A Formulation of the Channel Capacity of Multiple-Access Channel

The necessary and sufficient condition of the channel capacity is rigorously formulated for the $N$ -user discrete memoryless multiple-access channel (MAC). The essence is to invoke an elementary MAC where sizes of input alphabets are not greater than the size of output alphabet. The main objective is to demonstrate that the channel capacity of an MAC is achieved by an elementary MAC included in the original MAC. The proof is quite straightforward by the very definition of the elementary MAC. The second objective is to prove that the Kuhn–Tucker conditions of the elementary MAC are sufficient (obviously necessary) for the channel capacity. The latter proof requires two distinctive properties of the MAC: Every solution of the Kuhn–Tucker conditions is a local maximum on the domain of all possible input probability distributions (IPDs), and then particularly for the elementary MAC a set of IPDs for which the value of the mutual information is not smaller than the arbitrary positive number is connected on the domain. As a result, in respect of the channel capacity, the MAC in general can be regarded as an aggregate of a finite number of elementary MACs.