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06FMT2SOL

# 06FMT2SOL - Question 1[Multiple Choices 20 Consider a LT I...

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Unformatted text preview: Question 1: [Multiple Choices, 20%] Consider a LT I system with impulse response h(t) = Ll(t) — u(t — 2). Consider an input signal = cos(27rt) + sin(7rt) and denote the corresponding output as y(t). For convenience, we let :zzc(t) = cos(27rt) and \$80) = sin(7rt) and X (to), Xc(w), Xs(w), H (w), and Y(w) are the corresponding the Fourier transforms ofa:(t), watt), x505), h(t), and y(t) respectively. 1. [Outcome 4, 4%] What is the value of XC(7r)? (a) 0, (b) 0.5, (c) 0.500. 2. [Outcome 4, 4%] What is the value of Xs(7r)? (a) 0, (b) ——0.5j, (c) —0.500j. 3. [Outcomes 2 and 4, 4%] What is the value of H(0)? (a) 0, (b) 1, (c) 2. 4. [Outcomes 2, 4, and 5, 4%] What is the value of Y(1 ? (a) 0, (b) 1, (c) 2. 5. [Outcomes 4, and 5, 4%] How would you name the system? (a) a. low—pass ﬁlter, (b) a high-pass ﬁlter. Question 2: [Short—Answer, 15%] Please provide a short, one-sentence explanation of the following terms / theorems. 1. [Outcomes 1, 4, and 5, 3%] Fourier series / transformations convert the original signal 27(23) (or to another representation with different “basis signals.” What type of Signals are the “basis signals” of the Fourier series / transformation repre— sentation? 2. [Outcome 4, 3%] Suppose X (to) is the Fourier transformation of What is the physical meaning of X (0)? What is the physical meaning of X (100)? 3. [Outcome 4, 3%] What is the physical meaning of the Parseval’s theorem / relation— ship? 4. [Outcomes 4, and 5, 3%] An important feature of converting signals to their Fourier representations is that the response y(t) = h(t) >I< :r(t) of a LTI system becomes Y(w) = H(w)X What is the physical meaning of the last equation? 5. [Outcomes 1, 4, and 5, 3%] A LTI system of impulse response h(t) is invertible if and only if its corresponding Fourier transform H (w) 7é 0 for all m. Why is it so? Hint: Y(w) = H(w)X(w). /- Sinuyolﬂfnl tum/63‘ (Campley ego/redial fig/14(5) 2. = DC ODmPone/Etw X0013) " WDMZ'mLe/fig/ [170 Campone/ft~ 3‘ fetal energy Conserve st 41 The output of a smorde Wax/e hp at F: §fAtLSonA[ wlﬁk Mpliztwfe C/tMge HCW)’ M Phase climate 4m») 5‘ Because We, can OOAS'THACt ﬁlé flu/else Pin/«26. resf arise as H10») 2 PIE-{:5 :5 HifW)*YLW)= er) Question 3: [Work-out Question 25%] Consider a discrete signal of period 5 such that within one period, the values of the signal is 0 ifn=0 l ifn=1 :c[n[= 2 ifn=2. 0 ifn=3 0 ifn=4 1. [Outcome 4, 7%] Find out the Fourier series representation of Note that this question is asking a complete representation so you not only have to specify the coefﬁcient ak but also have to specify the corresponding frequency. 2. [Outcome 4, 5%] Find out the value of 2:20 ak. 3. [Outcome 4, 6%] Find out the value of 2:20 [ak|2. Note: (1k may be a complex number and | ~ | is the corresponding absolute value of a complex number. 4. [Outcomes 4, and 5, 7%] Let y[n] = — :r[n — 1]. Find out the Fourier series representation of Again, one needs to specify both the coefﬁcients Bk and the corresponding frequency. 2K :9 ﬁle Corrapondfrér 'R/Q/ﬁuehC/é)” [ix «Qt—oil .— We, {l _ Question 4: [Work-out Question 20+5%] Consider a serial concatenation of two LTI systems as follows, h2(t) 3/“) where h1(t) = e“3tI/{(t) and h2(t) = 63—57105). 1. [Outcomes 2, 3, 4, and 5, 10%] Let h(t) = h1(t) * h2(t) denote the impulse response of this serially concatenated system. Find H 2. [Outcomes 4, and 5, 10%] Suppose x(t) = em, and we know that the output y(t) = Aej(2t+0). Find out the gain factor A and the phase shift 6. Hint: Let C = A610 and rewrite y(t) = Ceﬂt, Now consider input—output relationship in the Fourier domain and ﬁnd the value of C. 3. [Bonus: Outcomes 4, and 5, 5%] Find h(t). Hint: One can obtain the result from H (w) or one can directly compute the convolution. H‘ (to): hictye'wale Question 5: [Work—out Question 25%] Consider a discrete moving average system of Window size 3: h[n] = 1/3(6[n] + 6[n ~— 1] + 6[n —- 1. [Outcomes 4, and 5, 10%] Find out the Fourier transform H (w) of Note: we are considering the discrete Fourier transform. 2. [Outcome 4, 10%] = cos(3n + 7r/2). Find out the Fourier transform X(w) of 3. [Outcomes 2, 3, 4, and 5, 5%] What is the output y[n] and its Fourier transform Y(w)? E i 9’5” (A) -=' CA a H() “rm 3 __ 2 ,L Q'dwn A 11 , - 3 § He) :5: S<w~3mm ""HWZ? WSW) m=~w 3“ : V0: Q'BjCi +3—Cos(3)):l’ 5(W23‘BRM) myw 3’ 331— + 303563) laws J91“ + *6 (3 3 )xq \ 2 .YE. 43,} L +63%}:- +§COSE~3)) Oi (Q Kim) ._____ (ink 2.335(5)) ﬁn (gnaw [~i)/§g 5 3 Q ’@ @MPWIYV'6\”%€ Canvduch‘gﬂ ...
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06FMT2SOL - Question 1[Multiple Choices 20 Consider a LT I...

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