Tomographic analysis of reflectometry data II: the phase derivative

A tomographic technique has been used in the past to decompose complex signals in its components. The technique is based on spectral decomposition and projection on the eigenvectors of a family of unitary operators. Here this technique is also shown to be appropriate to obtain the instantaneous phase derivative of the signal components. The method is illustrated on simulated data and on data obtained from plasma reflectometry experiments in the Tore Supra.

💡 Research Summary

The paper presents a novel application of tomographic signal analysis for the accurate estimation of the instantaneous phase derivative (i.e., instantaneous frequency) of individual components within a complex waveform. Building on the authors’ previous work, which demonstrated that a tomographic technique—based on projecting a signal onto the eigenvectors of a family of unitary operators—can separate overlapping reflectometry signals into orthogonal components, the current study extends the method to recover the phase evolution of each component and to compute its time derivative.

Theoretical framework
A signal (s(t)) is acted upon by a unitary operator (U(\theta)) parameterized by an angle (\theta). This operator effectively rotates the time‑frequency plane, generating a set of rotated versions of the original signal. For each (\theta) the signal is projected onto a complete orthonormal basis ({\psi_n}) that diagonalizes the operator, yielding complex coefficients
(c_n(\theta)=\langle\psi_n|U(\theta)s\rangle).
The magnitude (|c_n(\theta)|) indicates how strongly a particular basis vector contributes at a given rotation; peaks in the ((\theta,n)) plane correspond to distinct physical components (e.g., reflections from different plasma layers).

Phase reconstruction and derivative extraction
Once a component is identified, its complex envelope is reconstructed as (a_n(t)=A_n(t)\exp