Soft Covering Through the Lens of Hypothesis Testing

Abstract

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 $\hat{I}{\pm }_n(\ddot{I}„,R)$ and missed-detection (MD) probability ${\hat{I}}_n^2(\ddot{I}„,R)$, as functions of the decision threshold $\ddot{I}„$ 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 $\ddot{I}„\in (0,I(X;Y)-R)$. At $R=I(X;Y)$, this tradeoff interval collapses to the single point $\ddot{I}„=0$, 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 $\ddot{I}„=0$; moreover, for every $\ddot{I}„$ at least one exponent is zero: the FA exponent is zero for $\ddot{I}„\le 0$ (FA probability does not decay exponentially), and the MD exponent is zero for $\ddot{I}„\ge 0$ (and finite, channel-specific for $\ddot{I}„<0$; see Remark~\ref{rem:jump}). There is no interval of $\ddot{I}„$ where both exponents are simultaneously positive. A sharp phase transition in the MD exponent occurs at $\ddot{I}{„}^{\ast }=[I(X;Y)-R{]}_{+}$ for all rates.

25 pages, 7 figures, submitted for publication