Classifying Sums of Exponentially Damped Sinusoids Using an Associated Numerical Range

The matrix pencil method (MPM) is a well-known technique for estimating the parameters of exponentially damped sinusoids in noise by solving a generalized eigenvalue problem. However, in several cases, this is an ill-conditioned problem whose solution is highly biased under small perturbations. When the estimation is performed to classify the observed signal into two categories, the estimation errors induce several misclassifications. In this work we propose a novel signal classification criteria by exploiting the relationship between the generalized eigenvalue problem posed in the MPM and the numerical range of a pair of rectangular matrices. In particular, the classification test is formulated as a set inclusion problem, and no spectrum estimation is required. The technique is applied to a problem of electromagnetic scattering to classify dielectric materials using the scattering signal observed when a target is illuminated by an ultra-wideband signal. The performance of the classification scheme is assessed in terms of error rate and it is compared to another classification technique, the generalized likelihood rate test (GLRT).

Model Order Estimation for A Sum of Complex Exponentials

In this paper, we present a new method for estimating the number of terms in a sum of exponentially damped sinusoids embedded in noise. In particular, we propose to combine the shift-invariance property of the Hankel matrix associated with the signal with a constraint over its singular values to penalize small order estimations. With this new methodology, the algebraic and statistical structures of the Hankel matrix are considered. The new order estimation technique shows significant improvements over subspace-based methods. In particular, when a good separation between the noise and the signal subspaces is not possible, the new methodology outperforms known techniques. We evaluate the performance of our method using numerical experiments and comparing its performance with previous results found in the literature.