Optimal signal reconstruction based on time-varying weighted empirical mode decomposition☆
Graphical abstract
Introduction
Signal reconstruction is an important task for practitioners, especially in the presence of disturbances. The method used is also crucial for this task, so that it should be applied to any type of signal since a great majority of signals encountered in the real world are nonlinear and non-stationary in nature. Thus, at this point, the method of empirical mode decomposition (EMD) developed by Huang et al. [1] helps practitioners. The EMD is an intrinsically adaptive data-driven method that decomposes any signal into a finite set of oscillation modes, termed, intrinsic mode functions (IMFs) and a residual function, ordered from high to low frequency. While the IMFs represent fast to slow oscillations, the residual function represents a monotonic mean trend in the signal. The sum of oscillation modes yields the original signal completely [1].
The EMD method has found widespread applications in various fields such as engineering, medicine, geology, earth and environmental sciences. In practice, when the EMD is used for de-noising purposes, an important issue is to accurately reconstruct the original signal from its noisy measurements. Numerous research studies on this issue are reported in the realms of applications like biomedical engineering [2], [3], [4], [5], power systems [6], speech enhancement [7], communication [8], sensor data processing [9], [10], and multi-rate high-resolution signal reconstruction [11], [12], [13]. In these applications the method of EMD is used as a de-noising tool by performing some operations on IMFs. For example, in [5], [7], [11], noise removal is carried out through hard thresholding by either keeping or discarding IMFs in a signal recovery scheme. Reconstructing the original signal by this way is principally based on the studies reported in [14], [15], [16] where detailed statistical analysis of the IMFs of a signal disturbed by fractional Gaussian noise and white Gaussian noise is introduced. It is deduced from these researches that most of information in a signal are contained in the high-order IMFs, while the low-order IMFs are dominated by noise. Furthermore, methods in [17], [18], [19] apply the hard thresholding approach to the clustered noise dominant IMFs. In some studies [12], [13], signal reconstruction is fulfilled by processing the remained IMFs so as to increase the de-noising performance. On the other hand, instead of discarding the noise dominant IMFs, signal reconstruction is carried out by using all IMFs in terms of applying some operations like filtering on the first high-order IMFs [2], [3], [4] or on the IMFs clustered as noise-led [8], [9]. Moreover, in [6], [10], various thresholding approaches such as level dependent hard- or soft-thresholding, interval thresholding and wavelet thresholding are applied on each IMF for eliminating noise effect on it. Methods in [20], [21], [22] are used for these purposes.
Note that the EMD-based methods mentioned up to now are not based on any optimality criterion for the signal reconstruction task. To the authors’ knowledge, there are only a few attempts related to this issue. Recently, in [23], three algorithms optimal in the minimum mean-square error (MMSE) sense are developed where the EMD followed by the linear weightings of the oscillation modes is offered. The first algorithm, termed optimal EMD (OEMD), approximates a given signal via the combination of the linearly weighted IMFs. The second algorithm, termed bidirectional OEMD (BOEMD), considers the correlations between samples of each IMF. Thus, a filter bank involving a finite impulse response filter for each one of the IMFs is determined. These filters are then individually applied on the IMFs for approximating the signal, in which a set of filter coefficients for weighting the IMFs is determined in the MMSE sense. Although the BOEMD has a more flexible structure than the OEMD, it suffers from solving ill-conditioned system of linear equations which results in unstable filter estimation and then unsatisfactory signal reconstruction performance. To tackle with the potential ill-condition problem of the BOEMD, a regularized BOEMD (RBOEMD) algorithm is proposed in [23]. In spite of the fact that both the OEMD and RBOEMD provide satisfactory performance over the traditional EMD in applications like signal de-noising, block-based natures of these algorithms restrict their noise removal performance, especially for real world signals where the characteristics of IMFs change with time.
In this paper, we focus our attention on deriving a new optimal signal reconstruction algorithm in the EMD-based framework. In this regard, the EMD followed by time-varying weightings of the oscillation modes is proposed. The proposed algorithm, termed time-varying weighted EMD (TVW-EMD), tries to reconstruct any signal in the MMSE sense through the time-varying weighted fusion of the oscillation modes. Determination of the time-varying weights assigned to each one of the oscillation modes is the backbone of the proposed algorithm, for which deterministic regression approach [24] is used. Inspiring from this approach, time-varying weights are determined by expanding them onto a finite set of orthonormal basis functions. Here we employed the discrete cosine transform (DCT) basis function because of its satisfactory reconstruction performance. The determination problem of the time-varying weights for all oscillation modes is therefore reduced to the problem of estimating time-invariant expansion coefficients in the MMSE sense. The signal recovery performance of the proposed algorithm is evaluated on real-life signals in the presence of different time-varying disturbances, and its effectiveness is compared with the existing counterparts so-called OEMD and RBOEMD.
The paper is outlined as follows. Section 2 is devoted to the preliminaries including notations, a short description of the EMD, and the problem statement. The algorithm is proposed in Section 3. Simulations are given in Section 4 for evaluating the performance of the proposed algorithm. Finally, concluding remarks are reported in Section 5.
Section snippets
Preliminaries
In this section, notations and a short description of the EMD in the derivation of the proposed algorithm are given. The problem statement is also addressed.
Proposed algorithm
As mentioned in Section 1, the EMD method has found widespread applications in reconstructing a signal from its measurements affected by various types of disturbances such as noise and interference. A great number of suggestions are made in relation to this issue, but only a few of them are based on an optimality criterion. To improve the reconstruction capability of the traditional EMD under noise constraints, two algorithms, termed OEMD and RBOEMD, originated from using the EMD followed by
Simulation examples
In order to demonstrate the performance of the proposed algorithm, we consider two examples involving real-life signals. In the first example, we study on the reconstruction of ECG signal from its baseline wander noise corrupted counterparts. The second example is related to the reconstruction of EEG signal from its ECG contaminated versions. We have resorted to PhysioBank database [25] to provide signals. From this database, the noise-free ECG and EEG recordings are extracted from the MIT-BIH
Conclusions and future works
A new signal reconstruction algorithm in the EMD-based framework is proposed. The algorithm, termed, MMSE-based TVW-EMD originates from the EMD followed by time-varying weightings of the oscillation modes arising from the traditional EMD of a signal. The proposed algorithm can be considered as the generalized version of the existing counterparts named as OEMD and RBOEMD, whereby instead of processing oscillation modes in a block-wise manner, the oscillation modes are processed over their
Aydin Kizilkaya received his B.Sc. and M.Sc. degrees (both in electrical and electronics engineering) from the Karadeniz Technical University and Pamukkale University, respectively, Ph.D. degree (in electronics and communication engineering) from the Technical University of Istanbul, Turkey. His research interests include digital signal processing and multidimensional signal processing.
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Aydin Kizilkaya received his B.Sc. and M.Sc. degrees (both in electrical and electronics engineering) from the Karadeniz Technical University and Pamukkale University, respectively, Ph.D. degree (in electronics and communication engineering) from the Technical University of Istanbul, Turkey. His research interests include digital signal processing and multidimensional signal processing.
Mehmet Dogan Elbi received his B.Sc. and M.Sc. degrees (both in electrical and electronics engineering) from the Pamukkale University. He is currently a research assistant and a Ph.D. student at the Pamukkale University. His research interests include digital signal processing, nonlinear modeling, and machine learning.
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Reviews processed and approved for publication by Editor-in-Chief.