Robust rank conditions beyond the independence assumption

Establish whether the sufficiency condition for full rank proved in Theorem \ref{theorem:M=N=2K=2J} extends beyond the mutual-independence and continuous-density assumptions on the UE–AP channels, DL precoders, and DL data, and characterize the nondegenerate channel configurations for which the regression matrix remains full rank.

Background

The theoretical rank analysis assumes that all elements of the UE–AP channel matrices, DL precoding matrices, and DL data matrices are mutually independent random variables with positive density over the complex plane. Numerical experiments indicate that full rank can persist when this assumption is violated, such as under zero-forcing precoding or correlated Rayleigh fading, except in degenerate cases such as rank-one channel matrices.

The paper does not provide a theoretical extension covering these more realistic dependencies and channel correlations. A rigorous result would clarify when the proposed identifiability guarantees remain valid in practical propagation and precoding scenarios.

References

These results indicate that the sufficiency condition in Theorem \ref{theorem:M=N=2K=2J} may extend beyond Assumption~\ref{assum:VS}, except in degenerate cases where $\A$ loses rank (such as the $0\circ$ scenario above). A comprehensive numerical study (and theoretical extension) is required to make this precise, which we leave for future work.

— Leveraging Slowly Time-Varying AP-AP Channels for Interference Mitigation in Dynamic TDD  (2609.11669 - Andersson et al., 10 Sep 2026) in Section 8.5, subsection “Numerical Stability and Robustness”