Singular Detection in Noncoherent Communications
Marc Vilà-Insa, Jaume Riba
TL;DR
The paper addresses noncoherent MIMO detection and fundamental error floors by deriving necessary and sufficient conditions for asymptotically vanishing error (ASD) in two regimes: large arrays and high SNR. It adopts classical detection theory and Hilbert-space tools to connect ASD to unique identification and to Grassmannian geometry of codewords, yielding a j-divergence criterion for the large-array case and a subspace-distinction criterion for the high-SNR case. The main contributions are two propositions: (i) ASD in the large array regime occurs when the j-divergence J_{Nr} diverges with Nr for all codeword pairs, and (ii) ASD at high SNR occurs iff codewords span distinct subspaces, i.e., distinct column spaces. These results generalize prior work to arbitrary K and Nt, link noncoherent code design to Grassmannian structures, and provide fundamental benchmarks for massive MIMO and energy-efficient noncoherent transmission schemes.
Abstract
This paper proposes a general analysis of codeword detection in noncoherent communications. Motivated by the existence of error floors in various regimes, fundamental characteristics of signal design are investigated. In particular, the necessary and sufficient conditions for asymptotically singular detection (i.e. error-free in the limit) are derived from classical results in detection theory. By leveraging tools from linear algebra and the theory of Hilbert spaces, we are able to characterize asymptotic singularity in two main scenarios: the large array and high SNR regimes. The results generalize previous works and extend the notion of unique identification, as well as re-contextualize the geometry of Grassmannian constellations from an alternative perspective.