Channel codes are constructed by introducing redundancy in a structured fashion. A canonical means of doing...

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Channel codes are constructed by introducing redundancy in a structured fashion. A canonical means of doing this is by introducing parity checks. In this problem, we see how one can make inferences based on three bits b₁,b2, b3 which satisfy a parity check equation: b₁ b₂b3 = 0. Here denotes an exclusive or (XOR) operation. (a) Suppose that we know that P[b₁ = 0] = 0.8 and P[b₂ = 1] = 0.9, and model b₁ and be as independent. Find the probability P[b3 = 0]. = Pb=1 (b) Define the log likelihood ratio (LLRs) for a bit b as LLR(b) log Setting L LLR(b), i = 1, 2, 3, find an expression for L3 in terms of L₁ and L2, again modeling b₁ and b as independent. Channel codes are constructed by introducing redundancy in a structured fashion. A canonical means of doing this is by introducing parity checks. In this problem, we see how one can make inferences based on three bits b₁,b2, b3 which satisfy a parity check equation: b₁ b₂b3 = 0. Here denotes an exclusive or (XOR) operation. (a) Suppose that we know that P[b₁ = 0] = 0.8 and P[b₂ = 1] = 0.9, and model b₁ and be as independent. Find the probability P[b3 = 0]. = Pb=1 (b) Define the log likelihood ratio (LLRs) for a bit b as LLR(b) log Setting L LLR(b), i = 1, 2, 3, find an expression for L3 in terms of L₁ and L2, again modeling b₁ and b as independent.