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 of the formulation is to invoke an {\em 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. Moreover it is proved that the Kuhn-Tucker conditions of the elementary MAC are strictly sufficient and obviously necessary for the channel capacity. The latter proof requires some steps such that for the elementary MAC every solution of the Kuhn-Tucker conditions reveals itself as local maximum on the domain of all possible input probability distributions and then it achieves the channel capacity. 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 MAC's.