Fast computation of the exact FIM for deterministic signals in colored noise

A fast algorithm for computing the exact finite-sample Fisher information matrix (FIM) for the parameters of a deterministic signal observed in Gaussian AR noise is derived. In the case of a harmonic signal with random phases, closed-form expressions for the finite-sample posterior Cramer-Rao bound (PCRB) are established. It is shown that the fast algorithm is also useful for computing the conditional CRB when the additive noise is a non-Gaussian AR process. It is seen that the asymptotic CRB may deviate significantly from the exact CRB even when the data length is moderate, whereas the PCRB, which is easy to compute, provides a better approximation. Theoretical results are illustrated via numerical evaluation of the different lower bounds