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(ELEC211)[2011](f)midterm~=_rrzkjdj^_30649.pdf
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Fall 2011 ELEC 2100 Mid-Term Suggested Solution
Question 1 (25 points)
a) Magnitude = 4 Phase = 2
. ..
b) | z2| .
2 .
2 . z2 . tan 1(1) . ..
1
24 c)
time
d)
0.8
0.6
0.4
0.2
-0.2
-0.4
-0.6
-0.8

-1
-5 0 5
n





t
. st . 2s
2 . st .e . 21. e
g) H(s) .. e dt ...
0 .. s ..0 s
. j2.
1. e
h) s . j. . H( j.) .
j.
1 j0.5. t 1 . j0.5. t
i) x (t) . cos0.5.t . e . e
3 22
. j. j.
1. e 41. e 4
H( j./ 2) .. H(. j./ 2) . ..
j./2 j.. j./2 j.
41 j0.5. t 41 . j0.5. t 24
y (t) . e . e . (2 j)sin(0.5.t) . sin(0.5.t)
3 j. 2 j. 2 j..
Question 4 (30 points) Question 2 (25 points)
01 02 03 04 05 06 07
C B D A C D B

08 09 10 11 12 13 14
D C D C A D A

15 16 17
B B D


2.a) .o .
7
2. 2. 2. 2.
jn j (3) nj (4) nj (6) n
777 7
b) x[n] . 0.5 e . je . je . 0.5 e
2. 6. 6. 2.
jn jn . jn . jn
. 2.. . 6..
777 7
c) x[n] . 0.5 e . je . je . 0.5 e . cos. n.. 2sin. n. . 7 .. 7 .
2
12 2 . 1 . 2

N . 1
d) . | x[n]| .. | ak | . 2 . .. 2 (1) . 2.5 Nn . 0 k ..N .. 2 .
e) This is a DT lowpass filter and the cutoff frequency is 2/8. All the frequency components of x[n] are blocked. Therefore, y[n] = 0
f) b .a where M = 1
kk . M
b . a . 0, b . a . 0.5, b . a . 0, b . a .. j
1021 3243
b . a . j , b . a . 0, b . a . 0.5
54 6506
4. 8. 10.
jnjn jn
77 7
g) x1[n] . 0.5 e . je . je . 0.5 . y1[n] . 0.5

Question 3 (20 points)
a) Yes. This is because the signal is real and even so that its FT would be also real and even (from table 4.1). . 1 t . |2| . 2 t . |2| . sin(2.) .2
b) x(t) .. * . . X ( j.) . 8 ..
0 otherwise 0 otherwise .
.. .. . sin(2.) .2 . j 4.
c) X ( j.) . 8 .. e1 ...
35
30
25

| X ( j.)|
1 20
15
10

5 0 -4-3-2-1 0123 4
16
. X1( j.)

Slope=.4 .44

d)

8


6.40461014
t
e) Yes. The spectrum of y(t) is continuous so that it would be an aperiodic signal.


h) 3
Y(j...
....
....

....

..
.......................................................................

........................................................................
1 . i) Y4( j.) . Y ( j.) . 5 ..(.. k / 5)
k ...
Y4
(j...
....
.... ....................................... ..