Dilation Operator Approach and Square Root Velocity Transform for Time/Doppler Spectra Characterization on SU(n)

Abstract

We propose in this work the use of Dilation theory for non-stationary signals and their time/Doppler spectra to embed the underlying spectral measure on the Special Unitary group SU(n). The Dilation theory gives access to rotation-like matrices built in with partial correlation coefficients. Due to the non-stationary condition, the time/Doppler spectra is associated with a path on SU(n). We use next the Square root Velocity Transform which has been proven to be equivalent to a first order Sobolev metric on the space of shapes. Because the metric in the space of curves naturally extends to the space of shapes, this enables a comparison between curves’ shapes and allows then the classification of time/Doppler spectra.