On the Design of Optimal Noisy Channel Scalar Quantizer with Random Index Assignment

The general approach in a noisy channel scalar quantizer design is an iterative descent algorithm, which guarantees only a locally optimal solution. While sufficient conditions under which the local optimum becomes a global optimum are known in the noiseless channel case, such sufficient conditions were not derived for the noisy counterpart. Moreover, efficient globally optimal design techniques for general discrete distributions in the noiseless case exist; however, they seem not to extend to the noisy scenario when a fixed index assignment is assumed. Recently, the design of noisy channel scalar quantizer with random index assignment (RIA) was proposed using a locally optimal iterative algorithm. In this paper, we derive sufficient conditions for the uniqueness of a local optimum, which, thus, guarantee the global optimality of the solution. These sufficient conditions are satisfied for a log-concave probability density function which is, additionally, symmetric around its mean. Furthermore, we show that, assuming an RIA, the globally optimal design for general discrete sources can also be carried out efficiently.