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ECE 110 Exam Three FA2008 Solutions - ECE 110 Professors...

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Unformatted text preview: ECE 110 Professors Brunet and Loui November 17, 2008 HOUR EXAMINATION #3 1) Write your: ., . Last Name (use capital letters): 90!.» i [Q m?) ( VQSS First Name (use capital letters): Signature: MESS») 2) Write your name and section at the back of the test. DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD Make sure to write your name AGAIN at the top of every page of your exam. A. Write or print clearly. Answer each problem on the exam itself. If you need extra paper, there is an extra sheet at the end of this exam. Clearly identify the problem number on any additional pages. The decimal/ binary/ hexadecimal table, the Flip-flop characteristic tables, the ASCII Code, the Morse Code alphabet, the USPS Code, and numbers and properties for log base 2 are also attached to the exam. B. In order to receive partial or full credit, you must show all your work, e.g., your solution process, the equati0n(s) that you use, the values of the variables used in the equati0n(s), etc. You must also include the unit of measurement in each answer. Students caught cheating on this exam will earn a grade of F for the entire course. Other penalties may include suspension and/or dismissal from the university. Problem 1 (20p0ints) b) [6 p15. ] Using your results in a) and assuming LOAD = 1, fill out the table below for consecutive clock pulses. Initially Q0(O) = 0 and Q1(0) = 0. clock pulse 0 1 l 2 l 3 Q0 0 l O 0 Q1 0 O l l l l X 1 l l O l 0 Problem 1 (20p0ints) b) [6 ptS. ] Using your results in a) and assuming LOAD = l, fill out the table below for consecutive clock pulses. Initially Qo(0) = O and Q1(0) z 0. clock pulse 0 I l I 7 Q0 0 Q1 0 Q1.) Q0 9 X H.(nui Problem 2 (15 points) You will design and analyze a mod-6 counter whose state sequence is (q2 q1 qo) = 101, 100, 011, 010, 001, 000, 101, 100, a) [8 pts.] Using a 3-bit down—counter with LOAD function, introduce one or more gates and constant inputs (0 or 1) to produce the specified counting sequence. X0 3—bit ‘10 L c 3 _, 000 C X] down 611 (fiifii 3t) “' 3 ‘ counter 612 L :— ‘ M01 “$0. I LOAD b) [7pts.] Suppose the clock period is 3 ms. What are the periods of q0, q1, and (12? Period ofqo = E Period qu1= (? mg Period qu2 Z (9 mg 67‘ ‘5H‘32i o {ngmrnl’mfiafifirb MU’SJ‘D gmgt Problem 3 (15poz'ms) Define g(t) = cos(6 1t z) and h(t) = cos(14 7t t), where t is in seconds. The frequency of g(t) is 3 Hz, and the frequency of h(t) is 7 HZ. a) [4 pm] Find a sampling frequency f5] such that f3] > 7 Hz for which h(t) aliases to g(t). awe?) qjgfié)af3:3tt5, <76 (if-é) «313:745 szig,~¥kl MM 4;:(0H13/ b) [5 pm] Find a cosine whose frequency is higher than 7 Hz that also aliases to g(t) when sampled at the frequency f51_ <10§€QG ) c) [6 ptS. ] Find tw_o sampling frequencies fig such that 3 Hz <fi2 < 7 Hz for which h(t) aliases to g(t). Justify your answers briefly. ‘ ta; 4 1% me %= 14—71 New aromm ; s a afiw Mm = gfimwaiimm, F52 it?“ Em if lS—Zl? 2 (+6ng hfi dflfW «a {751:2 «1%. mo 3 M t came (ac/am. M.~C. 95mm? Problem 2 (15 points) You will design and analyze a mod-6 counter whose state sequence is (qz q} qo) = 101, 100, 011, 010, 001, 000, 101, 100, a) [8 ptS.] Using a 3-bit down—counter with LOAD function, introduce one or more gates and constant inputs (0 or 1) to produce the specified counting sequence. mm aw: 051;") dyDWOl. go *‘o ?’ +95- 3 —bit down q] counter qz q 2. q‘ qg clock LOAD l o ‘ \ 0 f’ a l 9 \ ° 0 a " o o ’ I b) [7pts.] Suppose the clock period is 3 ms. What are the periods of qo, q], and q2? Rx3ms 6x5!“ Period of go = 6M3 Period ofql = ‘8 ms Period ofqz = a\ X 0-3 § is Problem 3 (15 points) Define g(t) = cos(6 7t 1) and Mr) : cos(14 7: t), where t is in seconds. The frequency of g(t) is 3 Hz, and the frequency of h(t) is 7 Hz. a) [4pts.] Find a sampling frequency 121 such that fsl > 7 Hz for which h(t) aliases to g(t). '5 ‘r \o O W $3.: 1011?: (\O..°r:?>) WV 3 b) [5 pm] Find a cosine whose frequency is higher than 7 Hz that also aliases to g(t) when sampled at the frequency fl]. $5: \O+3 ANT; :3 Cosfciblflrl c) [6 pts. ] Find 1;va sampling frequencies fig such that 3 Hz <f32 < 7 Hz for which h(t) aliases to g0). OJustify your :nswers biefly. a, R}: 5 H527 €1th 56:1an 2 2— 3. 9‘ a $5 ‘ \ o 3 of 3 a:4—Ht,c1 3 WM W) WT»: WW W CUB/62:93) “I > H. laud Problem 4 (20p0int5) A Qrefix—free code was designed for four symbols A, B, C and R. The codeword for C has been lost, but you know that the average length of the code is between 1.5 and 2.5 bits. Symbol A B C R Codeword 10 O ? 1101 Relative frequency 3/16 1/4 1/2 1/16 a) [5 pts] Give the encoding for the name BARB 8 A “R ‘B i OlOHmO b) [8ptS.] Let x be the length of the codeword for C. Using the average length of the code, determine the possible values for x. Show your work. weenie-en. 0 combo A B C. ’D ._ c 4 89;. :L. _. I4._.+._-?...x " 754L73+ Ia *Ie ” (do Since [.55 ”(l-Ex 522-4? Shem 7565 6‘" Edi?) 24$ 144-ng4'0 Xfi—SZ er 3 c) [7pts.] Determine the codeword for C. EX lain our reasonin in words. WC. EMU/Wk Problem 4 (20 points) A prefix—free code was designed for four symbols A, B, C and R. The codeword for C has been lost, but you know that the average length of the code is between 1.5 and 2.5 bits. Symbol A B C R Codeword 10 0 ? 1101 Relative frequency 3/16 1/4 1/2 1/16 :: ‘Hlé a 8/(6 a) [5 pls.] Give the encoding for the name BARB LAN 1 OlOHOlO b) [8pm.] Let x be the length of the codeword for C. Using the average length of the code, determine the possible values for x. Show your work. Q \84—14.\+37L wszxa-gx‘zx ><_$_‘z__ 154 ‘ +81 49.: =9 94 g mung 40 16 ’ >> 1048x4326 =5 I.a54x$8.95 =99LMR&9V3 Eli‘s c) [7 ptS.] Determine the codeword for C. Exglain your reasoning in words. I x? at: £1 3Lomml' Slam} with 0 (math: f5; 3) sinugflxdzt‘x free . WY— s‘voxls wit-k \ -. tunnel" let. to (mdin 1 49? A) ER / :00an be \I (grew 0? Cockng ‘ =9 1 mt let 3 m as} out \\ (mm w“) HO 1 tunnel Small is Q‘ mawmgefis WWW \\\ pa? \\0\ (udingfi‘kg) Wlpui Problem 5 (15 points) A facsimile (fax) machine digitizes an image using pixels that are 1/8 mm wide and 1/4 mm high. Each pixel color is quantized with one bit, white (1) or black (0). a) [6 pts. ] Determine the time, in seconds, required to fax a document that is 8.5 inches wide and 11 inches high, without data compression, over a telephone line that allows digital data to be transmitted at 30,000 bits per second. Use the conversion factor 1 inch = 25.4 mm. (Hint: The width is 1728 pixels.) flag“; “in XZSMM/hzlllgftxzflé (mmfldtfiflb “5‘ wm/fixéfl - 4 NEE W6c%frftx€9- T (72%,:(“8 66h 1M0; : M :éLI—Ll- 49.62; 30, 000 Win/Md $4.4m 1 b) [5 pm] Suppose a run-length