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

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

Recovery of periodicities hidden in heavy-tailed noise

We address a parametric joint detection-estimation problem for discrete signals of the form $x(t) = \sum_{n=1}^{N} \alpha_n e^{-i \lambda_n t } + \epsilon_t$, $t \in \mathbb{N}$, with an additive noise represented by independent centered complex random variables $\epsilon_t$. The distributions of $\epsilon_t$ are assumed to be unknown, but satisfying various sets of conditions. We prove that in the case of a heavy-tailed noise it is possible to construct asymptotically strongly consistent estimators for the unknown parameters of the signal, i.e., the frequencies $\lambda_n$, their number $N$, and complex amplitudes $\alpha_n$. For example, one of considered classes of noise is the following: $\epsilon_t$ are independent identically distributed random variables with $\mathbb{E} (\epsilon_t) = 0$ and $\mathbb{E} (|\epsilon_t| \ln |\epsilon_t|) < \infty$. The construction of estimators is based on detection of singularities of anti-derivatives for $Z$-transforms and on a two-level selection procedure for special discretized versions of superlevel sets. The consistency proof relies on the convergence theory for random Fourier series. We discuss also decaying signals and the case of infinite number of frequencies.Comment: This e-print differs in style, formatting, pagination, and small non-mathematical details from the version of this paper accepted for publication in "Mathematische Nachrichten" (Wiley-VCH). In comparison with the previous version, the following changes have been made: The introduction section 1 was split into two sections 1 and 2. The new section 5 was added. The algorithm 4.1 was adde

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

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. Keywords: Two-dimensional random fields; model order selection; least squares estimation; strong consistency

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

The problem of jointly estimating the number as well as the parameters of twodimensional sinusoidal signals, observed in the presence of an additive colored noise field is considered. We begin by establishing the strong consistency of the non-linear least squares estimator of the parameters of two-dimensional sinusoids, when the number of sinusoidal signals assumed in the field is incorrect. Based on these results, we prove the strong consistency of a new family of model order selection rules