Approximated kernel density estimation for multiple TDOA detection

The Generalized State Coherence Transform (GSCT) has been recently proposed as an efficient tool for the estimation of multidimensional TDOA of multiple sources. The transform defines a multivariate likelihood of the TDOA through a non-linear integration of complex-valued states, representing the acoustic propagation of multiple sources. In the previous works the non-linearity was heuristically motivated leading to a difficult interpretation of the resulting likelihoods and of a correct choice of the parameters. Modeling the time-delays of the acoustic propagation of multiple sources with a multivariate multimodal distribution, a non-parametric kernel density estimator may be derived, which intrinsically accounts for spatial aliasing. From the theoretical analysis it follows that with an appropriate frequency-dependent non-linearity the GSCT likelihood approximates the true kernel density. Theoretical discussion is confirmed by experimental results which show that the proposed nonlinearity dramatically improves both resolution and smoothness of unidimensional and bidimensional likelihoods.