Soft Covering Through the Lens of Hypothesis Testing

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 $\alpha_n(\tau,R)$ and missed-detection (MD) probability $\beta_n(\tau,R)$, as functions of the decision threshold $\tau$ and the codebook rate $R$. The derived single-letter formulas of the exponents $\EFA(\tau,R)=-\lim_{n\to\infty}\frac{1}{n}\ln\alpha_n(\tau,R)$ and $\EMD(\tau,R)=-\lim_{n\to\infty}\frac{1}{n}\ln\beta_n(\tau,R)$ 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 $\tau \in (0, I(X;Y)-R)$. At $R = I(X;Y)$, this tradeoff interval collapses to the single point $\tau = 0$, 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 $\tau = 0$; moreover, for every $\tau$ at least one exponent is zero: the FA exponent is zero for $\tau \le 0$ (FA probability does not decay exponentially), and the MD exponent is zero for $\tau \ge 0$ (and finite, channel-specific for $\tau<0$; see Remark~\ref{rem:jump}). There is no interval of $\tau$ where both exponents are simultaneously positive. A sharp phase transition in the MD exponent occurs at $\tau^* = [I(X;Y)-R]_+$ for all rates.