The goal of the receiver is to decide correctly the data. Of course it is possible that because of noise, the decision

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A communication system transmits a bit (0 or 1) of information by transmitting a signal on four different frequencies. The received signal strength is different on the four frequencies and is described as follows.

RO = BS0 + no

R1 = Bs + T1

R2 = BS2 + 112

• Here Ro, R1, R, represent the received signal on the four frequencies.

• The random variable B is either +1 or -1 depending on the bit. For example if the information bit is 0 then B = +1 whereas if the information bit is 1 then B = -1

• The variables no, ma, 12, represent random (Gaussian or normal) noise with mean O and variance o. The noise no, n, and 12 are independent.

• The receiver combines the received signals as R= ao Ro + a_Ri + az R2 and then compares R with 0. The receiver's decision for B is called B. If R > 0 the receiver decides that B = +1. If R 50 the receiver decides B =-1.

• The goal of the receiver is to decide correctly the data. Of course it is possible that because of noise, the decision is incorrect and we say an error occurred.

(a) Determine the conditional mean of R given B = +1 as a function of ao, 21, and S0, S1, S2.

(b) Determine the conditional variance of R given B +1 as a function of 20, 21, 22 and so, S1, S2.

(c) Determine the conditional density of R given B = +1 as a function of 20, 21, 22 and S0, S1, S2.

(d) Determine the conditional probability of error given B = +1 as a function of 20, 21, 22, 23 and S0, S1, S2. The answer should be in terms of the Q function.

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