4. A communication system transmits a bit of information by transmitting a signal that is either...

 

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4. A communication system transmits a bit of information by transmitting a signal that is either +1 or -1 with equal probability. Let b be the random variable representing the data bit. The receiver has 3 antennas. The received signal on the three antennas are Y - b+ N Y2 = 26+ N Y3 = 3b+ N3 where N1, N2, and N3 are independent Gaussian random variables with mean 0 and variance 1 and independent of b. The receiver forms the decision statistic Y = Y+2Y2 + 3Y3. (a) The receiver decides the data bit is b = +1 if Y> 0. The receiver decides the data bit is b = -1 if Y < 0. Determine the probability of error of the receiver. Express your answer in terms of the Q function. (b) Suppose the noise (N1, N2, N3) is not independent but has covariance matrix CN= 1 0.5 0 0.5 1 0.5 0 0.5 1 Determine the probability of error. Express your answer in terms of the Q function. 4. A communication system transmits a bit of information by transmitting a signal that is either +1 or -1 with equal probability. Let b be the random variable representing the data bit. The receiver has 3 antennas. The received signal on the three antennas are Y - b+ N Y2 = 26+ N Y3 = 3b+ N3 where N1, N2, and N3 are independent Gaussian random variables with mean 0 and variance 1 and independent of b. The receiver forms the decision statistic Y = Y+2Y2 + 3Y3. (a) The receiver decides the data bit is b = +1 if Y> 0. The receiver decides the data bit is b = -1 if Y < 0. Determine the probability of error of the receiver. Express your answer in terms of the Q function. (b) Suppose the noise (N1, N2, N3) is not independent but has covariance matrix CN= 1 0.5 0 0.5 1 0.5 0 0.5 1 Determine the probability of error. Express your answer in terms of the Q function.