In the measurement of Electroencephalogram (EEG), differences in voltages from distinct sites of the brain are recorded over a period of time (Hernández-Ronquillo et al., 2020; Sun et al., 2020). The EEG signal contains abundant information related to physiological, psychological and pathological activities of the brain. Therefore, the analysis of EEG signals is of vital importance in the clinical diagnosis and treatment of many diseases, such as Alzheimer’s disease (Faiman et al., 2021), depression (Miao et al., 2021), idiopathic epilepsy and psychogenic non-epileptic seizures (Sharma G. et al., 2021). On the other hand, EEG also plays an important role in the research of brain-computer interface (BCI). BCI is referred to as a non-invasive way of human-machine interface between the brain and exterior devices (Dagdevir and Tokmakci, 2021). BCI does not rely on conventional neurotransmission pathways (Deshpande et al., 2017). Nowadays, many brain imaging methodologies have been adopted by the researchers, such as EEG, magnetic resonance imaging (MRI) and functional MRI (fMRI; Hong et al., 2018). Among the above mediums, EEG is regarded as the most convenient one due to its mobility and lower cost.

In medical applications, owing to non-stationary characteristics of EEG features and differences among individual measurements, the analysis of EEG signals is comparatively complicated and difficult (An et al., 2021). The traditional signal analysis method based on fast Fourier transform (FFT) cannot meet the requirements of EEG signal analysis. In order to reduce excessive professional requirements for medical staff to identify EEG, deep learning (DL) based methods, such as convolutional neural network (CNN; Cao et al., 2019; Prathaban and Balasubramanian, 2021), are introduced to ensure intelligent understanding of EEG signals, especially for large scale datasets. Aiming at achieving robust recognition ability, researchers have designed novel DL neural networks with distinguished types of network architecture, input formulation, and activation function (Saegh et al., 2021). Li proposed an FFT-based deep feature learning method for EEG classification (Li and Chen, 2021). Shankar proposed a DL based epileptic seizure detection algorithm by using 2D recurrence plot images generated from EEG signals for specific brain rhythms (Shankar et al., 2021). Jana studied an efficient seizure prediction technique based on the combination of CNN and a novel technique of channel reduction (Jana and Mukherjee, 2021). According to the above studies, researchers also attached significant attentions in deriving special features from raw EEG signals because it can help improve the accuracy and efficiency of the DL neural networks.

In spite of the versatility of EEG signal in clinical applications and brain imaging science, it is reported that the salient features within the EEG signals are weak in energy. On the other hand, a variety of artifacts are likely to be incorporated during acquisitions of EEG signals. To investigate the essential physiological information corrupted by these noisy artifacts, pre-processing based on signal processing techniques is indispensable. Among the many sources of interference, the power line artifact (PLA), at the frequency of 50 or 60 Hz, is almost inevitable in “field” or mobile EEG measurements outside the lab (Leske and Dalal, 2019). The presence of PLA can significantly affect the analysis and extraction of essential features from EEG recordings. Although shielding of the environment has been recognized as an effective way to reduce PLA in the lab, it is often impractical in scenarios of natural environments. Therefore, alternative solutions, hardware based or software based, must be adopted to address this PLA problem. In comparison, the advantages of software-based solutions are more prominent because they are more flexible and easier to be realized. In state-of-the-art researches, signal processing tools, such as digital notch filters (DNFs; Piskorowski, 2013), short time Fourier transform (STFT; Huang et al., 2019), wavelet transform (WT; He et al., 2015; Huang et al., 2021; Sharma S. et al., 2021), and empirical mode decomposition (EMD; Taran et al., 2018), have been utilized to remove the PLA. However, according to theoretical investigations, the above techniques can be interpreted as digital filters with specific passing band in the frequency domain (Wu and Huang, 2004). Owing to the problems of energy leakage and picket-fence effect (PFE) in canonical Fourier transform, such kinds of digital filters will cause non-ignorable distortions in filtered results, whereas the same problems also occur in hardware-based solutions. To overcome this side effect, a feasible way is to construct a compensation signal with high precision. According to engineering experiences, PLA can be modeled as a sinusoidal component consisting of a simple harmonic wave. While, a sinusoidal component can be uniquely determined by harmonic parameters of amplitude, frequency and phase. Therefore, to retrieve the harmonic parameters become the essential task in PLA removal.

