Time-domain identification of spectral signal subspaces of rational type with application to the broadband sources localization problem

The estimation problem for rational-type spectral signal subspaces in the time domain is discussed. Two methods are presented. The first is the transcription of the methods proposed by K.M. Buckley and L.J. Griffiths (1988) to the rational modeling context. It is based on the fact that a one-to-one correspondence exists between the polynomial vector-valued functions of the noise spectral subspace and the eigenspace corresponding to the smallest eigenvalue of the covariance matrix of the data. In the second method, the authors exploit the fact that the noise spectral subspace admits rational-type orthonormal bases related to the causal rational transfer functions with coisometric values on the unit circle, the action of which on the data produces a minimum energy output. This optimization problem is solved by using an efficient parametrization of these transfer functions in conjunction with a gradient algorithm, the initial conditions of which are calculated by using the first method. By contrast with the first approach, it is shown that the corresponding estimator is related to a quadratic reconstruction criterion which can be interpreted as a maximum-likelihood criterion in a particular case