Through refined asymptotic analysis based on the normal approximation, we study how higher-order coding performance depends on the mean power $Γ$ as well as on finer statistics of the input power. We introduce a multifaceted power model in which the expectation of an arbitrary (but finite) number of arbitrary functions of the normalized average power is constrained. The framework generalizes existing models, recovering the standard maximal and expected power constraints and the recent mean and variance constraint as special cases. Under certain growth and continuity assumptions on the functions, our main theorem gives an exact characterization of the minimum average error probability for Gaussian channels as a function of the first- and second-order coding rates. The converse proof reduces the code design problem to minimization over a compact (under the Prokhorov metric) set of probability distributions, characterizes the extreme points of this set and invokes the Bauer's maximization principle. Our results for the multifaceted power model serve as more precise benchmarks for practical modulation schemes with multiple amplitude levels, probabilistic shaping and nonuniform constellation geometries.

翻译：基于正态近似精细渐近分析，我们研究了高阶编码性能如何依赖于平均功率Γ以及输入功率的精细统计特性。我们提出了一种多层面功率模型，在该模型中约束任意（有限）个归一化平均功率的任意函数的期望值。该框架推广了现有模型，将经典的最大功率与期望功率约束，以及近年来提出的均值与方差约束作为特例纳入其中。在函数满足特定增长性与连续性的假设下，我们的主定理精确刻画了高斯信道平均错误概率最小值与一阶、二阶编码速率之间的函数关系。逆证明将码书设计问题简化为紧集（在普罗霍洛夫度量意义下）上概率分布的最小化，刻画了该集合的极值点，并应用了鲍尔最大化原理。针对多层面功率模型得出的结果，为多幅度电平、概率整形及非均匀星座构型等实际调制方案提供了更精确的性能基准。