L-values for modulated symbols

If a mapping with $M_b$ bits per symbol is used, $M_b$ individual L-values per symbol have to be computed. Thus, we extend the calculation of the channel L-value as follows. $L_{ch}(x_{m}|z) = \ln\frac{\sum_{\mathbf{x}\in\mathbb{X}_{m,1}} p(z|\widehat{\mathbf{y}}) \cdot \exp\left[ \sum_{x_k=1, k\neq m} L_A(x_k)\right]}{\sum_{\mathbf{x}\in\mathbb{X}_{m,0}} p(z|\widehat{\mathbf{y}})\cdot \exp\left[ \sum_{x_k=1, k\neq m} L_A(x_k)\right]}$ with $\mathbb{X}_{m,1}$ being the set of the $2^{M_b-1}$ symbols $\mathbf{y}$ having $x_m = 1$. $\mathbb{X}_{m,1} = \{\mathbf{x}|x_m= 1\}, \mathbb{X}_{m,0} = \{\mathbf{x}|x_m = 0\} \text{ respectively.}$ Thus, the L-value of bit $m$ is $L(x_{m}|z) = \underbrace{\ln\frac{P[x_m=1]}{P[x_m=0]}}_{L_A(x_m)} + L_{ch}(x_{m}|z)$