Let's consider channels of the form \begin{equation}\label{eq:channelmodel2} Y_{1}= X e^{j \theta_{1}}+ V_1 \end{equation}
\begin{equation}\label{eq:channelmodel3} Y_{2}=X e^{j \theta_{2}}+ V_2 \end{equation}
Where $X$ is the channel input symbol. $Y_1$ and $Y_2$ are outputs. $V_{i}$ is complex white Gaussian noise with (known) variance $\sigma^{2},$ i.e., $\sigma^{2}/2$ per dimension. Both channels parameters $\theta_1$, $\theta_2$ and $V_{1}$ and $V_{2}$ are independent of each other. In simple words, it is a single input multiple output (SIMO) system. I am interested to compute $I(X; Y_1, Y_2)$. Can you please guide me on whether I am doing right or wrong? \begin{equation}\label{eq:channelmodel4} \begin{aligned} I(X; Y_1, Y_2)& = h( Y_1, Y_2) - h(Y_1, Y_2 | X) \\ &= h(Y_1) + h(Y_2|Y_1) - h(Y_1| X) - h(Y_2 | X , Y_1)\\ &\overset{\text{by independence}}{=} h(Y_1) + h(Y_2) - h(Y_1| X) - h(Y_2 | X )\\ &= I(X; Y_1) + I(X; Y_2) \end{aligned} \end{equation}