Finding the total capacity of two communication channels

I have the transition matrices of two communication channels. I am able to find the capacity of each by performing an optimization calculation, however I need the total capacity of the two channels. The channels are not weakly symmetric so I don't think I am able to simply add the two capacities. I think what I need to do is perform the optimization calculation over the combined channels but I'm not sure how to do this. I think I need to construct a transition probability matrix but I'm not sure how to go about this. If anyone can give some advice on this I'd be grateful.

Thanks

• There seems to be some information missing here. Are the channels coupled? If not, the capacity is additive. See for example Wyner, Capacity of Product of Channels. – Yuval Filmus Aug 9 '18 at 13:51
• For general channels, you can use the Blahut–Arimoto algorithm to computer the channel capacity. – Yuval Filmus Aug 9 '18 at 13:53
• @YuvalFilmus what do you mean by coupled in this context? – Wilky94 Aug 9 '18 at 13:57
• We can describe the operation of a channel as accepting an input $X$, a random variable $N$, and deterministically outputting $Y = f(X,N)$. In your case, you have $X_1,X_2,N_1,N_2,Y_1,Y_2$, and the question is whether $N_1,N_2$ are independent. – Yuval Filmus Aug 9 '18 at 13:59
• @YuvalFilmus In that case I believe they are independent so they must be additive, thanks for your comment! – Wilky94 Aug 9 '18 at 15:41