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Unformatted text preview: Midterm #3 of ECE301, Prof. Wang’s section
8—9pm Thursday, November 17, 2011, BB 170, . Please make sure that it is your name printed on the exam booklet. Enter your student ID number, e—mail address, and signature in the space provided on this
page, NOW! . This is a closed book exam. . This exam contains multiple choice questions and workout questions. For multiple
choice questions, there is no need to justify your answers. You have one hour to
complete it. The students are suggested not spending too much time on a single
question, and working on those that you know how to solve. . There are 16 pages in the exam booklet. Use the back of each page for rough work. . Neither calculators nor help sheets are allowed. Name:
Student ID:
E—mail: Signature: Question 1: [15%, Work—out question, Outcomes 3, 4, and 5] 1. [15%] A discretetime signal is described as follows 2 ifn=0 = 1 ifn=1or —1 (1)
0 ifn=2 and is periodic with period 4. Find the Fourier transform X(ej‘*’) and plot it for the range of w = ~27r to 27r. Question 2: [25%, Workout question, Outcomes 3, 4, and 5] 1. [6%] $(t) = Sing), plot x(t) for the range 0ft = ~27? to 27F. Carefully mark the height of the main lobe and the locations it intersects the horizontal axis. 2. [4%] Find the Fourier transform X ( jw) and plot it for the range of w = —12 to 12. 3. [15%] y(t) = $05) sin(3t) sin(6t). Find the Fourier transform Y(jw) and plot it
for the range of w = —12 to 12. If you do not know the answer to the previous
question, you can write Y(jw) in terms of X(jw). You will get 12 points if your
answer is correct. /\ 35 (“’53 $3 at m a 5:533
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its expression. 2. [6%] Is X(jw) periodic? Compute the value of X(jw)dw.
3. [10%] Compute the Fourier transform X (67”) of = ne”"L{ 4. [6%] Is X (6”) periodic? Compute the value of 22:1 726‘". (Hint: You need to use
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(g “53” 3min; {is ‘ m”? V r; Question 4: [28%, Work—out question, Outcomes 3, 4, and 5] Consider an AM system,
which sends the input signal $(t) over a cos carrier of frequency % a: 0.637 H2. More speciﬁcally, we denote the input signal as 3305) and use y(t) to denote the AM
modulated signal, which will be sent out by the AM transmitter. 1. [4%] What is the value of the carrier frequency we with the unit being (rad/sec)?
Write down the input/ output relationship (equation) between 11:05) and y(t). 2. [12%] Consider the receiver end. To demodulate the original signal from y(t), the
receiver ﬁrst construct w(t) == 2  y(t) ~ cos(wct) and then passes w(t) through a
low—pass ﬁlter with cutoff frequency Wcutoff. Suppose we also know that = cos(t), plot the Fourier transforms X(jw), Y(jw),
and W(jw) for the range of u) = —6 to 6. 3. [12%] Suppose that Prof. Wang forgot the expression of the impulse response of
a lowpass ﬁlter, and decided to pass w(t) through an LTI system with frequency
response 1+0.5w if—2<w§0
H(jw)= 1—0.5w if0<w§2 . (2)
0 otherwise Let z(t) denote the output after passing w(t) through this system. Find the expres
sion of If you do not know the answer of Q42, you can assume that w(t) = 220300 2""lejnt
and solve this question. You will still get 12 points if your answer is correct. fig C’U g 7: 3”?“ X l '2‘; i3" i “a” "
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 Spring '06
 V."Ragu"Balakrishnan

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