On Kronecker and Linearly Structured Covariance Matrix Estimation

The estimation of covariance matrices is an integral part of numerous signal processing applications. In many scenarios, there exists prior knowledge on the structure of the true covariance matrix; e.g., it might be known that the matrix is Toeplitz in addition to Hermitian. Given the available data and such prior structural knowledge, estimates using the known structure can be expected to be more accurate since more data per unknown parameter is available. In this work, we study the case when a covariance matrix is known to be the Kronecker product of two factor matrices, and in addition the factor matrices are Toeplitz. We devise a two-step estimator to accurately solve this problem: the first step is a maximum likelihood (ML) based closed form estimator, which has previously been shown to give asymptotically (in the number of samples) efficient estimates when the relevant factor matrices are Hermitian or persymmetric. The second step is a re-weighting of the estimates found in the first steps, such that the final estimate satisfies the desired Toeplitz structure. We derive the asymptotic distribution of the proposed two-step estimator and conclude that the estimator is asymptotically statistically efficient, and hence asymptotically ML. Through Monte Carlo simulations, we further show that the estimator converges to the relevant Cramér-Rao lower bound for fewer samples than existing methods.