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 = 2^N 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 s_i 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 {P_0 j | j = 0, 1,..., 7}, the probability that the receiver output is s_j when s₀ was sent. By symmetry, use the set {P_{0 j}} to determine P_ij for all i and j.

(c) Let b(i) = [b₂(i) b₁(i) b₀(i) ]denote the input bit sequence that is mapped to s_i . Let d_ij 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 i_jk. For k = 0, 1, let H_k 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?