Calculation of the Fisher information matrix for multidimensional data sets

Data sets that are acquired in many practical systems can be described as the output of a multidimensional linear separable-denominator system with Gaussian measurement noise. An important example is nuclear magnetic resonance (NMR) spectroscopy. In NMR spectroscopy, high-accuracy parameter estimation is of central importance. A classical result on the Cramér–Rao lower bound states that the inverse of the Fisher information matrix (FIM) provides a lower bound for the covariance of any unbiased estimator of the parameter vector. The calculation of the FIM is therefore of central importance for an assessment of the accuracy with which parameters can be estimated. It is shown how the FIM can be expressed using the matrices that determine the system that generates the data set. For uniformly sampled data, it is shown how the FIM can be expressed through the solutions of Lyapunov equations. The novel techniques are demonstrated with an example arising from NMR spectroscopy.