Detrended fluctuation thresholding for empirical mode decomposition based denoising

Abstract

Signal decompositions such as wavelet and Gabor transforms have successfully been applied in denoising problems. Empirical mode decomposition (EMD) is a recently proposed method to analyze non-linear and non-stationary time series and may be used for noise elimination. Similar to other decomposition based denoising approaches, EMD based denoising requires a reliable threshold to determine which oscillations called intrinsic mode functions (IMFs) are noise components or noise free signal components. Here, we propose a metric based on detrended fluctuation analysis (DFA) to define a robust threshold. The scaling exponent of DFA is an indicator of statistical self-affinity. In our study, it is used to determine a threshold region to eliminate the noisy IMFs. The proposed DFA threshold and denoising by DFA–EMD are tested on different synthetic and real signals at various signal to noise ratios (SNR). The results are promising especially at 0 dB when signal is corrupted by white Gaussian noise (WGN). The proposed method outperforms soft and hard wavelet threshold method.

Introduction

The empirical mode decomposition (EMD) is an alternative method to analyze non-linear and non-stationary signals [1]. EMD breaks signal down into a finite number of amplitude and frequency modulated (AM/FM) zero-mean oscillations called intrinsic mode functions (IMFs). In contrast to wavelet decomposition, IMFs are expressed as the signal dependent semi-orthogonal basis functions via an iterative algorithm called sifting. However, they have fluctant frequency spectrum caused by mode-mixing effect. There are several attempts to reduce fluctuation or express an IMF with a single component [2]. On the other hand, it is another challenging study to explain the meaning of each IMF or determine which IMF refers to noisy oscillations, which is the generalized task of the EMD based denoising. While noisy IMFs may be determined manually observing the periodicity of the oscillations in the required range [3], some automated methods have been studied. Wu and Huang [4] deployed a hypothesis test method to find out the relevant information level of the IMFs, which is reported to perform poorly for low frequency oscillations. Information theoretical based approaches such as mutual information and relative entropy are applied to find noisy oscillations [5], [6]. In addition to these time-domain characteristics of IMFs, frequency domain characteristics of the EMD is investigated and modeled as a dyadic filter bank resulting from the decomposition of white Gaussian noise (WGN ) [7] or fractional Gaussian noise (fGn) [8], [9]. From this point of view, if the energy distribution of IMFs for noise-only signal is known, the discrepancy between energy of noisy-signal IMFs and noise-only IMFs indicates the presence of the relevant informative oscillations.

Our suggestion is to determine noisy IMF resulting from the decomposition of noisy-signal using a reliable metric. Detrended fluctuation analysis (DFA) [10] is a successful method to measure long-range dependency for non-stationary time series [11], [12]. The special cases $\alpha =0.5$, $\alpha =1$ and $\alpha =1.5$ correspond to completely uncorrelated white noise, pink noise and Brownian noise. When $0<\alpha <0.5$, the signal is called “anti-correlated”, in which large fluctuations are likely to be followed by small ones. While it increases from 0.5 to 1, temporal correlations are persistent. If $\alpha >1$, the correlations do not exhibit power-law behavior [13]. The slope, α can also be considered as an indicator of roughness [14]: the larger the value, the smoother time series or slower fluctuations. From this point of view, DFA can be used as a robust metric to identify noisy IMFs. The proposed method is to determine noisy IMFs resulting from the decomposition of noisy-signal using a reliable metric which is independent of a comparison or referencing with the signal. The IMFs are tested by DFA to measure their statistical properties, and the DFA based threshold is applied to exclude IMFs which contain mostly noise. The suggested method is tested on synthetic and real EEG signals to show its denoising capability comparing to wavelet threshold methods.

The remainder of the paper is organized as follows: Section 2 provides a short description of EMD. Signal denoising and thresholding are described in Section 3. Section 4 summarizes the DFA and explores its thresholding capability. The suggested DFA–EMD based denoising is presented in Section 5. Consequently, in Section 6 simulation results of the DFA–EMD method are examined, and the conclusions are drawn in Section 7.