IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 50, NO. 8, AUGUST 2002 1831 Regularity and Strict Identifiability in MIMO Systems Terrence J. Moore, Brian M. Sadler, Member, IEEE, and Richard J. Kozick, Member, IEEE Abstract—We study finite impulse response (FIR) multi-input multi-output (MIMO) systems with additive noise, treating the fi- nite-length sources and channel coefficients as deterministic un- knowns, considering both regularity and identifiability. In blind estimation, the ambiguity set is large, admitting linear combina- tions 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 regu- larity (i.e., to a full-rank FIM and, thus, a meaningful Cramér–Rao bound). Exploiting the null space of the FIM, we show how param- eters must be specified to obtain a full-rank FIM, with implications for training sequence design in multisource systems. Together with constrained Cramér–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. Index Terms—Error statistics, identification, MIMO systems, parameter estimation. I. INTRODUCTION M ANY signal processing scenarios are well modeled by convolutive multi-input, multi-output (MIMO) finite impulse response (FIR) models. The MIMO model naturally includes the single-input case with both multiple (SIMO) and single outputs (SISO), with and without channel memory. Per- haps the minimal set of assumptions is based on modeling both the finite-length inputs and finite memory channel coefficients as deterministic unknowns, which is the viewpoint of this paper. In particular, we consider both regularity and identifiability issues in the blind estimation context when both the channel and the input are entirely unknown, and we assume that the channel is constant over the observation interval (block stationary model). In the blind scenario, the MIMO ambiguity set is large, ad- mitting linear combinations of the sources. Because of this, the MIMO Fisher information matrix (FIM) is not full-rank (i.e., regularity is not inherent). We find the FIM maximum rank (equivalently, the minimum nullity) and develop general nec- essary and sufficient conditions on the channel and input to achieve the maximum rank. The complex MIMO FIM has nul- lity , where is the number of sources. Theorem 2 shows Manuscript received October 16, 2001; revised March 8, 2002.The associate editor coordinating the review of this paper and approving it for publication was Dr. Rick S. Blum. T. J. Moore and B. M. Sadler are with the Army Research Laboratory, AMSRL-CI-CN, Adelphi, MD 20783 USA (e-mail: bsadler@arl.army.mil). R. J. Kozick is with the Department of Electrical Engineering, Bucknell Uni- versity, Lewisburg, PA 17837 USA (e-mail: kozick@bucknell.edu). Publisher Item Identifier 10.1109/TSP.2002.800416. that it is possible to have more unknowns than equations and still obtain the minimum nullity of the FIM (as long as the dif- ference between unknowns and equations is ). Theorem 2 also shows that in order to achieve the minimum FIM nullity in the blind scenario, it is necessary to have more diversity chan- nels than sources, i.e., . When , then the FIM nullity grows with increasing data size , but this is not true when . Thus, when , the degrees of uncer- tainty might be overcome with a fixed amount of training. Given the general lack of invertibility of the FIM in the blind case, it is of interest to know what minimal set of parameters could be made known for the resulting FIM to be full-rank and, therefore, result in a useful Cramér–Rao bound (CRB). The- orem 4 shows how to specify at least parameters to achieve a full-rank FIM and thereby develop a corresponding CRB. This indicates the minimal MIMO training set. In addition to directly specifying parameters, there are other pathways to achieving regularity and, hence, a useful CRB. Constrained CRBs provide one approach, enabling inclusion of many forms of side information such as training, constant modulus sources, and array calibration. Within the constrained CRB framework, parameter specification is a special case of ar- bitrary nonlinear equality constraints. These constraints, which have been discussed elsewhere [11], [12], [16], [21], provide a means for obtaining appropriate CRBs when more information on the sources or channels is available. See [12] and [21] for examples of convolutive MIMO CRBs, including training (semi-blind) and constant modulus (CM) source cases. The memoryless MIMO case is considered in [11] for calibrated arrays and in [10] for uncalibrated arrays. In Section V, we also consider invariants, which are useful in the SIMO context [16] but appear to be of limited utility in the MIMO case. Regularity in the FIM implies identifiability (ID), although the converse is not always true, e.g., see [4]. In the SIMO case, Hua and Wax have established the equivalence of these [6], as well as equivalence with a form of identifiability that be- comes apparent through the cross relation method for SIMO channel identification [20]. We consider strict identifiability in the MIMO case and develop necessary and sufficient conditions. This builds on the work of Abed-Meraim and Hua [1], where some MIMO results for strict ID are presented (without proofs); see also the review article [2]. On the whole, the conditions for the FIM nullity to be a min- imum, and for strict ID, are very nearly equivalent. We show that for MIMO systems, the necessary conditions for the nullity of the FIM to be a minimum, and strict ID, are equivalent, ex- cept for required data length. Sufficient conditions are also very nearly identical. The necessary results provide a lower bound on the data size , whereas the sufficient conditions provide an upper bound. The strict ID analyses are worst case with respect 1053-587X/02$17.00 © 2002 IEEE