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<p>The peak detection plays an important role in the signal processing community. The conventional approach typically identify peaks by computing the first-order derivative of a signal. However, noise-induced phase distortion often renders these detected peaks inaccurate. Recent studies have derived the fractional stationary points for single-frequency periodic signals using the Riemann Liouville derivative initialized at negative infinity, establishing an affine linear relationship between the time difference of consecutive points and the fractional order. Nevertheless, many practical signals are rarely composed of a single frequency. This paper extends the theoretical framework to locate fractional stationary points in periodic signals containing multiple frequency components. Besides, the Riemann Liouville fractional derivative with the initial point at the infinity for the non-zero DC term is unbounded. Hence, this paper also derives the Riemann Liouville fractional derivative with the initial point at the zero for the periodic signals with the single frequency component. Crucially, the proposed method allows for the determination of fractional stationary points without performing computationally expensive numerical differentiation, thereby significantly reducing the computational load. The method’s effectiveness is validated using both synthetic and practical biomedical signals, including EEG with 512&#xa0;Hz sampling rate and EOG signal with 500&#xa0;Hz sampling rate. Specifically, for the tested EEG and ECG signals, the mean squared error between the proposed method and the classical Riemann Liouville method is maintained below 0.0035 and 0.0018, respectively, while the Correlation Coefficients exceed 0.986 and 0.992 across various fractional orders. These results confirm that the proposed method can effectively identify fractional stationary points with significantly reduced computational power while maintaining high accuracy.</p>

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Riemann Liouville Based Fractional Derivative of Periodic Signals with More Than One Frequency Component and Fast Detection of Their Fractional Stationary Points

  • 期刊论文

  • 发表日期：

    2026年3月28日

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  • Jiaqi Liu,
  • Shiqi Lin,
  • Zhaoheng Zhou,
  • Bingo Wing-Kuen Ling

摘要

The peak detection plays an important role in the signal processing community. The conventional approach typically identify peaks by computing the first-order derivative of a signal. However, noise-induced phase distortion often renders these detected peaks inaccurate. Recent studies have derived the fractional stationary points for single-frequency periodic signals using the Riemann Liouville derivative initialized at negative infinity, establishing an affine linear relationship between the time difference of consecutive points and the fractional order. Nevertheless, many practical signals are rarely composed of a single frequency. This paper extends the theoretical framework to locate fractional stationary points in periodic signals containing multiple frequency components. Besides, the Riemann Liouville fractional derivative with the initial point at the infinity for the non-zero DC term is unbounded. Hence, this paper also derives the Riemann Liouville fractional derivative with the initial point at the zero for the periodic signals with the single frequency component. Crucially, the proposed method allows for the determination of fractional stationary points without performing computationally expensive numerical differentiation, thereby significantly reducing the computational load. The method’s effectiveness is validated using both synthetic and practical biomedical signals, including EEG with 512 Hz sampling rate and EOG signal with 500 Hz sampling rate. Specifically, for the tested EEG and ECG signals, the mean squared error between the proposed method and the classical Riemann Liouville method is maintained below 0.0035 and 0.0018, respectively, while the Correlation Coefficients exceed 0.986 and 0.992 across various fractional orders. These results confirm that the proposed method can effectively identify fractional stationary points with significantly reduced computational power while maintaining high accuracy.