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quiz2 - Page 1 of 17 6.02 Fall 2012 Quiz 2 Name DEPARTMENT...

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6.02 Fall 2012, Quiz 2 Page 1 of 17 Name: DEPARTMENT OF EECS MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.02: Digital Communication Systems, Fall 2012 Quiz II November 13, 2012 × ” your section Section Time Recitation Instructor TA D 1 10-11 Victor Zue Ruben Madrigal D 2 11-12 Victor Zue Cassandra Xia D 3 12-1 Jacob White Kyu Seob Kim D 4 1-2 Yury Polyanskiy Shao-Lun Huang D 5 2-3 Yury Polyanskiy Rui Hu D 6 3-4 Jacob White Eduardo Sverdlin Lisker Please read and follow these instructions: 0. Please write your name in the space above and × your section. 1. Two two-sided “crib sheets” and a calculator are allowed. No other aids. 2. There are 27 questions (mostly short!) in VI sections, and 17 pages in this quiz booklet. 3. Your total allotted time is 120 minutes . 4. Please write legibly. Explain your answers, not just when we explicitly ask you to! If you find a question ambiguous, write down your assumptions. Show your work for partial credit. 5. Use the empty sides of this booklet if you need scratch space. If you use the blank sides for answers, make sure to say so! PLEASE NOTE: SOME STUDENTS WILL TAKE A MAKE-UP QUIZ LATER THAN YOU. PLEASE DON’T DISCUSS THIS QUIZ WITH ANYONE IN THE CLASS, UNLESS YOU’RE SURE THEY HAVE TAKEN IT WITH YOU TODAY. Do not write in the boxes below 1-6(*/19) 7-10 (*/19) 11-17 (*/25) 18-21 (*/13) 22-24 (*/12) 25-27 (*/12) Total (*/100)
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I 6.02 Fall 2012, Quiz 2 Page 2 of 17 Bring in The Noise A particular digital communication scheme sends signals over a channel that behaves essentially as a linear, time-invariant (LTI) system at baseband. The binary source at the transmitter generates the symbols “0” and “1” with equal probability. The characteristics of the channel and the timing at the receiver are such that the receiver is able to obtain M 1 good samples in each bit slot. The good samples in a bit slot corresponding to a “0” are all of the form y [ n ] = V 0 + w [ n ] while in a bit slot corresponding to a “1” they are all of the form y [ n ] = V 1 + w [ n ] . Here w [ n ] denotes a noise term that is a Gaussian random variable of mean 0 and variance σ 2 , and is independent across samples, i.e., the samples at the receiver are perturbed by additive white Gaussian noise. For on-off signaling, V 0 = 0 and V 1 = V . The linearity of the channel then ensures that for bipolar signaling V 0 = V and V 1 = V . The receiver decides on whether a “0” or “1” was sent by comparing the average value of the M samples in any bit slot with a threshold voltage V th . If the average is below V th , the receiver decides a “0” was sent; if the average is above V th , the receiver decides a “1” was sent. Suppose we are using bipolar signaling and M = 1 , i.e., only a single sample is taken in each bit slot. Then you’ve seen that the probability of the receiver making an error in deciding whether a “0” or “1” was sent in any particular bit slot, i.e., the bit error rate (BER), is minimized by choosing V th = 0 , with the corresponding BER being given by V BER bipol = 0 . 5 erfc .
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