
-6 -4 -2 0
0 1 2 3
Time (sec)
z b
(t
)
-6 -4 -2 0
0 1 2 3
Time (sec)
z a
(t
)
-6 -4 -2 0
0 1 2 3
Time (sec)
z c
(t
)
-6 -4 -2 0
0 1 2 3
Time (sec)
z e
(t
)
13) Which of the above signals represents z(−t)?
z t
a( )
z t
b( )
z t
c( )
z t
d( )
z t
e( )
14) Which of the above signals represents z(− +t 1)
z t
a( )
z t
b( )
z t
c( )
z t
d( )
z t
e( )
15) The function
( )
4 2t t j j
x t
=
e
+
e
isa) not periodic
16) The function
x t
( )
=
cos( ) sin(2
t
+
π
t
)
isa) not periodic
b) periodic with fundamental period 1 second c) periodic with fundamental period
π
seconds d) periodic with fundamental period 2π
seconds17) The function ( ) sin ( 1) 2
x t = ⎛⎜
π δ
t⎞⎟ t−⎝ ⎠ +t can be simplified as
a) x t( )=2 b) x t( )= +1 t c)x t( )=
δ
(t− +1) t d) x t( )=δ
(t− +1) 118) The integral
10
0
( 1) ( 2)d
δ λ
−δ λ
−∫
λ
d
can be simplified as a) 0 b) 1 c) none of these
19) The integral
5
1
(
2)
t
δ λ
−−
∫
λ
can be simplified as a) 2 b)t
c) 2 (δ
t−2) d) tδ
(t−2)20) The integral can be simplified as
2
1
(
t
3)
d
δ
−
−
∫
t
a) 1 b) 0 c) 3 d)
δ
(t−3)21) The integral
u t
(
1) (
u t
2)
e dt
t can be simplified as∞
−
−∞
+
−
∫
a) b) c) d) none of these
1
t
e dt
∞−
−
∫
2t
e dt
∞ −∫
2 1 te dt
−−
∫
22) The integral
u
( 1
λ
)
e
| |λd
λ
∞−
−∞
− −
∫
can be simplified asa)
1 | |
e
λd
λ
−−
−∞
∫
b) | |
1
e
λd
λ
∞−
−
∫
c) | |
1
e
λd
λ
∞−
23) The function x t( )below can best be represented by the function
a) ( ) 5 2 t x t = Π ⎜ ⎟⎛ ⎞
⎝ ⎠ b)
1 ( ) 5
2 t x t = Π ⎜⎛ − ⎞⎟
⎝ ⎠
c) ( ) 5 4 t x t = Π ⎜ ⎟⎛ ⎞
⎝ ⎠ d)
1 ( ) 5
4 t x t = Π ⎜⎛ − ⎞⎟
⎝ ⎠
-5
3
t
x(t
24) The function x t( ) below can best be modeled by the function
a) x t( )=u t( + +1) u t( − −1) u t( −2) b)x t( )=u t( + +1) 2 (u t− −1) 2 (u t−2) c) x t( )=u t( + +1) u t( − −1) 2 (u t−2) d)x t( )=u t( + +1) 2 (u t− −1) 3 (u t−2)
-1
1 2
2
1
t
( )
x t
25) The average power in the signal
( )
j tx t
=
ce
ω is a) 0 b)| |
2
c
c)
|c|
2 d)2
|c|
2
26) The average power in the signal
x t
( )
=
A
cos(
ω θ
t
+
)
is a)2
A
b)
A
c)A
2 d)2
2
A
27) The signal
x t
( )
=
A
cos(
ω
t u t
)[ ( )
−
u t
(
−
10)]
is28) The signal
x t
( )
=
2 cos(2 )
t
+
j
2sin(2 )
t
is a) an energy signal b) a power signal c) neither29) The signal
x t
( )
=
2 ( )
u t
−
u t
(
− −
1) 2 (
u t
−
2)
is a) an energy signal b) a power signal c) neither30) The signal
( )
t( )
x t
=
e u t
isa) an energy signal b) a power signal c) neither
31) Fill in the following table with a Y (yes) or N (no) for each of the system models given. Assume
−∞ < < ∞
t
for all of the systems.System Model Linear?
Time-Invariant?
Causal? Memoryless?
