Spectral and convergence analysis of the Discrete ALIF method

The Adaptive Local Iterative Filtering (ALIF) method is a recently proposed iterative procedure to decompose a signal into a finite number of “simple components” called intrinsic mode functions. It is an alternative to the well-known and widely used empirical mode decomposition method, and it has proved to be a powerful and promising algorithm. However, so far, no convergence analysis is available for the ALIF algorithm both in the continuous and discrete settings. In the present paper we focus on the discrete version of the ALIF method and we tackle the problem of studying its convergence properties. Using recent results about sampling matrices and the theory of generalized locally Toeplitz sequences — which we extend in this paper — we perform a spectral analysis of the ALIF iteration matrices, with a special attention to the eigenvalue clustering and the eigenvalue distribution. Based on the eigenvalue distribution, we formulate a necessary condition for the convergence of the Discrete ALIF method. Moreover, we provide a simple criterion to construct appropriate filters satisfying this condition. We also present several numerical examples in support of the theoretical study. Our contribution represents a first significant step toward a complete convergence analysis of the ALIF method, an analysis which appears to be rather difficult from a mathematical viewpoint as the ALIF iteration matrices possess a peculiar structure that, to the best of the authors' knowledge, has never been investigated in the literature.