examI_s08 - EE 497B EXAM I 19 February 2008 Last Name...

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Unformatted text preview: EE 497B EXAM I 19 February 2008 Last Name (Print): SOIIJ'LIO/IS First Name (Print): ID number (Last 4 digits): DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO Problem Weight Score w 1. You have 2 hours to complete this exam. 2. This is a closed book and notes exam. You may use one 8.5” X 11” note sheet and problem sets and their solutions. 3. Calculators are allowed. 4. Solve each part of the problem in the space following the question. If you need more space, continue your solution on the reverse side labeling the page with the question number; for example, Problem 1.2 Continued. NO credit will be given to solutions that do not meet this requirement. 5. DO NOT REMOVE ANY PAGES FROM THIS EXAM. Loose papers will not be accepted and a grade of ZERO will be assigned. 6'. The quality of your analysis and evaluation is as important as your answers. Your reasoning must be precise and clear; your complete English sentences should convey what you are doing. To receive credit, you must show your work. Problem 1: (25 Points) 1. (15 points) A power amplifier is programmed to shut down if both events A and B (such as overheating or impedance mismatch) occur simultaneously. The probability that A will occur is 0.001, and the probability of B occurring is 0.002. It is also known that P[B|A] = 0.01. o (5 points) What is the probability of transmitter shut-down? ' emn‘t thaw/miter- .Sl'lo'LSJ-ou/I :: i. n06} - PPObaJOI by ‘lI-f‘otnSMItter Jbv’lfloaopvn :- P l: A n £22.91. 5° oycw'n pmmn ; (E91 Frehfl .use. PE'BIHJ = PDQ o (5 points) Are the events A and B independent? _5 frfi] FEE] 3 0.001- 0.002 Bexause f’El‘l’l 91:61 '75 {E14 08'], are, no'l: InLQCpen‘th. 0 (5 points) Determine P[A|B]. 2. (10 points) Write MATLAB code to generate a vector G of 160 exam scores such that all scores are between 40 and 90, and are equally likely to occur. . ranl(flgo,|]) anemia; a. VeQ—Lov 0-! 160 DumL’QY‘S) em» 0km rdngomdsi WW o m9 I- 5031‘. (‘an ([léoJ 1'33 generouleé on veclzar o“: ’66 Wméersj 64.0% close/l Fanflomél ba’buean O anQ. SO. G: 50* ran9.<[160,1']7 4— ‘lo a léo numbers) eaglvx alum/I r‘aonm Problem 2: (25 Points) Figure 1 shows a binary communication channel with two input symbols {(1, b} and two output symbols {0,1}. The input probabilities are PM = 0.6 and P[b] : 0.47 While two of the channel transition probabilities are p = 0.2 and q = 0.3. Figure 1: A binary communications channel. 1. (4 points) What are the transition probabilities a and fl? 2. (8 points) Let P[mi flrj] = P[mi, rj] denote the probability that the input symbol is mi and the output symbol is 73-. Determine the probabilities P[a,0],P[a,,1],P[b,0], and P[b, 1]. 9: ram] .- 2; Prmcflp = Prmrtglm FEM] U3)? 1934.. PQSUL-E: PE“) ‘7]: 9E5] [TOM-1: I’D-TIP = (0.6) (0.2) 1' 042 I :1 O. ‘43 Pro» fl = Pro-j PL”) m = f’Ea’Jec. = (0.6)(0-8) (ON) (0.73 "' 077'? “3150:! -— Pm Ptom = P314; = befl—J = VHF) Pflllfl= [71373; 7— (o.~t)(o.3) 3. (6 points) Suppose that the receiver maps the output symbol 0 to the input symbol a, that is, the receiver states that the symbol a was sent when a 0 appears on the output channel. For this decision rule, what is the probability of correct detection, P[C]? “ll-eons)” rJlG-i aw 1:23 sag La3 Serra; kw Us say Lb? m1, Pro] 2 Prom 91:03 + PEI 1H1 Pun = Pro, 0:: + PU) 5'] ‘3 0,12. +- 047. PECJ '—‘- 0. 2H 4. (6 points) Suppose that the receiver maps the output symbol 0 to the input symbol b, that is7 the receiver states that the symbol b was sent when a 0 appears on the output channel. For this decision rule, what is the probability of correct detection, P[C]? - 41-60510!) Talk. RM» 7.01 s? Ulswlz‘, Kcv U3 8? 1&3 59075 PEc] = 95°le PEb] + PEMleij = Pfoab'] + (91—1)“? 