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We propose a novel algorithm based on the matrix pencil method to estimate the parameters of a class of signals modeled as damped power-law phase signals. This class arises primarily from the electromagnetic probing of dispersive geological and civil engineering materials as a consequence of the universal dielectric response. When stratified media are considered, direct application of conventional matrix-shifting methods is hindered not only by the nonlinear frequency dependency which destroys the desired shift-invariance property of the data matrix, but also by the stratified structure of the medium which introduces a cumulative effect. In this regard, the proposed algorithm restores recursively the Vandermonde structure of one mode vector at a time by means of a spline-interpolation technique and then orthogonally projects it to filter out its contribution before passing to another. The algorithm is tested on simulated and experimental data resulting from the probing of a stratified dispersive medium, and its performance is assessed against the Cramér-Rao lower bound. For the example of experimental data, collected from concrete cores by means of a cylindrical transition line, the permittivities at the reference frequency and the dispersion indices are determined using the new algorithm and compared with those of a nonlinear optimization scheme.