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Unformatted text preview: ECE 301, Final exam of the session of Prof. ChihChun Wang
Friday 1pm—3pm , May 7, 2010, CL50 224. . Enter your name, student ID number, email address, and signature in the space provided on this page, NOW! . This is a closed book exam. . This exam contains both multiplechoice and work—out questions. 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 24 pages in the exam booklet. Use the back of each page for rough work. The last pages are all the Tables. You may rip the last pages for easier reference.
Do not use your own copy of the Tables. Using your own copy of Tables
will be considered as cheating. . Neither calculators nor help sheets are allowed. Name: Sy/mfgms
Student ID:
E—mail: Signature: Question 1: [20%] No need to write down justiﬁcations for this question. 1. [2%] x(t) = Is 23(75) periodic? 2. [2%] 31(15): Singgggm). ls y(t) an odd signal? 3. [3%] Continue from the previous question. Is y(t) of ﬁnite power? Is y(t) of ﬁnite
energy? 4. [3%] Consider an LTI system with impulse response h(t). If we feed this system
with an input a:(t) = ej2010tZ/{(t + 4) + 172%, the output is y(t) = ej(2010t_2010)Z/{(t +
3) + (t_11)2+2. Write down the expression of the impulse response h(t). 5. [3%] Consider a system with the input / output relationship ya) = mews. <1) 2—1
Is the system causal? Is the system linear? Is the system time—invariant?
6. [2%] = ej”. Is periodic? 7. [2%] = and X (69“) is its Fourier transform. ls X (69") periodic? 8. [3%] x(t) = 2202*“) k—fﬁﬂt — 0.37rk) and X(jw) is its Fourier transform. ls periodic? Is X ( jw) periodic? M~ (forawU/L [ms
. Z.
fcrréj 3‘ [L % Yes Per/7.1 Zl—Sla {3574/ a” é ) Liv; [9 (74,01! 3 2' M0. y[—£/: 7) ever, ‘S;n(ZVV00"€/ : /(é/
~L€06 574 (~er Iré/
LN (re) g. par,” & (in (Y/g A’no [IV 4, Alf) : d/éﬂ) fé‘ w J; I: A(G‘1’ C0w5w{ g /V(~l) : “new” Ma] .Lﬂwt >4Vd,r.'dnil/§ l 6‘ V [WK 7_ 75 («V/(A10 S /’3(’/fdc/’»<I I {S Pcff/Q/[é I kqﬂ/
X(j") f5 ﬁﬂ/I’JJ;C
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2/6): //vf/2. Wham?
l<=7$0 l Question 2: [10%] “ Consider x(t) = 2:0 6(t”—~ 1.5k), and ‘ 2t+1 if——0.5<t<0
h(t)= 1—2t if0<t<0.5 . (2)
0 otherwise 1. [2%]Draw $05) for the range —2 < t < 4.
2. [2%] Draw h(t) for the range —2 < t < 4. 3. [6%] Draw y(t) = 33(t) * h(t) for the range —2 < t < 4. «[6
1' IL
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NR) Question 5’: [10%] $(t) = 6—3tU(t) and h(t) é 8—3tU(t — 1). Find the expression of
3/05) = 9305) * Mt) y“): fa 2((197/ AW) JZ 04 I
Km) «2 m/ i 0
kill/7A6
0 l , e ,w/ .33
f/f)’j g e
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' '6 (P) Question 4: [15%] Consider the following difference equation: yin] = yin —11—§yin — 21+ 3min]  gm — 1]. <3) 1. [5%] Find the frequency response H (6”). 2. [5%] When the input is = 6M, ﬁnd out the output If you do not know
the answer H (6”) of the previous question, you can assume that H(€jw)= 1 ifw<g _ (4)
0 ifgglwl<7r 3. [5%] When the input is : ejn, ﬁnd out the output If you do not know the
answers to the previous questions, you can assume that H<ejw): 1 ifwl<325 ' (5)
0 ifggwl<7r WM {1 mm 52—6927 = WW3~ {#7 mm): W? , 3—323” W") ” j—€*)*’»L%6*JZ“ \) 7! Question 5: [15%]
Prof. Wang wanted to transmit an AMSSB signal. To that end, he wrote the following MATLAB code. % Initialialization duration=8; f_sample=44100;
t=(((04)*f_sample+0.5):((duration4)*f_sampleO.5))/f_sample; % Read two different .wav files
[x1, f_sample, N]=wavread(’x1’);
x1=x1’;
[x2, f_sample, N]=wavread(’x2’);
x2=x2’; % Step 1: Make the signals bandlimited. w__M=2000*pi; .L alkﬁ/e
h=1/(pi*t).*(sin(W_M*t)); ZWUM) 6 Q é
x1_new=ece301conv(x1, h);
x2_new=ece301¢onv(x2, h); K Step 2: Multiply X_new with a cosine wave.
