Second-order coding rate for m-class source-channel codes

The second-order coding rate for source-channel codes with m levels of unequal message protection (UMP) is derived. The second-order coding rate takes the form of an optimization problem which reduces to separate source-channel coding rate for m = 2. A procedure for solving the given optimization problem is proposed. The proposed procedure exploits the structure of the problem to reduce the optimization search space to one dimension. Numerical results for the BSC are obtained and it is shown, empirically, that the second-order coding rate of source-channel codes with m levels of UMP approaches the optimal joint source-channel coding rate as m grows.