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10FFinalSOL - Question 1[12.5%7 VVork—out question...

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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%, Work-out 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 discrete-time 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 mam-WWW 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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