  A transmitter without channel state information (CSI) wishes to send a
delay-limited Gaussian source over a slowly fading channel. The source is coded
in superimposed layers, with each layer successively refining the description
in the previous one. The receiver decodes the layers that are supported by the
channel realization and reconstructs the source up to a distortion. The
expected distortion is minimized by optimally allocating the transmit power
among the source layers. For two source layers, the allocation is optimal when
power is first assigned to the higher layer up to a power ceiling that depends
only on the channel fading distribution; all remaining power, if any, is
allocated to the lower layer. For convex distortion cost functions with convex
constraints, the minimization is formulated as a convex optimization problem.
In the limit of a continuum of infinite layers, the minimum expected distortion
is given by the solution to a set of linear differential equations in terms of
the density of the fading distribution. As the bandwidth ratio b (channel uses
per source symbol) tends to zero, the power distribution that minimizes
expected distortion converges to the one that maximizes expected capacity.
While expected distortion can be improved by acquiring CSI at the transmitter
(CSIT) or by increasing diversity from the realization of independent fading
paths, at high SNR the performance benefit from diversity exceeds that from
CSIT, especially when b is large.
