Detection Below the Threshold-to-Noise Ratio of Three

Characterize and correct direct Mahalanobis maximum-likelihood symbol detection for modulo-folded, quantized observations when the folding threshold-to-noise ratio satisfies \(\lambda/\sigma_{\mathrm{lpf}}<3\), where wrap events are frequent and the single-Gaussian likelihood approximation and unfolded-noise covariance approximation fail.

Background

The proposed detector approximates the exact wrapped-Gaussian likelihood by retaining only the zero-wrap term and uses the covariance of the unfolded, filtered noise together with quantization-noise variance. The paper establishes this approximation for the regime $\lambda/\sigma_{\mathrm{lpf}}\ge 3$, where wrap events are sufficiently rare and the folded-noise covariance remains close to the unfolded-noise covariance.

For smaller threshold-to-noise ratios, wrap events occur often enough that the single-term lattice truncation is inaccurate, while the covariance of the folded noise differs materially from that of the unfolded noise. The paper therefore leaves unresolved both the characterization of detection performance and the development of corrected likelihood and covariance models in this regime.