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

• Published in: IEEE transactions on information theory 2014-07, Vol.60 (7), p.4266-4285
• Main Authors: Pastore, Adriano, Koch, Tobias, Rodriguez Fonollosa, Javier
• Format: Article
• Language: English
• Subjects: Allocations; Applied sciences; Channel capacity; Channels; Communication channels; Detection, estimation, filtering, equalization, prediction; Enginyeria de la telecomunicació; Entropia (Teoria de la informació); Entropy; Entropy (Information theory); Errors; Exact sciences and technology; Fading; Fading channels; Flat fading; Gaussian; Gaussian processes; Imperfect channel-state information; Information theory; Information, signal and communications theory; Informàtica; Lower bounds; Mutual information; Noise; Normal distribution; Processos gaussians; Random variables; Receivers; Signal and communications theory; Signal to noise ratio; Signal, noise; Systems, networks and services of telecommunications; Telecommunications; Telecommunications and information theory; Transmission and modulation (techniques and equipments); Upper bound; Variables; Variance; Àrees temàtiques de la UPC
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2014.2321567

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 X 1 and X 2 and by applying Médard's lower bound first to bound the mutual information between X 1 and Y while treating X 2 as noise, and by applying it a second time to the mutual information between X 2 and Y while assuming X 1 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.