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Log-Likelihood Ratios (L-values) for BPSK

With Bayes's rule \[ P[y=+1|z] = \frac{p(z|y=+1)}{p(z)} P[y=+1] \] and channel output PDF $p(z|y)$ conditioned on the transmitted symbol $y$ we obtain \[ \underbrace{L(y|z)}_{\text{a posteriori L-value}} = \ln\frac{p(z|y=+1) P[y=+1]}{p(z|y=-1) P[y=-1]} = \underbrace{\ln\frac{P[y=+1]}{P[y=-1]}}_{\text{a priori L-value } L_A(y)} + \underbrace{\ln\frac{p(z|y=+1)}{p(z|y=-1)}}_{\text{channel L-value } L_{ch}(y|z)} = L_A(y) + L_{ch}(y|z).\] The channel L-value tells us what we learn about the transmitted symbol $y$ based on the channel observation.

For an AWGN channel with zero mean and variance $\sigma^2$ the channel L-value computes as \[ L_{ch}(y|z) = \ln\frac{\exp(-\frac{(z-1)^2}{2\sigma^2})}{\exp(-\frac{(z+1)^2}{2\sigma^2})} = \frac{2}{\sigma^2}z. \]