A grid-based nonlinear approach to noise reduction and deconvolution for coupled systems

Highlights

• Nonlinear denoising by leveraging the reference signal of the same dynamical system.

• Deconvolution method to reconstruct time-resolved data.

• Time-series data with noise amplitude of 100% are denoised and deconvolved.

Abstract

To varying degrees, all experimental measurements are corrupted by real-world noise sources including electronic noise in the acquisition system, far-field perturbations in the surrounding environment, and local physical phenomena. This paper presents a grid-based nonlinear analysis technique for causally coupled systems which leverages the availability of a high-fidelity reference signal to effectively denoise a target measurement signal. The foundation of this approach, essentially an ensemble-averaging procedure in multidimensional phase space, is strongly motivated by work from dynamical systems theory. Furthermore, a straightforward extension of this technique allows for recovery of a time-resolved representation from the underresolved measurement signal. Both the nonlinear noise reduction and this temporal deconvolution extension are applied to signals from three different coupled dynamic systems (sine wave system, Lorenz system, and Hall-effect thruster) to demonstrate effectiveness on periodic, chaotic, and experimental systems.

Introduction

Since the goal of experimental measurements is to more fully understand the state of a physical system, dealing with the ubiquitous presence of noise is a constant challenge for experimental design. From a strictly theoretical perspective, noise is completely independent from the system being measured; however, nature provides virtually limitless coupling pathways which prevents absolute decoupling of the two. Therefore, this paper is limited to considering experimental noise as having a sufficiently weak coupling as to not significantly affect the dynamics of the physical system under study. This distinction causes challenges in the context of the infinitesimal sensitivity of chaotic systems and will therefore require the additional constraint of application to bounded systems. Nevertheless, independent of the distribution of noise (e.g. random, Gaussian, etc.), so long as it has an algebraic mean of zero, measurement of sufficient amount of data is the underpinning of virtually all strategies to effectively denoise experimental measurements.

The traditional approach to noise reduction in a single signal is to simply assume that it is faster than the true signal; this opens up a huge variety of linear “smoothing” techniques. Two simple examples of linear smoothing include performing an FFT decomposition of the signal and reconstructing it with only low frequency components of the decomposition (i.e. choosing a cutoff frequency) or convoluting the signal with some sort of filter (e.g. top-hat filter) in time. A variety of nonlinear noise reduction technique have also been successfully proposed. The majority of this work has focused on denoising through projection of time-series data onto a high-dimensional phase space (i.e. inspired by time-delay embedding) and subsequent averaging/filtering of the resulting manifold [1], [2], [3], [4], [5], [6], [7], [8], [9].

Beyond single signal filtering techniques, it is also possible to leverage additional information embedded in other signals gathered from the same system to perform noise reduction. A simple, yet highly effective example of this was demonstrated by Sternickel et al. [10], who used two signals measured simultaneously. One of the measurement contained a clean signal contaminated by noise, and the other only contained only the noise. This “local” approach to denoising has been used broadly and with great success, but only works if the measurements are taken at the same spatial volume and only if the system state can be accurately synchronized to the acquisition of the desired measurement.

The crux of this paper is to demonstrate a new strategy for leveraging the availability of high-fidelity reference probe from one part of the system to denoise low-fidelity measurement signal sampled from a different spatial location in the system. This technique has particular promise for quasi-periodic bounded systems commonly associated with chaotic limit-cycle behavior, and with sufficient data, a straightforward extension also allows for temporal deconvolution of the measurement signal. The structure of the paper is as follows. In Section 2, a short discussion of the relevance of quasi-periodic bounded systems and previous denoising strategies for a prototypical chaotic limit-cycle system, the Hall-effect thrusters (HETs), is discussed. Next, Section 3 discusses the motivation for high-dimensional phase-space representations from dynamical systems theory and the underlying basis for temporal deconvolution. This is followed by Section 4, which provides details on both algorithms and numerical implementations used in this work. In Section 5, both nonlinear noise reduction and temporal deconvolution are applied to signals from three different coupled dynamic systems (sine wave system, Lorenz system, and Hall-effect thruster) and discussion of their performance relative to linear filtering methods is provided. Finally, in Section 6, further research areas and other potential applications of these methods are identified and discussed.