( )
3sin(
1) (
1)
y t
=
t
+
x t
−
( ) 1
2 t y t =x⎛⎜ − ⎞⎟
⎝ ⎠
(
)
( )
1
y t
=
x
−
t
2
( )
( )
sin( ) ( )
y t
+
t y t
=
t x t
32) The average power in the signal
x t
( )
=
ce
j tω is a) 0 b)| |
2
c
c)
|c|
2 d)2
|c|
2
33) The average power in the signal
x t
( )
=
A
cos(
ω θ
t
+
)
is a) | |2 A
b) |A| c)
A
2 d)2
2
A
34) The average power in the signal
x t
( )
=
ce
j tω+
de
j2ωt is a) 0 b) | | | |2 2
c + d
c)
|c|
2+
|
d
|
2 d)2 2
|c|
|
|
2
2
d
35) The average power in the signalx t( )=Acos(
ω θ
t+ )+Bcos(2ω φ
t+ ) is a) | | | |2 2
A + B
b) |A|+|B| c)
A
2+
B
2 d)2 2
2
2
A
B
+
36) Assume we know a system is a linear time invariant (LTI) system. We also know the following input x t( ) – output y t( ) pair:
0 1
2 3 4 5
6
-1
2
3
4
1
( )
x t
t
( )
newx
t
0
1
2
3
4
5 6
-1
2
3
4
1
( )
y t
t
If the input to the system is now
x
new( )
t
2
3
4
1
Which of the following best represents the output of the system? a)
y t
a( )
b)y t
b( )
c)y t
c( )
d)y t
d( )
0
1
2
3
4
5
6
-t
1
0 1
2 3 4 5
6
-1
2
3
4
1
( )
ay t
t
0
1
2
3
4
5 6
-1
2
3
4
1
( )
by t
t
0 1
2 3 4 5
6
-1
2
3
4
1
( )
cy t
t
0
1
2
3
4
5 6
37) The impulse response of the mathematical model of a systemy t( )=2 (x t−1) is a) h t( )=
δ
( )t b) h t( )=2 ( )δ
t c) h t( )=2 (δ
t−1) d) h t( )=2 (u t−1)38) The impulse response of the mathematical model of a system
1
( )
( )
t
y t
x
λ λ
d
−
−∞
=
∫
isa) h t( )=u t( ) b) h t( )=1 c) h t( )=u t( −1) d) h t( )= −t 1
39) The impulse response of the mathematical model of a system
1
( )
(
2)
t
y t
λ λ
x
d
λ
−
−∞
=
∫
−
isa) h t( )=2 (u t−1) b) h t( )=2 (u t−2) c) h t( )=2 (u t−3) d) h t( )=2 ( )u t
40) The impulse response of the mathematical model of a system ( ) ( 2) t
y t
λ λ
x dλ
∞
=
∫
− isa) h t( )=2 ( )u t b) h t( )=2 (2u −t) c) h t( )=2 (u t−2) d) h t( )=u t( )
41) The impulse response of the mathematical model of a system y t( )+2 ( )y t =3 ( )x t is
a)
h t
( )
=
3
e u t
−2t( )
b)h t
( )
=
3
e u t
2t( )
c) h t( )=3 ( )u t d) h t( )=6 ( )u t42) The impulse response of the system
1
( )
( )
t
h t
=
e u t
−h t
2( )
=
2 (
δ
t
−
1)
( )
x t
( )
v t
y t
( )
is
a)
h t
( )
=
2
e u t
−t( )
b)h t
( )
=
2
e
−tδ
(
t
−
1)
c)h t
( )
=
2
e
− −(t 1)u t
(
−
1)
d)h t
( )
=
2
e
− −(t 1)u t
( )
43) Consider an unknown system. When the input to the system is x t( )=2 cos(2 )t the output of the system isy t( )=2 cos(2 )t +cos(4 )t . Is the system linear?
44) Consider the following input/output pair for an unknown system.
-1 -0.5 0 0.5 1 1.5 2
-1 -0.5 0 0.5 1
t (sec)
x
(t
)
t
he i
nput
-1 -0.5 0 0.5 1 1.5 2
0 1 2 3
t (sec)
y
(t
)
t
he out
put
Which of the following is true: a) The system is linear b) The system is not linear
c) It is not possible to determine if the system is linear based on the information given.
Problems 45 -48 refer to the following linear time invariant (LTI) system, with impulse response shown below on the left, and input
( )
h t x t( ) shown below on the right. The output of the
system, y t( ), is the convolution of the impulse response with the input, y t( )=h t( )∗x t( ).
0 1
2 3 4 5
6
-1
2
3
4
1
( )
h t
t
0
1
2
3
4
5 6
-1
2
3
4
1
( )
x t
t
45) Is this LTI system causal? a) Yes b) No
Answers:
1) xb 2) xe 3) xa 4) xc 5) xd 6) xc (xe is compressed and shifted)
7) xb 8) yb 9) ya 10) c=2 11) c=0.5 12)yb 13) zd 14) zb
15) d 16) a 17) c 18) a 19) b 20) b 21) b 22) a (or c) 23) d 24) c
25) c 26) d 27) a 28) b 29) b 30) c
31a) L, not TI, C, not M 31b) L, not TI, not C, not M 31c) L, not TI, not C, not M
31d) L, not TI, C, not M 32) c 33) d 34) c 35) d 36) b
