On the Gaussian Cramér-Rao Bound for Blind Single-Input Multiple-Output System Identification: Fast and Asymptotic Computations

The Cramér-Rao Bound (CRB) is a powerful tool to assess the performance limits of a parameter estimation problem for a given statistical model. In particular, the Gaussian CRB (i.e., the CRB obtained assuming the data are Gaussian) corresponds to the worst case; giving the largest CRB among a large class of data distributions. This makes it very useful in practice since optimizing under the Gaussian data assumption can be interpreted as a min-max optimization (i.e., minimizing the largest CRB). The Gaussian CRB is also the corresponding bound of Second-Order Statistics (SOS)-based estimation methods, which are frequently used in practice. Despite its practicality, computing this bound might be cumbersome in some cases, particularly in the case where the input is assumed deterministic and has a large number of samples. In this paper, we address this computational issue by proposing a fast computation for the deterministic Gaussian CRB of Single-Input Multiple Output (SIMO) blind system identification. More precisely, we exploit circulant matrix properties to reduce the cost from cubic to quadratic with respect to the sample size. Moreover, we derive a closed-form formula for the asymptotic (large sample size) Gaussian CRB and show how it can be computed using the residue theorem.