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(elec211)[2010](sum)midterm~hlchanab^_10306.pdf
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ELEC 211 Mid-Term Summer 2010 Date: July 14, 2010 Time: 10:00 C 11:30
Student Name : ________________________________ Student ID : _______________________
-This is a close books and close notes exam.
-
No note sheet, cheat sheet or draft paper is allowed.

-
No use of mobile phones, computers, PDA or calculators is allowed.

-
No use of electronic devices with communication capabilities is allowed

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


-Students caught cheating will receive zero marks and referred to the university for discipline.
Question Max. Points Points
1 25
2 20
3 25
4 30
Total 100

Table 3.1, 3.2, 4.1, 4.2, 5.1 and 5.2 are provided on the last page of this exam paper.
j. j j. j
e + ee . e
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) = ake ak = x(t) e dt 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
1
j t . j t
CTFT x(t) = X ( j) ed X ( j) = x(t) e dt
2. .
1 j jtj. jn
DTFT x[n] = X (e ) ed X (e ) = x[n] e
2
2 n =.
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 =. j22
Question 1 (25 points)
(a)
(4 pts) In discrete-time Fourier series (DTFS), we decompose a DT periodic signal x[n] with period N into a set of orthogonal complex sinusoids. For N= 4, specify this set of orthogonal complex sinusoids.

(b)
(3 pts) Which complex sinusoid among the set in Part (a) represents the complex sinusoid at the highest frequency? Also, plot this complex sinusoid as a function of n(i.e. real part versus n and imaginary part versus n).


For parts (c), (d), (e) below, consider the following discrete-time periodic signal x[n] :

. C 4 C 3 C 2 C1 0 1 2 3 4 .
(c)
(2 pts) What is the fundamental frequency (in rad/s) of x[n]?

(d)
(6 pts) Let ak be the discrete-time Fourier series coefficients of x[n]. Determine the value of :

(i) a1(ii) a2 (iii) a4

(e)
(2 pts) From the result in (d), and based on the fact that x[n] is real, determine the value of a3without use of the analysis equation.


[] + xn[.
xn]
(f) (3 pts) The even part of a DT signal can be obtained by . Plot the even part of x[n].
2
He [ny[n]
(g) (5 pts) x[n] as specified above is the input to an LTI system ( j ) and the output y] is found to be as shown:

2
.


.
x[n]


n

C 2 C1 0
1 2

3
. 2

( j )
He
Which of the followings is the only possible sketch of
?
( j )
He
Briefly explain why the other three choices cannot