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Unformatted text preview: Question 1: [12.5%7 VVork—out question] Consider a discrete time signal 3
0 otherwise xlnl { Suppose : x[—n — 1]. n if0£n§2
if3§n£100. 1. [7.5%] Find the discrete time Fourier transform of 2. [5%] Find the discrete time Fourier transform of If you do not know the answer to the ﬁrst sub—question, you can assume get full credit if your answer is correct. (n + 1)2“"Z/{ and you will still Question 2: [12.5%, Workout question] Consider a continuous time signal $(t) = sin(7rt).
We pass it through an impulse train sampler. Let :rp(t) denote the result of impulse train
sampling. 1. [2%] Suppose the sampling frequency is 2 HZ. Draw $p(t) for the range —1.25 < t <
2.75. 2. [3%] Continue from the previous question, Draw Xp(jw) for the range —47r < w <
47v. (The unit of w is radian/sec, not HZ.) 3. [3%] How do we reconstruct the original signal from 33,,(25) using the optimal recon—
struction (also known as the ideal bandlimited interpolation). You need to specify
the cutoff frequencies of your ﬁlter(s) and any necessary scaling terms. 4. [2.5%] Suppose the sampling frequency is now reduced to 1 HZ. Draw Xp(jw) for
the range —47r < w < 477'. (The unit of w is radian/sec, not Hz.) 5. [2%] Continue from the previous question. What is the new xp(t)? And what is the
reconstructed signal 53(75)? Question 3: [12.5%, Work—out question] Consider a sampling plus discrete time signal
processing system as follows.
For any input signal :z;(t), it is sampled with sampling frequency 100 HZ, to generate
a discrete—time signal The discrete time signal is then processed through a discrete
time linear—time invariant system with impulse response hd = 0.5%! The output is
denoted by Then is passed through an ideal reconstruction to generate y(t).
Answer the following question: 1. [1.5%] What is the DTFT Hd(ej“’) of hd[n]? 2. [5%] What is the end—to—end frequency response H (jw) = 3. [3.5%] When the continuous time input is 33(t) = 63807”, ﬁnd the overall output y(t). 4. [2.5%] When the continuous time input is a:(t) = ej2007‘t, ﬁnd the overall output y(t). l Hit?) r a I p ‘ N Wjﬁ‘xczg/TCW .,.._.., O W Q Question 4: [12.5%, Work~out question] Consider a discretetime signal with the
expression of its Z—transform being X = 1
4+z2 left—sided. Answer the following questions. 1. 2.5%] Find out all the poles of X and draw them in a z—plane. 2.
3.
4. 2.5%] What is the ROC of the Z—transforrn X 2%] Does the discrete—time Fourier transform of exist? 2%] Is :c[5] equal to zero or not? 2%] Find out the value of You may have to use the time—reversal property
before applying the property table. 1.5%] Find out the expression of by Taylor’s expansion. 433+ “H W5) 3..., + 2. Suppose we also know that is ﬂ‘mzmwwmbxzm“mshmwam mamWWW Question 5: [12.5%, Work—out question] Consider two Signals: $05) = Ll(t + 2) — Ll(t — 2) and
100 if 0 <t
W) = Ft . (2)
50(1+ COS(m)) 1ft 3 0 Find y(t) = 1:05) >x< h(t). . m {I I g: g {2%) S) ZS": Mm) «(Yam 59 §<ﬁ QUE? (f H (Egg) 7‘79 (Em 1 4'2: “f
m i
gﬁﬂ‘: S 30 ( (.4; C34 “(t“9) [Cm w
,, L "TC ﬁlms) ‘ L Question 6: [20%, Multiple—choice question] Consider two signals h1(t) = cos(7rt) ' sin(t)
and MM] 2 e_”(1+j)Z/{[(n — 2)2] 1.
2. 5.
6. Suppose the above two signals are also the impulse responses of two LTl systems:
System 1 and System 2, respectively. 1.
2. 1% 1%] Is hg periodic? 1% 1%] Is System 1 causal?
1%] Is System 2 causal?
1%] Is System 1 stable?
1%] Is System 2 stable?
1.5%] Is System 1 invertible? 1%] ls System 2 invertible? it. Is [1105) periodic? ls h1(t) even or odd or neither?
ls hg even or odd or neither? ls h1(t) of ﬁnite energy? . Is hg of ﬁnite energy? (, ]7 WWW rig“
(Wei it) Question ’7: [12.5% Work—out question] 1. [6%] x(t) = cos(27rt) + sin(3/47rt) + ejmi. Find its Fourier series representation. 2. [6.5%] Consider a periodic signal y[n] with period 1000. We also know that 5 if 1000 S n < 1250
0 if 1250 S n < 1500 1 if 500 g n <1000
W] = Find the Fourier series representation of T9 1»: 2:1 u"WWMWWMWWyWWNWWwww.cwnww Question 8: [12.5% Work—out question] Let 2(t) : 6—271 Suppose we know the in— put / output relationship of a system satisﬁes y(t) = dig—gt) + 2(t) * Find the impulse
response of the above system. {7 ...
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 Spring '06
 V."Ragu"Balakrishnan

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