Sufficient Conditions for Regularity and Strict Identifiability in MIMO Systems

Abstract

We consider regularity and identifiability of convolutive multi-input multi-output (MIMO) systems with additive white Gaussian noise, modeling the source and finite impulse response (FIR) channel sequences as deterministic unknowns. In the blind context, the MIMO system is not locally identifiable; hence, its Fisher information matrix (FIM) is not regular. In fact, the dimension of the complex-valued blind FIM null space is at least the number of sources squared. Because the FIM is singular, additional information about the system is required to resolve the degrees of uncertainty and thereby obtain a valid Cramer-Rao bound (CRB); therefore, it is of interest to know under what conditions the blind FIM nullity reaches its lower bound. We develop sufficient conditions for the complex FIM to attain its minimum nullity, refining previous necessary conditions, and extending single-input multi-output (SIMO) results. We show that the sufficient conditions for the complex FIM to have minimum nullity are also equivalent to sufficient conditions for MIMO strict identifiability. These provide sufficient conditions on the richness of the sources, the required diversity, and the source lengths. Under these conditions, additional constraints, such as training, may be employed to yield an identifiable system with no ambiguities remaining.