In Problem 8.4.5, we used simulation to estimate the probability of symbol error. For transmitting a binary bit stream over an M-PSK system, we set each M = 2N and each transmitted symbol corresponds to N bits. For example, for M = 16, we map each four-bit input b3b2b1b0 to one of 16 symbols. A simple way to do this is binary index mapping: transmit si when b3b2b1b0 is the binary representation of i. For example, the bit input 1100 is mapped to the transmitted signal s12. Symbol errors in the communication system cause bit errors. For example if s1 is sent but noise causes s2 to be decoded, the input bit sequence b3b2b1b0 = 0001 is decoded as 
3210 = 0010, resulting in 2 correct bits and 2 bit errors. In this problem, we use Matlab to investigate how the mapping of bits to symbols affects the probability of bit error. For our preliminary investigation, it will be sufficient to map the three bits b2b2b0 to the M = 8 PSK system of Problem 8.3.6
(a) Determine the acceptance sets {A0,..., A7}.
(b) Simulate m trials of the transmission of symbol s0. Estimate {P0 j | j = 0, 1,..., 7}, the probability that the receiver output is sj when s0 was sent. By symmetry, use the set {P0 j} to determine Pij for all i and j.
(c) Let b(i) = [b2(i) b1(i) b0(i) ]denote the input bit sequence that is mapped to si . Let dij denote the number of bit positions in which b(i) and b(j) differ. For a given mapping, the bit error rate (BER) is
(d) Estimate the BER for the binary index mapping.
(e) The Gray code is perhaps the most commonly used mapping:
Does the Gray code reduce the BER compared to the binary index mapping?
Problem 8.4.5
Suppose a user of the multilevel QPSK system needs to decode only the third bit k of the message ijk. For k = 0, 1, let Hk denote the hypothesis that the third bit was k. What are the acceptance sets A0 and A1? What is P[B3], the probability that the third bit is in error?