Reduced Mean-Square Error Quadratic Inverse Spectrum Estimator

A new spectrum estimator is introduced. The new estimator exploits quadratic-inverse theory to attain improved mean-square error performance over the standard multitaper spectrum estimators. The standard, non-adaptive, eigenvalue weighted multitaper estimator is obtained by averaging a high-resolution inconsistent spectrum estimator over the estimator bandwidth. The improved performance of the proposed estimator results from replacing this average with a weighted average computed in the space spanned by the quadratic-inverse basis. The weighting, determined analytically, is chosen such that the resulting estimator minimizes the sum of the variance and the square of the in-band bias; neglecting bias due to spectral leakage and potential bias due to the possible incompleteness of the quadratic-inverse basis. For a white spectrum the neglected bias is found to be as small as that of a standard, non-adaptive multitaper spectrum estimator. The relative reduction of the mean-square error of the proposed spectrum estimator is validated by simulation for an ARMA(4,2) process, and results in a typical mean-square error reduction of 5% for large time-bandwidth parameters and 20% for a time-bandwidth parameter of four, when compared to the non-adaptive, non-eigenvalue weighted multitaper estimator. When compared to the adaptive multitaper spectrum estimator, larger mean-square error improvements are attainable. An expression for the theoretical probability density function for the proposed estimator is given. It is found to be as accurate as the asymptotic probability density function for the standard multitaper estimator.