We study the soft covering phenomenon through the lens of Neyman--Pearson hypothesis testing: given a channel output sequence $y^n$, can one decide whether it was produced when the channel was driven by a random codeword, or generated independently from the output marginal? We derive exact exponential decay rates for the jointly averaged false-alarm (FA) probability $α_n(τ,R)$ and missed-detection (MD) probability $β_n(τ,R)$, as functions of the decision threshold $τ$ and the codebook rate $R$. The derived single-letter formulas of the exponents $\EFA(τ,R)=-\lim_{n\to\infty}\frac{1}{n}\lnα_n(τ,R)$ and $\EMD(τ,R)=-\lim_{n\to\infty}\frac{1}{n}\lnβ_n(τ,R)$ are tight in the random coding sense. The analysis reveals a rich phase structure. For $R < I(X;Y)$, there is a genuine exponential tradeoff between the two error types over the interval $τ\in (0, I(X;Y)-R)$. At $R = I(X;Y)$, this tradeoff interval collapses to the single point $τ= 0$, where both error exponents simultaneously vanish, a fact which manifests the soft covering phenomenon in the Neyman--Pearson sense. For $R > I(X;Y)$, the same instantaneous collapse persists at $τ= 0$; moreover, for every $τ$ at least one exponent is zero: the FA exponent is zero for $τ\le 0$ (FA probability does not decay exponentially), and the MD exponent is zero for $τ\ge 0$ (and finite, channel-specific for $τ<0$; see Remark~\ref{rem:jump}). There is no interval of $τ$ where both exponents are simultaneously positive. A sharp phase transition in the MD exponent occurs at $τ^* = [I(X;Y)-R]_+$ for all rates.

翻译：本文通过奈曼-皮尔逊假设检验的视角研究软覆盖现象：给定信道输出序列$y^n$，能否判断该序列是由随机码字驱动的信道产生，还是由输出边际分布独立生成？我们推导了联合平均虚警概率$\alpha_n(\tau,R)$和漏检概率$\beta_n(\tau,R)$作为决策阈值$\tau$和码本速率$R$函数的精确指数衰减率。所得到的指数$\EFA(\tau,R)=-\lim_{n\to\infty}\frac{1}{n}\ln\alpha_n(\tau,R)$和$\EMD(\tau,R)=-\lim_{n\to\infty}\frac{1}{n}\ln\beta_n(\tau,R)$的单字母公式在随机编码意义下是紧致的。分析揭示了丰富的相结构：当$R < I(X;Y)$时，在区间$\tau\in(0,I(X;Y)-R)$内两种错误类型之间存在真正的指数权衡；当$R = I(X;Y)$时，该权衡区间坍缩至单点$\tau=0$，此时两个误差指数同时消失——这一事实体现了奈曼-皮尔逊意义下的软覆盖现象；当$R > I(X;Y)$时，相同的瞬时坍缩仍出现在$\tau=0$处；此外，对于每个$\tau$至少有一个指数为零：虚警指数在$\tau\le 0$时为零（虚警概率不呈指数衰减），漏检指数在$\tau\ge 0$时为零（且在$\tau<0$时为有限值且与具体信道相关，见注记~\ref{rem:jump}）。不存在使两个指数同时为正的$\tau$区间。对于所有速率，漏检指数在$\tau^*=[I(X;Y)-R]_+$处发生尖锐相变。