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A communication system transmits a bit {D 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. R3 = Baa + u R1 = 331 + H1 R2 = 332 + H: In Here Ra, R1, R3 represent the received signal on the four frequencies. I The random variable 3 is either +1 or -1 depending on the bit. For example if the information bit is (i then B = +1 whereas if the information bit is 1 then B=1. In The variables g, 311, :12, represent random (Gaussian or normal} noise with mean I] and variance air. The noise 139,131, and n; are independent. I The receiver combines the received signals as R = cuRu +e1R1 + {1232 and then compares R with I]. The receiver"s decision for B is called 3. If R 3: 0 the receiver decides that B = +1. HR E I] the receiver decides B = 1. o The goal of the receiver is to decide correctly the data. 01:" course it is pomible 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 do, a1, a2 and So, 81, 82. (b) Determine the conditional variance of R given B = +1 as a function of do, a1, a2 and So, $1, 82. (c) Determine the conditional density of R given B = +1 as a function of an, a1, a2 and So, 81, 52. (d) Determine the conditional probability of error given B = +1 as a function of do, 01, a2, a3 and so, $1, 82. The answer should be in terms of the @ function.(e) Evaluate the probability of error for the case so = 1, $1 = 2, $2 = 4, do = 4, 01 = 2, a2 = 1, and ON = 4. (Use Matlab command qfunc). (f) Evaluate the probability of error for the case so = 1, $1 = 2, $2 = 4 and do = 1, a1 = 1, a2 = 1 and on = 4. (Use Matlab command qfunc). (g) Evaluate the probability of error for the case so = 1, $1 = 2, s, = 4 and ag = So, 01 = $1, 02 = 82 and ON = 4. (Use Matlab command qfunc). (h) Which receiver (e, for g) gives better performance