From Distance to Angle: One-Shot Detection Under Additive White Cauchy Noise
2026-04-10 • Information Theory
Information Theory
AI summaryⓘ
The authors explore how to detect symbols sent over a channel corrupted by a special type of noise called Cauchy noise, using fixed sets of possible signals. They find that the way the decision boundaries are shaped is the same as with normal Gaussian noise, but the chances of making errors depend differently on the signal arrangement. In low-noise cases, error rates relate more to the overall geometry of the signal set, while in high-noise cases, angles between signals become more important. Their work shows a clear shift from distance-based to angle-based error behavior depending on the noise level.
additive white Cauchy noisefinite constellationmaximum-likelihood detectionVoronoi regionssymbol error probabilitydistance spectrumheavy-tailed noise4QAMdecision geometryrecession cone
Authors
Yen-Chi Lee
Abstract
We study one-shot detection under additive white Cauchy noise (AWCN) using finite constellations, with emphasis on the geometric mechanisms governing symbol-level reliability. Under isotropic Cauchy noise, the maximum-likelihood rule induces the same Euclidean Voronoi decision regions as in the Gaussian case, so the distinction lies not in the decision geometry itself but in how probability mass is distributed over these fixed regions. In the small-noise regime, we derive a reciprocal distance-spectrum upper bound for the symbol error probability, showing that reliability retains a longer-range dependence on the global constellation geometry than under additive white Gaussian noise. In the large-noise regime, we prove that the correct-decision probability converges to a limit determined solely by the angular measure of the associated Voronoi recession cone. These results formalize a regime-dependent transition from distance-based to angle-based reliability descriptors under heavy-tailed noise. The theory is further illustrated through an asymmetric four-point example exhibiting geometric collapse and a standard 4QAM sanity check.