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

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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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