On estimation of covariance matrices with Kronecker product structure.

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 valid covariance matrices. Examples of such problems are channel modeling for multiple-input multiple-output (MIMO) communications and signal modeling 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) method for the associated estimation problem has been proposed previously. It is asymptotically efficient but has the drawback of requiring an iterative search for the maximum of the likelihood function. Two methods that are fast and noniterative are proposed in this paper. Both methods are shown to be asymptotically efficient. The first method is a noniterative variant of a well-known alternating 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 the same, it performs somewhat worse than the first estimator in small samples. In addition, the Cramer-Rao lower bound for the problem is derived in a compact form. The problem of estimating the Kronecker factors and the problem of detecting if the Kronecker structure is a good model for the covariance matrix of a set of samples are related. Therefore, the problem of detecting the dimensions of the Kronecker factors based on the minimum values of the criterion functions corresponding to the two proposed estimation methods is also treated in this work.