A Rate-Splitting Approach to Fading Channels With Imperfect Channel-State Information

As shown by Médard, the capacity of fading channels with imperfect channel-state information can be lower-bounded by assuming a Gaussian channel input X with power P and by upper-bounding the conditional entropy h(X|Y, Ĥ) by the entropy of a Gaussian random variable with variance equal to the linear minimum mean-square error in estimating X from (Y, Ĥ). We demonstrate that, using a rate-splitting approach, this lower bound can be sharpened: by expressing the Gaussian input X as the sum of two independent Gaussian variables X₁ and X₂ and by applying Médard´s lower bound first to bound the mutual information between X₁ and Y while treating X₂ as noise, and by applying it a second time to the mutual information between X₂ and Y while assuming X₁ to be known, we obtain a capacity lower bound that is strictly larger than Médard´s lower bound. We then generalize this approach to an arbitrary number L of layers, where X is expressed as the sum of L independent Gaussian random variables of respective variances P_ℓ, ℓ = 1, ... , L summing up to P. Among all such rate-splitting bounds, we determine the supremum over power allocations P_ℓ and total number of layers L. This supremum is achieved for L →∞ and gives rise to an analytically expressible capacity lower bound. For Gaussian fading, this novel bound is shown to converge to the Gaussian-input mutual information as the signal-to-noise ratio (SNR) grows, provided that the variance of the channel estimation error H - Ĥ tends to zero as the SNR tends to infinity.