Jointly optimal rate and power allocation for multilayer transmission

In this paper we consider joint optimization of rate and power for communication systems that use multilayer source coding with successive information refinement, accompanied with a broadcast approach at the physical layer of the system. We analyze the problem under the assumption of Rayleigh fading channels where rates and power ratios of the source layers are jointly optimized based on channel statistics information, with the objective of maximizing the expected user satisfaction which is usually defined by a \rev{differentiable} concave increasing utility function of the total decoded rate. As special cases, we consider two utility functions; namely, the expected total decoded rate at the receiver and the expected rate distortion of a Gaussian source. We show that the optimal solution can be obtained using a two-dimensional bisection search for any number of layers. The outer bisection search is over the Lagrangian dual variable and the inner bisection search is over the decoding threshold of the \rev{last} layer. Moreover, we show that with a small number of layers, we can approach the performance upper bound that is achieved by transmitting an infinite number of layers.