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

Charles K. Chui, Ningning Han+1 more|arXiv (Cornell University)|Jan 31, 2020

Blind Source Separation Techniques16 references4 citations

TL;DR

This paper proposes a theory-inspired deep neural network that extracts instantaneous frequencies and recovers individual signal components from non-uniformly sampled, blind-source composite signals without traditional training. Based on a rigorous mathematical framework, the method precisely identifies the number of components and achieves super-resolution decomposition, outperforming EMD, SST, and prior SSO methods in accuracy and robustness, even with noise or extrapolation beyond data boundaries.

ABSTRACT

This paper is concerned with the inverse problem of recovering the unknown signal components, along with extraction of their instantaneous frequencies (IFs), governed by the adaptive harmonic model (AHM), from discrete (and possibly non-uniform) samples of the blind-source composite signal. None of the existing decomposition methods and algorithms, including the most popular empirical mode decomposition (EMD) computational scheme and its current modifications, is capable of solving this inverse problem. In order to meet the AHM formulation and to extract the IFs of the decomposed components, called intrinsic mode functions (IMFs), each IMF of EMD is extended to an analytic function in the upper half of the complex plane via the Hilbert transform, followed by taking the real part of the polar form of the analytic extension. Unfortunately, this approach most often fails to resolve the inverse problem satisfactorily. More recently, to resolve the inverse problem, the notion of synchrosqueezed wavelet transform (SST) was proposed by Daubechies and Maes, and further developed in many other papers, while a more direct method, called signal separation operation (SSO), was proposed and developed in our previous work published 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 neural network, based directly on a discrete sample set, that may be non-uniformly sampled, 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 with their IFs. Our neural networks are inspired by theory, designed so that they do not require any training in the traditional sense.

Motivation & Objective

• To solve the inverse problem of recovering signal components and their instantaneous frequencies from discrete, possibly non-uniform samples of a blind-source composite signal.
• To overcome the limitations of existing methods like EMD, SST, and traditional SSO, which fail to reliably resolve the inverse problem under general conditions.
• To develop a deep network architecture that is mathematically grounded and does not require classical backpropagation training.
• To enable accurate short-term prediction and extrapolation of signal components beyond the observed time interval.
• To achieve super-resolution decomposition that precisely identifies the true number of components without prior knowledge.

Proposed method

• The method is based on a theoretical framework derived from the adaptive harmonic model (AHM), which models non-stationary signals as a sum of amplitude- and frequency-modulated sinusoids.
• It extends the signal separation operation (SSO) into a deep neural network architecture, designed directly from discrete, non-uniform samples.
• The network is constructed using analytic signal extensions via the Hilbert transform and polar representation of analytic functions to extract instantaneous frequencies.
• The architecture is pre-fabricated using theoretical insights, eliminating the need for backpropagation-based training.
• The method ensures localization and stability by embedding mathematical constraints into the network structure, enabling robust component separation.
• It supports both reconstruction and short-term prediction of signal components, including extrapolation into unobserved time intervals.

Experimental results

Research questions

• RQ1 Can a deep neural network be constructed without traditional training to accurately recover signal components and their instantaneous frequencies from non-uniformly sampled blind-source data?
• RQ2 Does the proposed method outperform EMD, SST, and prior SSO approaches in resolving the inverse problem of signal decomposition?
• RQ3 Can the network automatically determine the correct number of signal components without prior knowledge of the composite signal’s structure?
• RQ4 How well does the method perform in the presence of noise or when extrapolating beyond the observed time interval?
• RQ5 Can the method achieve super-resolution decomposition, especially when instantaneous frequencies are closely spaced?

Key findings

• The proposed method successfully recovers all signal components and their instantaneous frequencies from non-uniformly sampled data, with a mean squared error (MSE) of $1.74 \times 10^{-2}$ for the fifth IMF in the bat echo-location example.
• In the extrapolation example, the method accurately predicted signal components and instantaneous frequencies over intervals $[-0.1, 0]$ and $[30, 30.1]$, demonstrating robust short-term prediction capability.
• For the bat echo-location signal, the method correctly identified five IMFs with MSEs below $1.74 \times 10^{-2}$, confirming high-fidelity recovery.
• The network achieved exact component count detection without prior knowledge, validating its ability to solve the inverse problem in a principled way.
• The method demonstrated superior performance over EMD and SST in resolving closely spaced instantaneous frequencies and handling non-uniform sampling.
• Theoretical analysis and numerical experiments confirm that the method is stable and effective even under noisy conditions, as proven in Theorem 2.1 and verified in Section 5.