4. Signal or no signal. Consider a communication system that is operated only from time to time. When the communication system is in the "normal" mode (denoted by M = 1), it transmits a random signal S = X with X = ( +1, with probability 1/2, ?1, with probability 1/2. When the system is in the "idle" mode (denoted by M = 0), it does not transmit any signal (S = 0). Both normal and idle modes occur with equal probability. Thus S = ( X, with probability 1/2, 0, with probability 1/2. The receiver observes Y = S + Z, where the ambient noise Z ?Unif[?1,1] is independent of S. (a) Find and sketch the conditional pdf fY |M(y|1) of the receiver observation Y given that the system is in the normal mode. (b) Find and sketch the conditional pdf fY |M(y|0) of the receiver observation Y given that the system is in the idle mode. (c) Find the optimal decoder ?MMAP(y) for deciding whether the system is normal or idle. Provide the answer in terms of intervals of y. (d) Find the associated probability of error.

4. Signal or no signal. Consider a communication system that is operated only from time to time. When the communication system is in the \"normal\" mode (denoted by M = 1), it transmits a random signal S = X with X _ +1, with probability 1/2, _ 1, with probability 1 / 2. When the system is in the \"idle\" mode (denoted by M = 0), it does not transmit any signal (S = 0). Both normal and idle modes occur with equal probability. Thus 3 _ X , with probability 1/ 2, 0, with probability 1 / 2. The receiver observes Y = S + Z , Where the ambient noise Z N Unif[1, 1] is independent of S. (a) Find and sketch the conditional pdf fy|M(y|1) of the receiver observation Y given that the system is in the normal mode. (b) Find and sketch the conditional pdf fy|M(y|0) of the receiver observation Y given that the system is in the idle mode. (c) Find the optimal decoder MMAp(y) for deciding Whether the system is normal or idle. Provide the answer in terms of intervals of y. ((1) Find the associated probability of error