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Unformatted text preview: Final Exam of ECE301, Prof. Wang’s section
Monday 3:20~5:20pm, December 12, 2011, PHYS 114. . 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 two hours 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. . Neither calculators nor help sheets are allowed. Name:
Student ID:
E—mail: Signature: Question 1: [15%, Work—out question]
Consider the following AMSSM system: ' “~23; is w Answer the following questions with carefully marked horizontal and vertical axes: 1. 2
3 [3%] Plot W(jw), the Fourier transform of w(t) for the range of —250 < w < 250.
. [3%] Plot V(jw), the Fourier transform of 11(t) for the range of —250 < w < 250.
. [3%] Plot Y(jw), the Fourier transform of y(t) for the range of —250 < w < 250.
. [3%] Plot Z(jw), the Fourier transform of 2(t) for the range of —250 < w < 250. . [3%] What is the range of the cutoff frequency Wcutoff of the lowpass ﬁlter for which
the demodulated signal 5?:(t) is identical to that of Mt)? Question 2: [14%, Work—out question] 1. [1%] Let x(t) = cos(7rt). What is the corresponding Nyquist frequency? Please use
the unit Hertz for your answer. 2. [4%] If we sample 30(15) with sampling period T = 1 seconds, we can convert the 2
continuous—time signal :r(t) into a discretetime signal Plot for the range of ——2 g n g 2. Plot only. There is no need to write down the actual expression. . [4%] If we use the impulsetrain sampling with sampling period T = % seconds, we
can convert m(t) into another signal 1121,05). Plot the Fourier transform Xp(jw) of
$p(t) for the range of ~5.57r 3 w 3 5.57r. Plot only. There is no need to write down the actual expression. . [5%] The discrete—time signal can be stored in a computer as an array. Suppose
the hard drive is corrupted and the signal values outside the range g 1 are erased
and set to zero. That is, the new signal becomes if 31 0 otherwise I (1) W] = Suppose we pass y[n] through the ideal reconstruction, also known as the band—
limited interpolation. Plot the reconstructed signal y(t) for the range of ~2 g t g 2.
Plot only. There is no need to write down the actual expression. Hint: If you do not know in the previous question, you can assume that =
L1[n — 1]. You will get 4 points if your answer is correct. ~ 1:.“ (“out ' W 7
is m v (Hp ﬁN}Z}Al§ 3:522, ml i 5;) {its { kg; (at) 2:»: 0m]: (QS‘Z‘UWQ
’ {org}: m m 4”: Cts< T is? r] at 3;... it] (m (m Q5 R: > a u M113] cm!» (t5 "(E ) r)” Mm E g” I V1 Question 3: [14%, Work—out question] Consider the following discrete—time processing
system for continuous—time signals. 'TgOiOl 5379C I  V , V. V. ,_ _‘ 5pm . iﬁmrogs airm E1 +0‘5 ’Xlitﬂ ‘
4” (3)5 @3174] V Gait£23130 Eon
., ( 0‘31??an res; rimarrive Hc 5: M do w @mﬂ rigs OF‘VS‘Q
1. [6%] Find the discretetime frequency res onse Hd(ejw) and plot Hd(ej“’) for he
range of —27r S w s 27r. 2. [8%] Find the continuous—time frequency response Hc(jw) plot Hc(jw) for the range
of —2007T < w < 20071”. Hint: If you do not know the answer to this question, you can answer the following
questions instead: If the input is 33(25) 2 5, what is the output? (ii) What type
of ﬁlter is the overall system? A low—pass ﬁlter? A high—pass ﬁlter? Or a band—pass
ﬁlter? Use a single sentence to justify your answer. If both your answers are correct,
you will still get 4 points. ‘. Y ,{TLU A, V63 )m 0&6? M6?” 2w: 3“)
avenge“) AS“ Q4;ng (7%) X j , ‘ ﬁg Ha >3 w Q “wt” (3,35 Question 4: [20%, Work—out question] Consider a discrete—time signal with the ex— pression of its Z—transforrn being X = “5:62 1. Suppose we also know that the discrete—time Fourier transform of exists. Answer the following questions. 1. [4%] Find out all the poles of X and draw them in a Z—plane.
2. [4%] What is the ROC of the Z—transform X 3. [4%] What is the value of Zfzwo Hint: You may want to ﬁnd the discretetime
Fourier transform X (63“) ﬁrst. 4. [8%] Find the expression of /. a ’
N3) 3(1+23")(i+33“’) ’2' ROC Camus +/\e unit Girdle ) Win/l“; Tfme— shift 1% he ﬁght {63)‘(‘1>‘('1§M‘n“’3
+3 (~l) (~3>AW:~A~I3> Question 5: [10%, Work—out question] Consider the following differential equation: gyﬁ) + 2513105) + W) = £2305) + 30(15) (2) Find out the output y(t) when the input is $(t) : cos(t) + Sin(\/§t). Hint: If you do it
right, there is no need to use partial fraction. We)? 32» ~+ i) (M tit/w W (m
x \ AB U3 ow we) wt “W i Question 6: [12%, Work—out question] Compute the continuoustime Fourier transform
H (jw) of the following signal. You can use a plot to represent your H (jw). No need to
write~down the actual expression: sin(%) sin(%) (3) and hg = max(sin((’n2 + 0.5)7r), 0)
1. [1%] Is h1(t) periodic? 2. [1%] Is h2[n] periodic? [7233
Was 3. [1%] Is h1(t) even or odd or neither? 5 1%] Is h1(t) of ﬁnite energy? 6. [1%] Is hg of ﬁnite energy? 4. [1%] Is hg even or odd or neither?
 1 [M3
N0 Question 7: [15%, Multiple—choice question] Consider two signals h1(t) = fft :—2t cos(7rs)ds Suppose the above two signals are also the impulse responses of two LTI systems: System 1 and System 2, respectively. 1. [1.25%] Is System 1 causal?
2. [1.25%] Is System 2 causal?
3. [1.25%] Is System 1 stable?
4. [1.25%] Is System 2 stable?
5. [1.25%] Is System 1 invertible? 6. [1.25%] Is System 2 invertible? [\[o
[\lo M.)
Ma
No
fVo ...
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This note was uploaded on 02/12/2012 for the course ECE 301 taught by Professor V."ragu"balakrishnan during the Spring '06 term at Purdue.
 Spring '06
 V."Ragu"Balakrishnan

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