# 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

## 2 Answers

If the channels are not coupled (i.e., their effect on the input is independent of each other) then their combined capacity is the sum of their individual capacities. See for example Wyner, Capacity of Product of Channels, who also considers the converse.

You have to consider different scenarios. The capacity of a broadcast channel is different than a multiple access channel. The total capacity also varies for symmetric and asymmetric channels. See "Network Information Theory" by Abbas El Gamal and Young-Han Kim

• I'm looking for a general method for not necessarily symmetric channels. I'm trying to construct a transition matrix for the total channel, which I believe would be a broadcast channel. For a simple case, consider two erasure channels. I want to find the total capacity of the two in parallel by constructing a transition matrix and optimizing. – Wilky94 Aug 10 '18 at 12:51
• Please see chapter 9 of Network Information Theory by Abbas El Gamal and Young-Han Kim on Vector Channels. – Hannah Aug 11 '18 at 20:35