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Nonstationary and nonlinear signals are ubiquitous in real life. Their decomposition and analysis is an important topic of research in signal processing. Recently a new technique, called Iterative Filtering, has been developed with the goal of decomposing such signals into simple oscillatory components. Several papers have been published regarding the analysis of this technique from a mathematical point of view. All these work start with the assumption that each compactly supported signal is extended outside the boundaries periodically. In this work we tackle the problem of studying the influence of different boundary conditions on the decompositions produce by the Iterative Filtering method.

Inspired by the theoretical results on optimal preconditioning stated by Ng, R.Chan, and Tang in the framework of Reflective boundary conditions (BCs), in this paper we present analogous results for Anti-Reflective BCs, where an additional technical difficulty is represented by the non orthogonal character of the Anti-Reflective transform and indeed the technique of Ng, R.Chan, and Tang can not be used. Nevertheless, in both cases, the optimal preconditioner is the blurring matrix associated to the symmetrized Point Spread Function (PSF). The geometrical idea on which our proof is based is very simple and general, so it may be useful in the future to prove theoretical results for new proposed boundary conditions. Computational results show that the preconditioning strategy is effective and it is able to give rise to a meaningful acceleration both for slightly and highly non-symmetric PSFs., Comment: 19 pages. arXiv admin note: substantial text overlap with arXiv:0710.5272