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(ELEC2100)(sum)midterm~=ld0433^_43159.pdf
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Student Name : ________________________________ Student ID : ________________
-This is a closed books and closed notes exam.
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No note sheet, cheat sheet or draft paper is allowed.

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No electronic device is allowed. No calculator.

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No communications or talking will be allowed during the exam time.

Question Max. Points Points
1 15
2 15
3 20
4 20
5 30
Total 100

Tables 3.1, 3.2, 4.1 and 4.2 are provided on the last page of this exam paper.
j.. . j. j.. . j.
ee ee
Eulers Relation cos .. sin ..
22 j Convolution Integral y(t) . x(. ) h(t .. ) d.. h(. ) x(t .. ) d.

.. ..
.. .. ..

Convolution Sum y[n] .. x[k] h[n . k] .. h[k] x[n . k]
k ... k ...

. jk t 1 . jk t 2
oo

CTFS x(t) . ae a . x(t) e dt .
kk o
..
T

k ... TT
jk n 1 . jk n 2
oo

DTFS x[n] .. ake ak .. x[n] e o .
k ..N . Nn ..N . N
2... ..
1 j t . j t
CTFT x(t) . X ( j.) ed. X ( j.) . x(t) e dt
.. ..
31 j. j 2 .
cos(. / 6) .

cos(. / 4) . cos(. / 2) . 0 cos(.) ..1 e ..1 e . 1 22

11 j .. j .
sin(. / 6) . sin(. / 4) . sin(. / 2) . 1 sin(.) . 0 e 2 . je 2 .. j
22

sin .

tan .. cos .
(a) (4 pts) Find the magnitude and phase of the following complex numbers:
.
0.5 . j 3
i. z1 . e ii. z2 ..1
. 2..
.. 0.2 . j . t . 4 .

(b)
(4 pts) Sketch the CT signal x(t) . Re{eu(t) } for . 2 < t < 6. n2.

(c)
(4 pts) Sketch the DT signal x[n] . Re{z (u[n . 2] . u[n .10])} where | z |.0.9 and . z . .

k . 2 k

(d) (3 pts) Sketch the DT impulse response h[n] . ..0.8..[n . k]
k .. 3

Question 2 (15 points)
t . 3

(a) (3 pts) For an LTI system H1, the output given by y(t) . x(.) d. where x(t) is the input. Sketch the
.
t . 1

impulse response h1(t) of this system and write down its mathematical expression.
(b) (3 pts) For system H1, sketch the output y1(t) when the input x1(t) is as shown below:

x1(t)
2
1

. 1
2 4 t

(c) (3 pts) A system H2 has the impulse response h2(t) . u(t .1) . u(t . 5) . Recall that the frequency response of an LTI system is given by the Fourier transform of its impulse response. For this system, use the table and the time-shifting property to determine its frequency response H 2( j.).
.
(d) (3 pts) What are the angular frequency, ordinary frequency, and period of the signal x2(t) . cos t ?
4
.

(e) (3 pts) For system H2, determine the output y2(t) if the input is x2(t) . cos t .
4

For the signal x1(t) in Question 2,

(a)
(4 pts) Sketch the even part and odd part of x1(t).

(b)
(6 pts) Using the time shifting property in Table 4.1 and the basic Fourier transform pairs provided in Table 4.2, determine X1( j.) the Fourier transform of x1(t).

.

(c) (3 pts) Sketch the signal g(t) ..x1(t . 8k) for | t | < 13.
k ...

(d)
(3 pts) g(t) is a periodic signal and has a Fourier series representation. Determine the value of b0 , the Fourier coefficient for the ze