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Regularity and strict identifiability in MIMO systems

We study finite impulse response (FIR) multi-input multi-output (MIMO) systems with additive noise, treating the finite-length sources and channel coefficients as deterministic unknowns, considering both regularity and identifiability. In blind estimation, the ambiguity set is large, admitting linear combinations of the sources. We show that the Fisher information matrix (FIM) is always rank deficient by at least the number of sources squared and develop necessary and sufficient conditions for the FIM to achieve its minimum nullity. Tight bounds are given on the required source data lengths to achieve minimum nullity of the FIM. We consider combinations of constraints that lead to regularity (i.e., to a full-rank FIM and, thus, a meaningful Cramer-Rao bound). Exploiting the null space of the FIM, we show how parameters must be specified to obtain a full-rank FIM, with implications for training sequence design in multisource systems. Together with constrained Cramer-Rao bounds (CRBs), this approach provides practical techniques for obtaining appropriate MIMO CRBs for many cases. Necessary and sufficient conditions are also developed for strict identifiability (ID). The conditions for strict ID are shown to be nearly equivalent to those for the FIM nullity to be minimized.