Optimal Power Allocation for $(N,K)$-Limited Access Channels

In this paper, we consider a system that consists of
$N$
independent parallel channels, where the receiver starts to decode the information being transmitted when it has access to at least
$K$
of them. We refer to this system as the
$(N,K)$
-limited access channel. No prior knowledge for the distribution about which transmissions will be received is assumed. In addition, both the channel inputs and channel disturbances can be arbitrary, except that the mutual information function for each channel is assumed strictly concave with respect to the input power. Hence, the channel capacity below which the code rate is guaranteed to be attainable by a sequence of codes with vanishing error can be determined by the minimum mutual information among any
$K$
out of
$N$
channels. We then investigate the power allocation that maximizes this minimum mutual information subject to a total power constraint. As a result, the optimal solution can be determined via a systematic algorithmic procedure by performing at most
$K$
single-power-sum-constrained maximizations. Based on this result, the closed-form formula of the optimal power allocation for an
$(N,K)$
-limited access channel with channel inputs and additive noises, respectively, scaled from two independent and identically distributed random vectors of length
$N$
is subsequently established, and is shown to be well interpreted by a two-phase water-filling principle. Specifically, in the first noise-power redistribution phase, the least
$N-K$
noise powers (equivalently, second moments) are first poured (as noise water) into a tank consisting of
$K$
interconnected unit-width vessels with solid base heights, respectively, equal to the remaining
$K$
largest noise powers. Afterward, those
$W$
vessels either with noise water inside or with solid base height equal to the new water surface level are subdivided into
$N-K+W$
vessels of rectangular shape with the same heights (as the water surface level) and widths in proportion to their noise powers. In the second signal-power allocation phase, the heights of vessel bases will be first either lifted or lowered according to the total signal power and channel mutual information functions, followed by the usual signal-power water-filling scheme. The two-phase water-filling interpretation then hints that the degree of “noisiness” for a general (possibly, nonadditive and non-Gaussian) limited access channel might be identified by composing the derivative of the mutual information function with its inverse.