Capacity-achieving Input Distribution Research Articles

Capacity bounds via duality with applications to multiple-antenna systems on flat-fading channels

A technique is proposed for the derivation of upper bounds on channel capacity. It is based on a dual expression for channel capacity where the maximization (of mutual information) over distributions on the channel input alphabet is replaced with a minimization (of average relative entropy) over distributions on the channel output alphabet. We also propose a technique for the analysis of the asymptotic capacity of cost-constrained channels. The technique is based on the observation that under fairly mild conditions capacity achieving input distributions "escape to infinity." The above techniques are applied to multiple-antenna flat-fading channels with memory where the realization of the fading process is unknown at the transmitter and unknown (or only partially known) at the receiver. It is demonstrated that, for high signal-to-noise ratio (SNR), the capacity of such channels typically grows only double-logarithmically in the SNR. To better understand this phenomenon and the rates at which it occurs, we introduce the fading number as the second-order term in the high-SNR asymptotic expansion of capacity, and derive estimates on its value for various systems. It is suggested that at rates that are significantly higher than the fading number, communication becomes extremely power inefficient, thus posing a practical limit on practically achievable rates. Upper and lower bounds on the fading number are also presented. For single-input-single-output (SISO) systems the bounds coincide, thus yielding a complete characterization of the fading number for general stationary and ergodic fading processes. We also demonstrate that for memoryless multiple-input single-output (MISO) channels, the fading number is achievable using beam-forming, and we derive an expression for the optimal beam direction. This direction depends on the fading law and is, in general, not the direction that maximizes the SNR on the induced SISO channel. Using a new closed-form expression for the expectation of the logarithm of a noncentral chi-square distributed random variable we provide some closed-form expressions for the fading number of some systems with Gaussian fading, including SISO systems with circularly symmetric stationary and ergodic Gaussian fading. The fading number of the latter is determined by the fading mean, fading variance, and the mean squared error in predicting the present fading from its past; it is not directly related to the Doppler spread. For the Rayleigh, Ricean, and multiple-antenna Rayleigh-fading channels we also present firm (nonasymptotic) upper and lower bounds on channel capacity. These bounds are asymptotically tight in the sense that their difference from capacity approaches zero at high SNR, and their ratio to capacity approaches one at low SNR.

Capacity-achieving input distributions for some amplitude-limited channels with additive noise (Corresp.)

An additive noise channel wherein the noise is described by a piecewise constant probability density is shown to reduce to a discrete channel by means of an explicit construction. In addition, conditions are found which describe a class of continuous amplitude-limited channels for which the capacity-achieving input distribution is binary.

Determination of capacity achieving input probabilities for a class of finite state channels with side information

The form of capacity achieving input distribution is specified for a class of finite state channels with side information available to the transmitter. It is shown that the best random codes consist of strategy function sequences whose elements are selected independently. It is a result of Theorem 4 (which is of independent interest) that the required distribution of the strategy functions is for the present class of channels specified by ah instead of the apparent a h parameters, so that a random selection of optimal codes is practically possible. As an application, an extremely simple capacity expression for finite state galois additive noise channels is obtained, and it is shown that the value of capacity will not increase if the transmitter is supplied with information about the channel states.