Kronecker Structured Covariance Matrix Estimation

The estimation of signal covariance matrices is a crucial part of many signal processing algorithms. In some applications, the structure of the problem suggests that the underlying, true, covariance matrix is the Kronecker product of two matrices. Examples of such problems are channel modelling for MIMO communications and signal modelling of EEG data. In applications it may also be that the Kronecker factors in turn can be assumed to possess additional, linear, structure. The maximum likelihood (ML) estimator for the problem has been proposed previously. It is asymptotically efficient but has the drawback of requiring an iterative search. Two methods that are both non-iterative and asymptotically efficient are proposed in this paper. The first method is derived from a well-known iterative maximization technique for the likelihood function. It performs on par with ML in simulations, but has the drawback of not allowing for extra structure in addition to the Kronecker structure. The second method is based on covariance matching principles, and does not suffer from this drawback. However, while the large sample performance is shown to be identical to ML, it performs somewhat worse in small samples than the first estimator. In addition, the Cramer-Rao lower bound (CRB) for the problem is derived in a compact form.