Strongly Consistent Model Order Selection for Estimating 2-D Sinusoids in Colored Noise

TL;DR

This paper introduces a new model order selection method for accurately estimating the number and parameters of 2-D sinusoidal signals in colored noise, demonstrating strong consistency even with incorrect initial assumptions.

Contribution

The paper presents a novel strongly consistent model order selection rule for 2-D sinusoidal signals in colored noise, with theoretical proofs of convergence.

Findings

Almost sure convergence of least squares estimates when under-estimating the number of sinusoids

Estimates contain correct parameters even when over-estimating the number of sinusoids

Proven strong consistency of the new model order selection rule

Abstract

We consider the problem of jointly estimating the number as well as the parameters of two-dimensional sinusoidal signals, observed in the presence of an additive colored noise field. We begin by elaborating on the least squares estimation of 2-D sinusoidal signals, when the assumed number of sinusoids is incorrect. In the case where the number of sinusoidal signals is under-estimated we show the almost sure convergence of the least squares estimates to the parameters of the dominant sinusoids. In the case where this number is over-estimated, the estimated parameter vector obtained by the least squares estimator contains a sub-vector that converges almost surely to the correct parameters of the sinusoids. Based on these results, we prove the strong consistency of a new model order selection rule.