Improved capacity lower bounds for fading channels with imperfect CSI using rate splitting

As shown by Medard (“The effect upon channel capacity in wireless communications of perfect and imperfect knowledge of the channel,” IEEE Trans. Inform. Theory, May 2000), the capacity of fading channels with imperfect channel-state information (CSI) can be lower-bounded by assuming a Gaussian channel input X, and by upper-bounding the conditional entropy h(XY, Ĥ), conditioned on the channel output Y and the CSI 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, by using a rate-splitting approach, this lower bound can be sharpened: we show that by expressing the Gaussian input X as as the sum of two independent Gaussian variables X₍₁₎ and X₍₂₎, and by applying Medard´s lower bound first to analyze the mutual information between X₍₁₎ and Y conditioned on Ĥ while treating X₍₂₎ as noise, and by applying the lower bound then to analyze the mutual information between X₍₂₎ and Y conditioned on (X₍₁₎, Ĥ), we obtain a lower bound on the capacity that is larger than Medard´s lower bound.