351_spring02_ex1_sol[1]

351_spring02_ex1_sol[1] - EE 351 ‘ v MIDTERM #3; »...

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Unformatted text preview: EE 351 ‘ v MIDTERM #3; » SPRING 2002 Name: ‘7%__g (/ NOTE: 1. Exam is comprised of 3 problems, each with multiple parts. 2. Point values are given in parantheses for each part. 3. You are allowed one 8 1/2 X 11 sheet of paper and a calculator. 4. You MUST show all work and write your answers in the spaces provided. 5. You have 2 hours to complete the exam. DO NOT WRITE BELOW THIS' LINE: Problem #1 (30 Points) : Problem #2 (36 Points) : __ Problem #3 (34 Points) : __ TOTAL (100 Points) : ______ Problem #1 (30 Points): Consider the simplified A/ D system below which performs sam- pling followed by quantization and encoding: Figure 1: Simplified A / D System a) If xc(t) = 98in(27r100t) + 23in(27r900t) and the sampling frequency is F, = 1000 Hz (T = 0.001 sec. / sample), determine $1 in its simplest form, i.e., resulting discrete—time signal components in x1[n] all should have unique frequencies. Comment on aliasing. (7 points) )(sm: 353%?1'3'1’“) +Zfih(2fl£“) [000’ [090’ M (iii fl-fivlfi=’7’5" => Io ' l0 - . * w]: MMZBL) +»25,~n( -25,‘n{ 273’") Afimfl ocrarr mm F5 7> 59/00 H; b) If the quantizer has the quantizer map given below, find and sketch the values of ‘x2[n] for [goo n = —1, 0, and 1. Note: If a data point falls on a quantization threshold, use the convention of selecting the larger level in magnitude. (6 points) c) The purpose of the encoder is to convert each quatization level to an N—bit unique binary code. Determine the minimum number of encoder bits N so that each quantization level would be expressed by a unique binary code. (2 points ) .ll lever 7% 2": H 3? (1) Find a different CT input signal, $62 (t), that would yield the same sample values that xc(t) had in part (a). Assume that the sampling frequency is unchanged, F, = 1000Hz. Your answer for xcz (t) must have different frequencies than xc(t) in part (a). (7 points) m; .21.: 3.1+27T: 21-12 :3; (ham-fl; 2277 .joop’zlflllW (0 )0 [O [9/ Tim“: mu W A u no‘l‘, Un‘fiue. e) Find a different sampling frequency F,3 so that if it were applied to the “original'xc(‘t)vin part (a) it would yield the same m1[n] in part (a). Is this answer unique? Why or why not? (8 points). w: .nrr : kg :2 MM 27? 19 wt: gfiqoo z—fiJrfikL E! F3 ’0 loo 2 iflgfi e;- m éfc £25k“? ANW 3 “All ' Problem #2 (36 Points): All parts below are independent of each other. x1[n]=[——1 2102.] a) A DT signal (1:1[n] is given as, T i) Find x2[n] = :31 [If + 2] (6 Points). VX;Cn+’Lj—': [:4 ‘L If 2.3 .2") X2 (“+732 [2 E; I 1 "‘0 ii) If x1[n]=X3["2n-2], find xaln] (613mm) Mow U00! ixavcio "CV50:— {SW 9(3an L3 2 +0 2 {MS Hm eke/fawn +2 yaw;ij Tkm Scale, by ’1 i - 2. 17. ~13 :33 PM "We Wai chmlw—aXuC-Ifl—C 0 T 4Skrmidzj 31/61?)an iv; 2 X’C:_(D:]ZI:ZOOO H9101] “Tm swig +0 L L; 1 a 3@3=x,[gzwmge [momva you Can a1.” VJ: 0km e, a? (arm.ch 3° b) Find evenpart of xlln] = [ ’1 i 1 0 2 l (5 Points). 0) Suppose that a DT signal is given as: y[n] = cos(0.17rn) + sin(0.1251rn) Determine if y[n] is periodic. If so, determine its fundamental period. (5 Points) W‘20‘l7/ZFL; 2.1—.22) lezo E? Z!” 20 N! MIC/é 002 3,0an _ 4.2}; 3/ 72:”;— 17‘ fl 'N‘L N"COA(Za/[é)-53N d) Assume that a CT signal xc (t) = 2sin(27r500t) + 3cos(27r200t) — 4cos(27r800t) is sampled with F,] = 1200 H 2. If reconstructed signal, a3,(t) , is given by; x,(t) = ~4cos(27r1600t) + 2sin(21rl700t) — 2sin(27r1900t) —- 4cos(2n'2000t) determine specifications of the reconstruction filter used to obtain ac, (t) . You can use any method you like to solve this problem. (7 Points) Problem #3 (34 Points): a) Consider the ideal sampling/ reconstruction scheme below: Ideal Lowpass Reconstruction Filter with cutoff frequency Fs/2 and Gain=T Ideal Sampler Xc (t) Xr (t) Sampling Freq.=Fs If the sampling frequency is F3 2 40000 H z (i.e, 40 kHz) and the input signal is given by xc(t) = 4cos(2%8000t) + 2cos(21r22000t), sketch X40), X,(Q) and X,(Q) in the given rangeon the graphs below. From the graph of X,(Q), determine the corresponding time— domain signal m,(t). Make sure to label all key amplitudes. (12 points) XC(Q ) F(kHz) 2 4 6 8101214161820222426 -26 ~22 -18 —14 —10 -6 -2 2 4 6 8 10 12 14 16 18 20 22 24 26 XI.(Q) 4:6 F(kHz) 2 4 6 8101214161820222426 b) Now consider the non—ideal sampling / reconstruction scheme below: Anti-aliasin Reconstruction Filter Gain=T 3 XP (t) Ideal Sampler XI (0 Sampling Freq. = Fs Assume that the sampling itself is ideal, but the anti—aliasing filter (Haa(9)) and reconstruction filter (HT(Q)) are fixed frequency non—ideal filters with frequency spectra shown below. Haa(Q) 141(9) _24 _20 ‘16 16 20 24 _24 40 46 16 20 24 If the sampling frequency is F,3 = 40000 Hz and the input signal is given by xc(t) = 4cos(2«8000t) + 2cos(27r220.00t), sketch Xp(Q) (NOT Xcm) ), X362) and X,(Q) in the given range on the graphs below. From the graph of X,(Q), determine the corresponding time-domain signal cc,(t). Make sure to label all key amplitudes. (14 Points) c) Suppose a signal bandlimited to 10 kHz but corrupted by high—frequency noise as shown below is put into this non—ideal sampling/ reconstruction system given in part (b): Signal + Noise Spectrum -F2 -F1 -10 10 F1 F2 Determine restrictions on noise frequencies F1 and F2 as well as the sampling frequency F, (F, is NO longer 40 kHz) so that the output of the non—ideal sampling/ reconstruction system is equal to the signal without the noise. Show all work and explain your reasoning. (8 Points) ? ANSWER: Restriction(s) on F1 Restriction(s) on F2: 0 “ f Restriction(s) on F,:_. ; 9; '1: ...
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351_spring02_ex1_sol[1] - EE 351 ‘ v MIDTERM #3; »...

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