1'” owls? Z: 0.18 5. (1 point) In order to optimize the performance of the receiver, which of the two decision rules should be employed? ' The 9'QOISIUM 1'ng #[Qltflty Qaraes‘li UWL’Q.’ 5% ‘5 0"” OV‘liIMaQ £Q—CIJIW1 rulaa. . m gar/£54; UAUQ’ Is 0,76J anfl, ocaorj 05W} that ~<ccw~§L ercmon Rev {03 Sag, UP) Jeni; U! Problem 3: (25 Points) 1. (10 points) A given code has four messages, a, [3, 7, 6, with probabilities 1/2, 1/4, 1/8, and 1/8 respectively; 0 (5 points) What is the information (in bits) conveyed by each message? o (5 points) What is the entropy of the code? entrqg = average. mgr/7‘0” ’9'“ ‘I Z 11ch Fix] ded $.17) 8 = Icaaijooj + rag.) Ptaj+ Ilfl?E/J+ ICSX PBS] ‘ LDC?) (2309-) + (33({p3+ 3U?) _’_‘|_ 9 \l 2. (15 points) A personal computer monitor displays pictures made up of pixels at a resolution of 1024 columns by 768 rows. Each pixel can take on one of 256 colors. 0 (3 points) How many different pictures are possible? I loZd-768 #Pic‘bures : (#cdorsjfiflms = (2-563 736‘qu :Hrpidbunes :: 25C 0 (5 points) If each of the possible pictures is equally likely7 what is the information (in Hartleys) from a single picture? PES'Vle P‘C‘tvm] = 256 786 '13:. l = I($\n6l€- pit-fire) : Qoalo Ffismale, PULL/re] o (5 points) Suppose one attempts to describe such a picture orally using 1000 words out of a vocabulary of 10,000 words. Assuming equiprobable words, What is the information (in Hartleys) of the 1000 word description? I [DE—SWIde WOrcfl-J '3 (a 000 .090 ZSanp/My EQ— f‘e’;(ac9 M 90%: fronale. looo—woro. Alaswyftloflj '7 .0000 00 .- lo a- gate (0000 \l 6 o 0 lg \3 0 3 Q Q C ‘ 45 '1: (Mahala, Fcfiyrg)» I 6:.le [000 or mot/Po, {5 Worth Inf/oh {her ’15qu C" Problem 4: (25 Points) 1. (10 points) A box contains n = 3 incandescent light bulbs. The probability that a given light bulb works is p = 0.9, independent of the status of any other light bulb. o (5 points) Suppose that all 72 light bulbs are connected in series7 and let SC represent the event that the series wired bulbs light when connected to a power source. Find an expression for P[SC] in terms of n and p. What is the numeric value of P [SC]? D 6‘2“) n watts} PEEL/lb n mule] PESC] = bedlb Lwovbsj Z Wm'lcbj ._ . "Japaan 3401‘“ {5C3 = {bulb l ("orb n (51b 2 Wm... o (5 points) Suppose that all n light bulbs are connected in parallel, and let PC represent the event that at least one of the bulbs lights when the parallel wired bulbs are connected to a power source. Find an expression for P[PC] in terms of n and p. What is the numeric value of P [PC]? ETC} = fan an»; Beremewl h bulb E’L‘FZJ -‘- PL mu; l .9.e[eo‘i=wa. n Mb 15am. -. “gopheng event-5 : PX (Mb I thfiin] “1be L fierce uo, M ~~ :- :I-P "P 'p FE PC] > PEG—7 «D- To Mercado, 305%”; mlxu/qi.‘€7/ imam 585432015 vsmaL ParajleQ ragwgucf. ‘2. (15 points) A memory module consists of nine chips. Each chip contains 71 transistors and functions properly if all of its transistors work. A transistor works with probability p independent of any other transistor. 0 (5 points) Let C denote the event that a given chip works. Specify P[C] in terms of p and n. BQ’CMQ, "the Chip Nah-9 onlfi NC on“ 0 tram/579:“: whack) o (5 points) Let k denote the number of working chips in a memory module. If an observation sequence consists of testng the nine chips within the memory module, how many sequences are there in which k chips work? o (5 points) Let Ck denote the event that k of the chips within the memory module work. What is P[Ck]? (59.64:)le tr‘qnsw'tar 'Pou’uy'eé ¢hQ, erflQngeo5% €990}: 0911f) Polling; are, OJ‘SO IreranOQn'é- ill: i K ©b‘p5 With}! Merrie? filong WorlLSS] n K n—lc 3 «DEC/p] : > (9th C1- rm) a. Sm (2. 5c 00(6 i "' W‘LWZ “7% Emma ...
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This note was uploaded on 07/23/2008 for the course EE 497B taught by Professor Schiano during the Spring '08 term at Pennsylvania State University, University Park.

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examI_s08 - EE 497B EXAM I 19 February 2008 Last Name...

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