Be thorough and clear

Four bits of information are communicated via a signal with one of 16 amplitudes and phases. The signal is received in the presence of noise, filtered mixed down to baseband (with cosine and sine) and sampled. The output of the sampling is a pair of random variables (X, Y) or a complex random variable Z = X + jY. Let Hi, i = 0, 1, ..., 15 be the event that a certain sequence of bits is sent as follows Ho H1 0 0 0 0 Ha 0 HA 0 1 0 1 H6 0 1 H- 0 1 H 8 1 0 Ho 1 0 0 1 H10 1 0 1 0 H11 1 3 H12 1 1 0 10 H13 1 1 0 1 H14 1 1 1 0 H15 1 1Assuming hypothesis i is true the output of the receiver is X = He(i)VE +N. Y = /s (i)VE+ N. where N. and N, are independent, zero mean Gaussian random variables with variance No/2. The receiver decides hypothesis H, is true if (X, Y) 6 Ri where R; is shown below (for the case E = 1). R2 R10 Ra RT 1 R15 -1 X .3 1 RI +-1 R13 RA Ro RA 4-3 R12 R.8(a) Determine an expression (involving the Q function) for the probability that the receiver makes an error in deciding which hypothesis was true. The answer should depend on the ratio Es/No where E, = E/4 since we are transmitting four bits of information using average energy E. Relate E to E. The energy used by signal 0, which has coordinates (-3VE,-3VE), is (-3VE) + (-3VE)' = 18E. The energy used by signal 1, which has coordinates (-3VE,-1VE), is (-3VE)'+ (-1VE)? = 10E. The energy used by signal 7, which has coordinates (-1VE,IVE), is (-IVE) + (1VE) = 2E. (b) Determine the probability that a bit error is made. The bits by and b, determine the X signal. The bits b, and by determine the Y signal. The X signal and Y signal are symmetric. If bo = 0 and bj = 0 a one bit error is made if -2 2. A two bit error is made if 0