A new class of multi-dimensional Teager-Kaiser and higher order operators based on directional derivatives

This work aims at introducing some energy operators linked to Teager-Kaiser energy operator (TKEO) (Kaiser in On a simple algorithm to calculate the energy of a signal, pp 381–384, 1990), its associated higher order versions and expanding them to multidimensional signals. These operators are very useful for analysing oscillatory signals with time-varying amplitude and frequency (AM–FM). We first propose a new mathematical expression of these operators using directional derivatives along any n-D vector and Kronecker powers (Proposition 1, Sect. 3). This mathematical formulation allows us to extend to n-D case some properties of the classical TKEO such as tracking of AM envelope and instantaneous frequency of a multi-dimensional AM–FM signal. In addition, we have introduced a new scalar function using the directional derivative along a vector to recover the “sign” of the frequency components. Applications of this model to a local n-D AM–FM signal and the related demodulation errors are presented. To show the effectiveness and the robustness of the new class of operators in term of envelope and frequency tracking, results obtained on synthetic and real data are compared to multi-dimensional energy separation algorithm (Maragos and Bovik in J Opt Soc Am A 12:1867–1876, 1995) and to our previously developed method (Salzenstein and Boudraa in Signal Process 89(4):623–640, 2009). Finally, the performances of these methods are investigated in the presence of an additive noise. F. Salzenstein (B) Laboratoire INESS, UMR CNRS 7163 Université de Strasbourg, Strasbourg, France e-mail: [email protected] A. O. Boudraa IRENav, Ecole Navale/Arts et Métiers ParisTech (EA 3634), BCRM Brest, CC 600, 29240 Brest, France e-mail: [email protected] T. Chonavel Lab-STICC, UMR 3192, TELECOM Bretagne, Technopole Brest Iroise, 29238 Brest, France e-mail: [email protected] 123 ha l-0 07 04 16 8, v er si on 1 4 Ju n 20 12 Author manuscript, published in "Multidimensional Systems and Signal Processing (2012) ." DOI : 10.1007/s11045-012-0179-2 Multidim Syst Sign Process