Model Order Selection for Complex Sinusoids in the Presence of Unknown Correlated Gaussian Noise

We consider the problem of detecting and estimating the amplitudes and frequencies of an unknown number of complex sinusoids based on noisy observations from an unstructured array. In parametric detection problems like this, information theoretic criteria such as minimum description length (MDL) and Akaike information criterion (AIC) have previously been used for joint detection and estimation. In our paper, model selection based on extreme value theory (EVT), which has previously been used for enumerating real sinusoidal components from one-dimensional observations, is generalized to the case of multidimensional complex observations in the presence of noise with an unknown spatial correlation matrix. Unlike the previous work, the likelihood ratios considered in the mutlidimensional case cannot be addressed using Gaussian random fields. Instead, chi-square random fields associated with the generalized likelihood ratio test are encountered and EVT is used to analyze the model order overestimation probability for a general class of likelihood penalty terms including MDL and AIC, and a novel likelihood penalty term derived based on EVT. Since the exact EVT penalty term involves a Lambert-W function, an approximate penalty term is also derived that is more tractable. We provide threshold signal-to-noise ratios (SNRs) and show that the model order underestimation probability is asymptotically vanishing for EVT and MDL. We also show that MDL and EVT are asymptotically consistent while AIC is not, and that with finite samples, the detection performance of EVT outperforms MDL and AIC. Finally, the accuracy of the derived threshold SNRs is also demonstrated.