Example: L-value for 4-PAM

Let us derive the L-value of the first bit ($m=1$) of a 4-PAM modulated signal. The constellation and the mapping of input bits $x_1$ and $x_2$ are according to the following figure.

The first bit is 1 for $y=+1$ and $y=-1$, and 0 for $y=+3$ and $y=-3$. Thus, the L-value of bit 1 is calculated as $L(x_{1}|z) = \ln \frac{p(z|y=+1) \cdot P[x_1=1] \cdot P[x_2 = 1] +p(z|y=-1)\cdot P[x_1=1] \cdot P[x_2 = 0] }{p(z|y=+3)\cdot P[x_1=0] \cdot P[x_2 = 1] +p(z|y=-3)\cdot P[x_1=0] \cdot P[x_2 = 0] }.$ For the given AWGN-channel it holds $L(x_{1}|z) = L_A(x_1) + \ln \frac{ \exp(\frac{-(z-1)^2}{2\sigma^2}) + \exp(\frac{-(z+1)^2}{2\sigma^2}) \cdot \exp \left[ L_A(x_2) \right] }{ \exp(\frac{-(z-3)^2}{2\sigma^2}) + \exp(\frac{-(z+3)^2}{2\sigma^2}) \cdot \exp \left[ L_A(x_2) \right] }.$