Asymptotically optimal water-filling in vector multiple-access channels

Dynamic resource allocation is an important means to increase the sum capacity of fading multiple-access channels (MACs). In this paper, we consider vector multi-access channels (channels where each user has multiple degrees of freedom) and study the effect of power allocation as a function of the channel state on the sum capacity (or spectral efficiency) defined as the maximum sum of rates of users per unit degree of freedom at which the users can jointly transmit reliably, in an information-theoretic sense, assuming random directions of received signal. Direct-sequence code-division multiple-access (DS-CDMA) channels and MACs with multiple antennas at the receiver are two systems that fall under the model. Our main result is the identification of a simple dynamic power-allocation scheme that is optimal in a large system, i.e., with a large number of users and a correspondingly large number of degrees of freedom. A key feature of this policy is that, for any user, it depends on the instantaneous amplitude of channel state of that user alone and the structure of the policy is "water-filling." In the contest of DS-CDMA and in the special case of no fading, the asymptotically optimal power policy of water-filling simplifies to constant power allocation over all realizations of signature sequences; this result verifies the conjecture made in Verdu and Shamai (1999). We study the behavior of the asymptotically optimal water-filling policy in various regimes of number of users per unit degree of freedom and signal-to-noise ratio (SNR). We also generalize this result to multiple classes, i.e., the situation when users in different classes have different average power constraints.