Problem 4.67 A binary transmission system transmits a signal X (1 to send a 0 bit;...

  

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Problem 4.67 A binary transmission system transmits a signal X (1 to send a 0 bit; +1 to send a "1" bit). The received signal is Y = XN where noise N has a zero-mean Gaussian distribution with variance . Assume that "0" bits are three times as likely as "1" bits. (a) Find the conditional pdf of Y given the input value: fy(y|X fx (y|X = 1). == +1) and (b) The receiver decides a "0" was transmitted if the observed value of satisfies fy(y|X = 1)P[X = 1] > fy(y|X = +1)P[X = +1] and it decides a "1" was transmitted otherwise. Use the results from part (a) to show that this decision rule is equivalent to: If Y T decide 1. (c) What is the probability that the receiver makes an error given that a +1 was transmitted? a -1 was transmitted? Assume = 1/16. (d) What is the overall probability of error? Problem 4.67 A binary transmission system transmits a signal X (1 to send a 0 bit; +1 to send a "1" bit). The received signal is Y = XN where noise N has a zero-mean Gaussian distribution with variance . Assume that "0" bits are three times as likely as "1" bits. (a) Find the conditional pdf of Y given the input value: fy(y|X fx (y|X = 1). == +1) and (b) The receiver decides a "0" was transmitted if the observed value of satisfies fy(y|X = 1)P[X = 1] > fy(y|X = +1)P[X = +1] and it decides a "1" was transmitted otherwise. Use the results from part (a) to show that this decision rule is equivalent to: If Y T decide 1. (c) What is the probability that the receiver makes an error given that a +1 was transmitted? a -1 was transmitted? Assume = 1/16. (d) What is the overall probability of error?