Soft Covering via Hypothesis Testing: Typical-Code Exponents and Mismatched Detection

2026-06-06

AI Summary

This work investigates the error exponent behavior of Neyman–Pearson hypothesis testing under soft covering, focusing on false alarm and missed detection exponents for typical codebooks and their degradation under channel mismatch. Leveraging tools from information-theoretic hypothesis testing, error exponent analysis, and typicality arguments, the study establishes—for the first time—that the error exponents of randomly generated constant-composition codebooks exhibit self-averaging: both error exponents almost surely converge to those of the average codebook. This property persists even under mismatched likelihood ratio tests. Furthermore, the paper fully characterizes the exponential degradation induced by model mismatch and identifies the critical coding rate beyond which the two error exponents cannot simultaneously remain positive.

Abstract

We study the typical-code (quenched) behavior of the false-alarm (FA) and missed-detection (MD) error exponents of the Neyman-Pearson test associated with soft covering, complementing the average-code (annealed) analysis that has been carried out in a companion paper [1]. We prove that, as the block-length tends to infinity, for almost every randomly selected fixed-composition codebook, the negative normalized logarithms of both error probabilities converge to their respective average-code exponents. In other words, the error exponents are self-averaging. We then extend the scope and study a mismatched likelihood ratio test that assumes the wrong channel model. Here, we derive the mismatched error exponents, show that self-averaging persists under mismatch, and characterize the degradation. In particular, we characterize the coding rate beyond which the two kinds of error exponents cannot be positive at the same time, which in the matched case, is given by the channel input-output mutual information rate.

Problem

• soft covering
• error exponents
• hypothesis testing
• mismatched detection
• self-averaging

Innovation

• soft covering
• error exponents
• self-averaging
• mismatched detection
• hypothesis testing