x1_h=x1_new.*cos(5000*pi*t); (ewfud
x2_h=x2_new.*cos(7000*pi*t); h1=1/(pi*t) .*(sin(5000*pi*t));
h2=1/(pi*t).*(sin(7000*pi*t)); @) 25M; g KW”? % Step 3: Keep one of the side bands
x1_sb=x1_hece301conv(x1_h, hl);
x2_sb=x2_hece301conv(x2_h, h2); % Step 4: create the transmitted signal
y=x1_sb+x2_sb;
wavwrite(y’, f_sample, N, ’y.wav’); 1. [2%] Is this system using the upper or the lower side band? 2. [6%] The frequency spectrums of X1 and x2 are described in the following ﬁgures. v.1 A43"! '6 ‘ :2 I, .‘ VL
ﬁx‘i atvrx >< i,. .
2' ‘e x (j I , a; {ﬂigde ) Plot the frequency spectrum of th and y. I<nomung that Prof VVang used.the above code to generate the “yANaV” ﬁk; a.student
tried to demodulate the output waveform “y.wav” by the following code. % Initialialization duration=8; f_sample=44100;
t=(((04)*f_sample+0.5):((duration4)*f_sample0.5))/f_sample; % Read the .wav files
[y’ f—Sample, N]=Wavread(’y’); y=y’:
% Create the low—pass filter. h_M=1/(pi*t).*(sin(2000*pi*t)); % demodulate signal 1
y1=4*y.*cos(5000*pi*t);
X1_hat=ece301conv(y1,h_M); wavplay<x1_hat,f_sample) % demodulate signal 2
y2=4*y.*cos(7000*pi*t);
x2_hat=ece301conv(y2,h_M); wavplay(x2_hat,f_sample) 3. [3%] Can the student demodulate x2 successfully without noise (also known as in—
terference)? Use one or two sentences to brieﬂy explain your answer. 4. [4%] Can the student demodulate X1 successfully without noise (also known as in
terference)? Use one or two sentences to brieﬂy explain your answer. e) “(r Fragmencic’; d'érve lWa 77/9514
(/K/‘(g)’ (J ( (010 V’“‘%/ < (' )
XL) My 1 ;[€ ofﬁl‘ﬂvl/ awfd’lf
'Z Question 6: [15%] 1. [1%] Given a signal m(t), write down the equation how to convert $(t) into its sample
values “[71] when the sampling period is 0.4 sec. 2. [2%] Suppose we know that xd[n] = 6[n — 2]. We use linear interpolation to recon
struct the original signal7 and denote the reconstructed output as 3305). Plot $105)
for the range —1 < t < 2. 3. [4%] If we use a band—limited interpolation to reconstruct the original signal, and
denote the reconstructed output as m2(t), write down the expression of 932(75) and
plot 372(15) for the range —1 < t < 2. 4. [8%] Consider the following digital signal processing system. ,. wwww w
a w [4g (3.14.175, 1‘: i4:: r 1 Suppose the h[n] has its DTFT being H<ejw): 1 if[w<g . (6)
0 ifgngl<7r When the input 30(75) : cos(7rt) + sin(0.57rt), what is the output y(t)? r. w}: WW 2. 76/6) 7cm) 1 : \/ D 3
(A6 >£
0.3/ 0‘7 0,8 (l an M): WSW) HA 2%) XJl—n‘} : (05(%m} 7L 5),, x
+ [jg/@7133“ (Cw /4//<77 Question 7: [15%] 1. [3%] = (—%)nZ/l Find the corresponding Ztransform X and plot the corresponding ROC, zeros, and poles. 2. [2%] X (z) = 22—2 + 10z3 and the corresponding R00 is the entire Z—plane except
for z = 0 and z 2 00. Find the corresponding 3. [5%] We know that X = (Tm and the Fourier transform of the corre—
3 sponding exists. Find 4. [5%] Suppose = 2"U[—n—1] and h[n] = (0.25)"U[—n—1]. Let y[n] = Find the Z—transform Y(z). 4 m): 2 MW!“ (i) i (“W
A: 'd
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2 21“): ZJZnilj j—(UJ/[nfzz
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X(2): M}; ww—«a 20c 5~</?/ ﬂ ’9“?
"é? k "72’! xx?) : ~31(¢§)ﬂ7/Lﬁrj;§(‘f/"@l['”"] ...
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This note was uploaded on 12/08/2010 for the course ECE 302 taught by Professor Gelfand during the Spring '08 term at Purdue UniversityWest Lafayette.
 Spring '08
 GELFAND

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