Model Selection for Sinusoids in Noise: Statistical Analysis and a New Penalty Term

Detection of the number of sinusoids embedded in noise is a fundamental problem in statistical signal processing. Most parametric methods minimize the sum of a data fit (likelihood) term and a complexity penalty term. The latter is often derived via information theoretic criteria, such as minimum description length (MDL), or via Bayesian approaches including Bayesian information criterion (BIC) or maximum a posteriori (MAP). While the resulting estimators are asymptotically consistent, empirically their finite sample performance is strongly dependent on the specific penalty term chosen. In this paper we elucidate the source of this behavior, by relating the detection performance to the extreme value distribution of the maximum of the periodogram and of related random fields. Based on this relation, we propose a combined detection-estimation algorithm with a new penalty term. Our proposed penalty term is sharp in the sense that the resulting estimator achieves a nearly constant false alarm rate. A series of simulations support our theoretical analysis and show the superior detection performance of the suggested estimator.