Robustness of EFD in Noisy Conditions: A Comparative Study with EWT for EEG and Synthesized Signal

The decomposition of EEG signals into constituent components is considered a crucial step to help in a better and more accurate signal analysis. Over the years, various signal decomposition methods have been developed, each offering approaches to representing signals.

A signal decomposition method breaks down a signal into constituent parts or sub-signals [1,2]. These individual components can then be studied separately to understand the underlying properties facilitating feature extraction, analysis, and learning [3]. Signal decomposition is essential for getting helpful information from complex signals in fields like biomedical engineering. By breaking down a complex signal into parts, such as non-stationary elements, we can improve algorithms to minimize noise and artifacts and better understand the main processes. However, the limitations of traditional signal decomposition methods have led to the development of alternative approaches in recent years. Fourier transform (FT), and wavelet transform (WT) are the two traditional and well-known methods for signal analysis. The EWT, introduced by Gilles [4], offers superior resolution for non-stationary and unstable signals, with the ability to prevent mode mixing [4]. The EFD technique, introduced in 2019, extends analysis capabilities to non-linear and non-stationary signals, overcoming mode mixing challenges for precise signal decomposition [5,6].

While EWT finds applications in different fields, including biomedical engineering, particularly in analyzing biological signals such as Electrocardiography (ECG) and EEG signals, EFD’s potential in different bio signal analyses is yet to be explored [7,8]. This aspect presents an intriguing area for future research and application, inspiring us to focus on this topic. Both EWT and EFD rely on data-based decomposition, which can limit their generalizability compared to methods based on predefined mathematical functions. Traditional signal analysis methods have severe limitations in handling non-stationary signals, uncertainty issues, and generalizing different bio signals. This issue makes us look for more robust and more reliable techniques.

This paper uses synthesized and real EEG data to compare two empirical decomposition methods, EWT and EFD. Also, we calculate the signal-to-noise (SNR), percentage root mean square error (PRMSE), and correlation (Corr) for the clean and noisy signal. Our research results indicate the EFD method’s superiority over the EWT. This achievement can fully demonstrate the robustness of EFD in noisy conditions.

The paper is structured as follows: Section 2 explains the EWT and EFD as the two decomposition methods. Section 3 covers the datasets utilized in the study, encompassing synthesized and real EEG signals. In Section 4, various metrics, i.e., SNR, PRMSE, and Corr are introduced. Proposed method and experimental results are given in Section 5. Finally, Section 6 concludes the paper.

Adaptive decomposition methods

This paper compares the EWT, which offers high resolution in both time and frequency domains, and the new method EFD, which extends the analysis capabilities to non-linear and non-stationary signals, as the two empirical decomposition approaches.

Empirical wavelet transform: EWT is a subset of WT offering high resolution in both time and frequency domains [9]. EWT’s ability to capture time and frequency-localized features and perform multi-resolution analysis makes it valuable for studying non-stationary signals like EEG. EWT decomposes a signal into oscillatory modes or wavelets adapted to the signal’s local characteristics, extracting wavelets empirically rather than relying on predefined functions. This involves identifying local extrema in the signal and constructing wavelets based on the local frequency content around these points. This allows the signal to be decomposed into components representing different scales or frequencies [4]. EWT generally employs an adaptive WT algorithm based on Fourier spectrum division. It involves two steps: I) segmenting the signal Fourier spectrum adaptively, and II) constructing the wavelet filter bank with adjustable cut-off frequencies based on the analyzed signal characteristics [10].

Empirical fourier decomposition: EFD draws on Fourier and EWT through two steps: I) refined segmentation and II) the establishment of a zero-phase filter bank within the Fourier spectrum [5,6]. First, an advanced segmentation technique based on the Fourier spectrum is introduced, and each segment represents a region of Fourier intrinsic band functions. Next, the EFD framework is developed for both continuous and discrete series. Finally, with the integration of the Hilbert transform, EFD is extended into a timefrequency representation. Each IMF captures a unique oscillatory mode, forming the empirical Fourier decomposition. This method is particularly effective in localized frequency analysis, allowing for the simultaneous capture of both low- and high-frequency components for a detailed examination of oscillatory patterns [5,6].