code is used in which the length of a run from one to seven is represented in binary with 3 bits, preceded by the color bit. For instance, a run of five black pixels is encoded Q 1 0 1. A run of twelve black pixels is encoded as consecutive runs of lengths seven and five: 0 1 l 1 Q l 0 1. Determine the compression ratio for an 8.5 inch row of pixels that all have the same color. drew eff (77$ P‘xaQ. Rm 240 ram «#5 7Fx22q-A) r9114 6% We; (M 06 6. (one :24“? rug 3: LL mil/TM Compression ratio = l 7 3 c) [4 pts] Propose a modification to the run-length code to increase the compression ratio for long runs of the same color. (Hint: What could code words 0 0 0 0 and l 0 0 0 mean?) n—Cwmk. Problem 5 (15 points) A facsimile (fax) machine digitizes an image using pixels that are 1/8 mm wide and 1/4 mm high. Each pixel color is quantized with one bit, white (1) or black (0). a) [6 ptS. ] Determine the time, in seconds, required to fax a document that is 8.5 inches wide and ll inches high, without data compression, over a telephone line that allows digital data to be transmitted at 30,000 bits per second. Use the conversion factor 1 inch = 25.4 mm. (Hint: The width is 1728 pixels.) \ 8.5 inches x \\ ‘mdwo 3 8.5 xQ§.4.x89txels x \\x&§.4x<}@\xels = was x m8 bll’s ( 1 m, 49ml) »W&m&§ 30,000 Time=l 64-- 4- $56st b) [5 pts] Suppose a run—length code is used in which the length of a run from one to seven is represented in binary with 3 bits, preceded by the color bit. For instance, a run of five black pixels is encoded Q l 0 l. A run of twelve black pixels is encoded as consecutive runs of lengths seven and five: Q l l 1 Q l 0 1, Determine the compressio ratio for an 8.5 inch r0 1 , ofpixels that all have the same color. original d6 0. ‘5 $3138 bl blgr 1 L738 lxels : 346x7+6 ‘ (Odingexfmele) Olll 0H\ Ot\\ 0\0\ =7 a47X4~bleS W (DNQRSM8V\= @ 2 fl Compression ratio = l l. :FS } c) [4 pts] Propose a modification to the run—length code to increase the compression ratio for long runs of the same color. (Hint: What could code words 0 0 0 0 and _l_ 0 0 0 mean?) 1‘? mm 000 T3 coda mmkfg )(tmh'gbr O {Shape/f” madNW)WE}M\l wmprem Wmove.‘ mo, ‘, \zafi Q‘de 7‘— Albxg 93x32, 333W =7 Q\bx4bz‘r$ 2(6 timeo wisp : LEEV— E93 — ‘hshwd a? \33 (we we»; “' 86+ ’ ‘9‘ C‘ > ma Problem 6 (15 points) For easy error detection and correction, it was decided to send a message of two ASCII symbols three times. The message below was received. There is one error. 10001100110010100011001101010001100110010 WWWWWQ __......_.a| 4% a) [5 pts.] Indicate which bitji/s wrong (circle it). Shoviyour work. ‘7 6 ottootO ammo ottomo b) [6 pts. ] Give the 14—bit message that was intended to be sent (sequence of bits). (000 no oUOOLO c) [4 pts. ] Give the two symbols that were intended to be sent. F2 tt—CWMC Problem 6 (15 points) For easy error detection and correction, it was decided to send a message of two ASCII symbols three times. The message below was received. There is one error. "'"" 1000110/0110010/1000110/0111’010/1000110/0110010/ a) [5 pts.] Indicate which bit is wrong (circle it). Show your work. Hf IOOOHO OttOOlO «mmQ‘M‘SwW looouOoulmo SW15“ W, looOHOouOOt’O 5% b) [6pts.] Give the 14-bit message that was intended to be sent (seguence of bits). loco no 0 no C>\O W c) [4 [91‘s,] Give the two mbols that were intended tob sent. it Tam} . ...
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