Under which condition is quantization optimal? We address this question in the context of the additive uniform noise channel under peak amplitude and cost constraints. We compute analytically the capacity-achieving input distribution as a function of the noise level, the average cost constraint, and the curvature of the cost function. We find that when the cost function is concave, the capacity-achieving input distribution is discrete, whereas when the cost function is convex and the cost constraint is active, the support of the capacity-achieving input distribution spans the entire interval. For the cases of a discrete capacity-achieving input distribution, we derive the analytical expressions for the capacity of the channel.

翻译：量化在何种条件下达到最优？本文在峰值幅度与成本约束下的加性均匀噪声信道框架中探讨这一问题。我们解析计算了容量可达输入分布随噪声水平、平均成本约束及成本函数曲率的变化关系。研究发现：当成本函数为凹函数时，容量可达输入分布呈离散态；而当成本函数为凸函数且成本约束激活时，容量可达输入分布的支撑集将覆盖整个连续区间。针对容量可达输入分布为离散态的情形，我们推导了信道容量的解析表达式。