Sparse representation (SR) is a comparatively new development of signal expansion (Li and Chen, 2008). It expresses an input signal in terms of linear combinations of atoms from a dictionary. It has achieved tremendous successes in various engineering applications (Wang et al., 2016; Yang et al., 2020; Zhang et al., 2021). Conventional SRs are implemented using iterative algorithms and may require large computational resources (Cao et al., 2021; Chen et al., 2021; An et al., 2022). If the efficiency of SR algorithm can be improved, it will effectively expand the scope of its use in medical engineering (Yang et al., 2019; Collazos-Huertas et al., 2020; Cury et al., 2020). In this paper, to overcome the signal distortion effect in PLA removal of EEG signals, we propose a novel approach based on the combination of SR and spectrum correction (SC). SC methods can identify the harmonic parameters based on the information in the frequency domain. It will be shown in this paper that for a sinusoidal component, the SC can be regarded as a numerical algorithm to realize SR via a non-iterative way. In ratio-based spectrum correction (RBSC) algorithms, the performance of harmonic parameter extraction is determined by signal-to-noise ratios of baseline spectral lines (BSSLs). Taking this phenomenon into account, a dual-step correcting algorithm (DSCA) is put forward. Firstly, a RBSC, using the rectangular window, is employed for preliminary identification of harmonic parameters. Secondly, two BSSLs with high SNR are calculated. Thirdly, another RBSC is employed to estimate the harmonic parameters with higher precision. On the basis of the estimated harmonic parameters, a compensation signal is reconstructed. PLA removal by subtracting the compensation signal from the original EEG signal can avoid the side effect of signal distortion. Analysis using numerical simulation and measured signal shows that the proposed method is effective, efficient, and can suppress waveform distortion.

The problems of energy leakage and PFE are inevitable for harmonic components whose digital samplings do not satisfy the full period sampling condition (Mewett et al., 2001). There is a view that these two side effects can be prevented by actively adjusting the sampling parameters. Because a variation of ± 2 Hz is likely to occur in actual power systems (Singhal et al., 2020), the parameter f_pla is an unknown variable. It is not practical to design DNFs perfectly suitable for all possible PLA noises. On the other hand, increasing the sampling length of EEG signal can improve the accuracy of harmonic parameter estimation. However, the process of brain electrical activity corresponding to the collected signals cannot be repeated, so we can only analyze and process the signals of limited length.

Wavelet transform and EMD outperform FFT by providing time-frequency representation of EEG signal. However, in either case, the above side effect is still unavoidable. Because no matter how they decompose the signal, they can only suppress part of the PLA, and the rest will still cause signal distortion. In principle, RBSC uses local spectral information to estimate the overall harmonic composition, so it can avoid the side effect.

The method proposed in this paper is an improvement of the classical RBSC. It not only improves the robustness of the algorithm in parameter correction, but also considers the efficiency of the algorithm. The complexity of the algorithm has only a small increase compared with the classical method. The method proposed in this paper is especially suitable for the case that the value of Δk is very small. When the value of this parameter is close to half the frequency resolution, the performance of the two methods is close. In general, the method proposed in this paper can be better applied to the complex measurement environment. However, in any case, the method in this paper has no less accuracy than the original RBSC in statistics. So we recommend that DSCA can completely replace the original RBSC.

Ratio-based spectrum correction only utilize two BSSLs to estimate harmonic parameters, and it can balance the efficiency and accuracy of the algorithm. In the literature, there have been some methods for harmonic parameter estimation using multiple BSSLs. These studies also show that their harmonic parameter estimation accuracy is indeed better than that of RBSC in the absence of noise. But this is not achievable in clinical applications. Therefore, the DSCA is proposed based on the RBSC. The calculation time of RBSC mainly includes the FFT transform of the signal and several complex arithmetic operations. The DSCA proposed in this paper mainly includes two RBSCs, so the efficiency is also very high.

For analysis of a simple harmonic component, the spectral correction is very similar with the spare representation theory in mathematical principle. A remarkable advantage is that it does not require a predetermined dictionary. The algorithm can generate a skinny dictionary containing only one atom to represent the actual sinusoidal component. Because the dictionary is not redundant, no iterative algorithms is needed. We can call the spectral correction as implicit sparse representation (ISR) for simple harmonic component analysis. The current research shows that the results obtained by using SR based on iterative numerical algorithm are very similar to those based on spectral correction method. Therefore, in the case of harmonic analysis, a method based on SC may be preferentially employed.

The data presented in the study are deposited in the ictal repository, accession number ictal1 (https://www.researchgate.net/publication/308719109_EEG_Epilepsy_Datasets).

MZ conceived and designed the classification method. Z-FL, RZ, and NX performed the experiment. Z-FL, RZ, Y-QD, and G-HL analyzed the data and wrote the manuscript. MZ and NX reviewed and edited the manuscript. All authors read and approved the manuscript.

J-LT, Z-FL, and RZ were employed by the company Shaanxi Aerospace Technology Application Research Institute Co., Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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