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This paper is concerned with the inverse problem of recovering the unknownsignal components, along with extraction of their instantaneous frequencies(IFs), governed by the adaptive harmonic model (AHM), from discrete (andpossibly non-uniform) samples of the blind-source composite signal.  None of the existing decomposition methods and algorithms, including the mostpopular empirical mode decomposition (EMD) computational scheme and its currentmodifications, is capable of solving this inverse problem.  In order to meet the AHM formulation and to extract the IFs of the decomposedcomponents, called intrinsic mode functions (IMFs), each IMF of EMD is extendedto an analytic function in the upper half of the complex plane via the Hilberttransform, followed by taking the real part of the polar form of the analyticextension.  Unfortunately, this approach most often fails to resolve the inverse problemsatisfactorily.  More recently, to resolve the inverse problem, the notion of synchrosqueezedwavelet transform (SST) was proposed by Daubechies and Maes, and furtherdeveloped in many other papers, while a more direct method, called signalseparation operation (SSO), was proposed and developed in our previous workpublished in the journal, Applied and Computational Harmonic Analysis, vol.30(2):243-261, 2016.  In the present paper, we propose a synthesis of SSO using a deep neuralnetwork, based directly on a discrete sample set, that may be non-uniformlysampled, of the blind-source signal.  Our method is localized, as illustrated by a number of numerical examples,including components with different signal arrival and departure times.  It also yields short-term prediction of the signal components, along withtheir IFs.  Our neural networks are inspired by theory, designed so that they do notrequire any training in the traditional sense.

Theory inspired deep network for instantaneous-frequency extraction and signal components recovery from discrete blind-source data