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(ELEC211)[2010](sum)final~=_8udkjdm^_39350.pdf
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Summer 2010 ELEC 211 Final
Problem 1 (Total = 20 points)
a. (3 pts) A periodic signal CT x(t) has period T = 0.2 and Fourier series coefficients a1 . a.1 . 2 a2 . a.2 . 1 a3 . a.3 ..1 ak. 0 for all other ks
Express x(t) as a sum of complex sinusoids and state the value of .o , the fundamental angular frequency of these complex sinusoids.
jk. t
o
b.
(3 pts) The Fourier transform of e is 2..(.. k.o ) . Sketch the Fourier Transform of x(t). Label all important dimensions.

c.
(3 pts) Let h(t) be the impulse response of an LTI system. The frequency response of this system is: . 1 . | . / 15 . | . 15 .... 15 . H ( j.)


H( j.) ..
. 0| . | . 15 . 1

.
. 15 0 15
Let x(t) be the input to this LTI system and let y(t) be the output. Determine y(t).

d. (4 pts) Assume that we implement h(t) using discrete-approximation, with a sampling interval of Ts = 0.05. That is, we implement hp(t) which is
..
hp (t) . h(t) ..(t . kTs ) .. h(0.05k) .(t . 0.05k)
k ...k ...
. 2. .. 2.k .
Sketch Hp ( j.) the Fourier transform of hp(t). ( Reminder: ..(t . kT ) . . ..... .. )
s k ... Ts k ... . Ts .
e.
(4 pts) Let x(t) be the input to hp(t) and let z(t) be the output. Determine z(t).

f.
(3 pts) Assume that we implement a discrete-time filter hd[n] such that hd[n] are the sampled values of h(t) at a sampling interval of 0.05; that is: hd [n] .h(0.05n)


Sketch Hd (ej. ) , the DTFT of d []. Carefully label all dimensions.
hn
Problem 2 (Total = 25 points)
Let x1(t) be a band-limited baseband signal with spectrum as shown below
X1(j...

.......................
a.
(2 pts) What is the largest sampling interval Ts allowed if we are to be able to recover x(t) from its samples.

b.
(2 pts) Assume that we modulate x1(t) using sinusoidal AM at a carrier frequency of 30 Hz, or .c. 60. rad/sec. The modulate signal is: y1(t) . x1(t)cos .ct . Draw the spectrum of y1(t). Label all important dimensions.

c.
(2 pts) Assume now that we modulate x1(t) using with-carrier AM at .c. 60. rad/sec, y.(t) . (x (t) . A) cos . t


11 c
.1()
Draw the spectrum of yt . Label all important dimensions.
d.
(2 pts) What is the reason of using with-carrier AM? .. cos 60.t if cos 60.t . 0

e.
(2 pts) Sketch the signal c (t) . max(cos60.t , 0) ..


0 if cos 60.t . 0
.
f. Let us determine the frequency domain representation of c. (t) . Notice that c. (t) is periodic with fundamental 2. 2. 1 .
frequency .c . 60. rad/sec, or period T .. . . We can consider c(t) as the product of .c 60. 30 cos 60.t with the periodic square wave r(t). That is, let c. (t). r(t) cos60.t
. 1| t | . T /4 1
where r(t) .. , r(t) is periodic with period T .
. 0 T /4 . | t | . T /2 30

i. (2 pts) Let ak be the Fourier series coefficients of cos60.t . Specify the value of ak for all k. sin k. /2
ii. (3 pts) From lecture notes, we the Fourier series coefficients of r(t) are given by bk .