On nonparametric spectral estimation

Petre Stoica,Tomas Sundin

doi:10.1007/bf01206681

Abstract

In this paper the Cramér-Rao bound (CRB) for a general nonparametric spectral estimation problem is derived under a local smoothness condition (more exactly, the spectrum is assumed to be well approximated by a piecewise constant function). Further-more, it is shown that under the aforementioned condition the Thomson method (TM) and Daniell method (DM) for power spectral density (PSD) estimation can be interpreted as approximations of the maximum likelihood PSD estimator. Finally the statistical efficiency of the TM and DM as nonparametric PSD estimators is examined and also compared to the CRB for autoregressive moving-average (ARMA)-based PSD estimation. In particular for broadband signals, the TM and DM almost achieve the derived nonparametric performance bound and can therefore be considered to be nearly optimal.

Highlights

The parametric approach to spectral estimation suers from a number of problems a fact that has motivated a renewed interest in the nonparametric approach
Answers to the following questions are of signi cant interest: (a) What is the best statistical performance in the class of nonparametric power spectral density (PSD) estimation methods, under some reasonable assumptions ?; (b) Is there any nonparametric PSD estimator that achieves the best statistical performance mentioned above?; (c) How do the best possible performances in the classes of parametric and nonparametric PSD estimation methods compare with one another?
Most papers in the literature do not address the above questions in any generality, but are limited to studies of speci c nonparametric PSD estimators, e.g., [2] [5]

Summary

INTRODUCTION

The parametric approach to spectral estimation suers from a number of problems (such as sensitivity to mismodeling) a fact that has motivated a renewed interest in the nonparametric approach. We show that two of the most successful nonparametric PSD estimators, viz. Thomson method (TM) [9] and Daniell method (DM) [3] can be interpreted as computationally convenient approximations to the nonparametric ML-based PSD estimator. This interpretation of the TM and DM provides new insights into the properties of these two methods and the relationship between them. To provide an answer to question (c) we compare the CRB for nonparametric PSD estimation derived here with the CRB for ARMA-based PSD estimation, in a number of cases

THE ML APPROACH

APPROXIMATE ML APPROACHES

THE CR BOUND

NUMERICAL EXAMPLE