  As shown by M\'edard, the capacity of fading channels with imperfect
channel-state information (CSI) can be lower-bounded by assuming a Gaussian
channel input $X$ with power $P$ and by upper-bounding the conditional entropy
$h(X|Y,\hat{H})$ by the entropy of a Gaussian random variable with variance
equal to the linear minimum mean-square error in estimating $X$ from
$(Y,\hat{H})$. 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 M\'edard'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 M\'edard'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 $P_{\ell}$, $\ell = 1,\dotsc,L$ summing up to $P$. Among
all such rate-splitting bounds, we determine the supremum over power
allocations $P_\ell$ and total number of layers $L$. This supremum is achieved
for $L\to\infty$ 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 $H-\hat{H}$ tends to
zero as the SNR tends to infinity.