Synthesized signals and EEG data set

This paper uses two datasets: the synthesized and real EEG signals.

The first synthesized signal comprises five sinusoids with different amplitudes and frequencies [10],

Figure 1a shows x(t) where the sampling rate is 256Hz.

The second synthesized signal includes two sinusoids with Linear Frequency Modulation (LFM) [11],

Figure 2a shows x(t) where the sampling rate is 256Hz.

The EEG dataset [12-14] consisted of pre-processed EEG recordings from 100 participants of HC, MCI, and AD.

Metrics SNR, PRMSE, and Corr are three metrics used in this paper to assess error removal’s effectiveness and the method’s performance. A higher SNR indicates better separation of the signal from noise, while a low SNR means that the signal is corrupted by noise and may be difficult to distinguish or recover. Lower (closer to zero) PRMSE values are desirable as they indicate less error or better performance after using a technique. Corr measures the strength and direction of the linear relationship between two signals. High Corr values indicate meaningful relationships between signals, such as linear relationships between variables, identifying time delays, or synchronization. In contrast, low values indicate a lack of relationship or independence between signals [15].

The method involves analyzing the signal (synthetic and real) using EWT and EFD to extract sub-bands. Evaluation criteria, including SNR, PRMSE, and Corr, are then calculated for the subbands before and after decomposition to determine the success rate of each method. Next, the additive white Gaussian noise (AWGN) is added to the original signal to compare the method’s noise robustness.

Figures 1&2 present two different synthesized signals (part a), their frequency spectrums (part b), and the corresponding sub-bands using EWT (part c) and EFD (part d). The Corr between the original sub-band signal before decomposition and the reconstructed sub-band after decomposition is also written for every sub-band. Generally, a sinusoidal signal is a smooth, repetitive oscillation described by sine or cosine functions, and an LFM signal has a linearly changing frequency over time.

Figure 1:a) Sinusoidal signal x(t), according to Eq. (1), is made up of five individual components,
b) the frequency spectrum of the sinusoidal signal, sub-bands decomposed by
c) EWT, and
d) EFD. For each sub-band, the Corr before and after decomposition is computed.

Figure 2:a) LFM signal x(t), according to Eq. (2), is composed of two individual components,
b) the frequency spectrum of the LFM signal, sub-bands decomposed by
c) EWT, and
d) EFD. For every sub-band, Corr before and after decomposition is written.

Parameter N specifies the number of sub-bands that should be extracted. For the sinusoidal signal, N is set to 5, and for the LFM signal, N is set to 2. The synthesized signal consists of several components that EWT and EFD decompose. According to Figures 1c&d, the obtained Corr of every sub-band for EWT and EFD are near, and there is no priority for the sinusoid signal. According to the results in Figures 2c&d, the obtained Corr based on EFD is significantly greater than EWT.

It was concluded that the EFD is preferred for non-stationary repetitive signals with frequency fluctuation compared to the well-known EWT method. We now analyze the real EEG signals to compare two empirical decomposition methods, EWT and EFD. Figures 3-5 present samples of EEG signals for HC, MCI, and AD, along with their corresponding frequency spectrums and the five sub bands named Delta, Theta, Alpha, Beta, and Gamma, using EWT and EFD. For the real EEG signal, N is set to 5. For the EWT, the number of N is set to one more value. Both methods exhibit similar computational complexity and processing time.

Figure 3:a) Sample signal of HC from the EEG dataset,
b) The frequency spectrum,
(c-d) The extracted sub-bands using EWT and EFD methods.

Figure 4:a) Sample signal of MCI from the EEG dataset,
b) The frequency spectrum,
(c-d) The extracted sub-bands using EWT and EFD methods.

