I have a question about the decoding of convolutional codes with the MAP (BCJR) algorithm. Let $\mathbf{u}$ denote the uncoded bits and $\mathbf{v}$ is the coded bits. Here is the point!
Let $\mathbf{v}$ is modulated with any arbitrary linear modulation (M-PSK, M-QAM) and transmitted through an AWGN channel. The receiver performs the following to obtain the log-likelihood ratios of the uncoded bits $\mathbf{u}$:
$$L\left(u_{l}\right) \equiv \ln \left[\frac{P\left(u_{l}=+1 \mid \mathbf{r}\right)}{P\left(u_{l}=-1 \mid \mathbf{r}\right)}\right]$$ which equals to $$L\left(\alpha_{k}\right)=\log \frac{ <u_l=0>\sum_{m'} \sum_{m}\alpha_{k-1}\left(m^{\prime}\right) \gamma_{k}\left(m^{\prime}, m\right) \beta_{k}(m)}{<u_l=1>\sum_{m^{\prime}} \sum_{m} \alpha_{k-1}\left(m^{\prime}\right) \gamma_{k}\left(m^{\prime}, m\right) \beta_{k}(m)}.$$ Here, $m$ and $m'$ stands for the trellis states, $<u_l = i>$ denotes that the lefthandside expression is under the condition of $u_l=i$.
Now, my question is: can I manipulate the MAP algorithm as given below: $$L\left(u_{l}\right) \equiv \ln \left[\frac{P\left(u_{l}=+1 \mid L(\mathbf{c})\right)}{P\left(u_{l}=-1 \mid L(\mathbf{c})\right)}\right]$$ where $L(\mathbf{c})$ is the channel LLRs. In this case, I could not find an answer how to write $\gamma_k$ values