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(ELEC211)2007_f_midterm.pdf
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ELEC 211 (Fall 2007)
Midterm Examination
October 30, 2007

12:00 pm C 1:20 pm
Time: 1 Hour and 20 Minutes Closed Book
There are 3 problems Total Points = 100
Question Marks
1
2
3
4
5
total 100

Useful Equations :

jk t 1 . jk t 2
00
CTFS x(t) = ake ak =Tx(t) e dt; 0 =
k =. TT
jk n 1 . jk n 2
00
DTFS x[n] = ake ak = x[n] e ; 0 =
k:<N > Nn:<N > N
1
jt . jt
CTFT x(t) = X ( j) ed X ( j) = x(t) e dt
2. .
Problem 1
.2t a) Let h(t) = eu(t) be the impulse response of an LTI system. Determine the frequency response
H ( j) of this system.
. j6t
b) Determine the output y(t) of the above system when the input is x(t) = e

.3t
c) Determine the output y(t) of the above system when the input is x(t) = eu(t)

d) Express the LTI system in the form of a linear differential equation (LDE).
.3t e) Assume that x(t) = eu(t) is again the input to the system described by the LDE in part d). It is
given, however, that the initial condition is such that the output y1(t) has an initial value of 1; that is,
y1(0) = 1. Is the output y1(t) the same as the output y(t) as determine in part c)? Briefly explain your answer.
j+ 4H ( j) =
f) For the causal and stable LTI system with the frequency response 2 + ,
6 . 5 j
determine | H ( j) | and H ( j) when = 0 and when
Problem 2
a) x[n] is a periodic discrete-time signal with period N=4.

. - 3 - 2 -1 0 1 2 3 4..
i. Find a0.
ii. Without finding the individual ak , determine the value of a0 + a1 + a2 + a3.
1
a =.
iii. Note that x[n] is real and even. It is known that 1 . Without using the analytic equation,
4
determine the value of determine the value a.1.
iv.
Continuing from part iii., determine the value of a3.

v.
Using results from i. to iv. or otherwise, determine the value of a2.


b) The signal in part a) is the input to an LTI system with unit sample response h[n]. The output y[n] is a constant as plotted below (i.e., y[n]=1): 1
.

.






n
. - 3 - 2 -1 0 1 2 3 4..
2

. jn
What is the value of h[n]e 4 ? Justify your answer.
n=.
c) For the periodic triangular wave x(t) shown below, determine its Fourier Series coefficients. (Hint: Use the integration property). x (t)

Problem 3
The impulse response h(t) of a continuous-time LTI system is given by:

.kT

h(t) =e (t . kT) where is real and e. = 0.5
k =0
a) Sketch h(t).
b) For the input signal x(t) shown below, sketch the output signal y(t) for t = 0 to t = 3T.
x (t)
1


t
0
T/2 T
c) Show that the frequency response of the system is
1

H ( j) =
. (+ j ) T
1 . e
d) The impulse response h(t) above can be viewed as the distortion caused by echoing in a concert hall. For music recordings made in the concert hall, it is desirable to have the original signal x(t) restored from y(t) by