Figure 5:a) Sample signal of AD patient from the EEG dataset,
b) The frequency spectrum,
(c-d) The extracted sub-bands using EWT and EFD methods.

Delta is from 0.5 to 4Hz, Theta is from 4 to 8Hz, Alpha is from 8 to 14Hz, Beta is from 14 to 30Hz and Lastly, Gamma is greater than 30Hz.

30Hz. Figures 3-5a&b shows the EEG signal in time and frequency domains, illustrating how EWT and EFD break down sub-bands. Extracting the five sub-bands is essential for understanding brain region synchronization.

In Figures 3-5c&d, EWT results in smoother, less detailed oscillations, reflecting typical EEG changes in AD, like reduced Alpha and Beta activity. In contrast, EFD offers a more detailed breakdown across all sub-bands, capturing small oscillations crucial for EEG analysis, particularly distinguishing between MCI and AD. EFD more clearly shows fluctuations and changes in amplitude and frequency, highlighting the disease’s impact on brain function and demonstrating its superiority in analyzing non-stationary, complex signals across both low and high frequencies with high resolution. Higher temporal resolution enables the capture of rapid changes in brain activity [16].

EEG signals and their frequency spectra are more complex than a simple sinusoidal signal (Figure 1) but less complex than an LFM signal (Figure 2). Like sinusoidal signals, they can be analyzed for their periodic patterns. Similar to LFM signals, they also need time-frequency domain evaluation. Two simulated signals-more controllable than real EEG signals-were initially used before applying these methods to actual EEG data to test the reliability of the decomposition methods.

In the subsequent section, the noise robustness of the EWT and EFD is assessed. For this purpose, the AWGN with SNR=0 dB (i.e., the power of signal and noise are equal) is introduced. The synthesized signals (sinusoid and LFM) and the real EEG signal are corrupted. The noisy signals are decomposed for the synthesized signals, and the correlation between the extracted and original components is computed, see Figure 6. As illustrated in Figure 6, EFD demonstrates superior performance in signal decomposition compared to EWT at both low and high frequencies [4,17]. Since real EEG signals’ true components are unknown, the signal is decomposed, the residual is disregarded, and the correlation between the reconstructed signal and the original clean signal is computed. As shown in Figure 7, EFD outperforms EWT according to the Corr metric. Table 1 demonstrates that EFD consistently outperforms EWT across all evaluated metrics, including SNR, PRMSE, and Corr. Table 2 compares the EFD and other methods. As presented in Table 2 [18-20], our method’s results for SNR, PRMSE, and Corr outperform those of other methods. So, whether the signal is clean or noisy, EFD is recommended for the empirical decomposition.

Figure 6:The original and noisy sinusoidal signal with SNR=0 dB was decomposed by
a) EWT and
b) EFD. The original and noisy LFM signal with SNR=0 dB was decomposed by
c) EWT and
d) EFD. The Corr between the estimated decomposed and the noisy sub-band is computed for every component.

Figure 7:The Corr between the original EEG signal of the HC sample and the estimated signal after decomposition with
a) EWT and
b) EFD is shown.
(c-d) The Corr between the original EEG signal and the estimated noisy signal (SNR=0 dB) after decomposition with EWT and EFD is also presented.

Table 1:Metrics (SNR, PRMSE, and Corr) are computed for the original and noisy reconstructed EEG (MCI and AD) dataset using EWT and EFD (best values are bold).

Table 2:Comparing results with other EEG works in the literature based on metrics (SNR, PRMSE, and Corr).

This study emphasizes the vital role of accurate signal decomposition. Different scenarios were considered to compare two empirical decompositions, EWT and EFD. The effectiveness of EFD opens new ways for any nonstationary signal analysis. Also, EFD’s potential in different bio signal analyses is to be explored. This aspect presents an intriguing area for future research and application.

The data that has been used is available at https://github.com/ tsyoshihara/Alzheimer-s-Classification-EEG/tree/master/data.

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