Analysis of Discrete Fourier Transform: Finite-Dimensional Hilbert Space in Electroencephalogram Signal

Fajar Ilham Maulana; Yani Ramdani; Gani Gunawan

doi:10.30871/jaic.v10i5.14024

Authors

Fajar Ilham Maulana Universitas Islam Bandung (UNISBA)
Yani Ramdani Universitas Islam Bandung (UNISBA)
Gani Gunawan Universitas Islam Bandung (UNISBA)

DOI:

https://doi.org/10.30871/jaic.v10i5.14024

Keywords:

Discrete Fourier Transform, Hilbert Space, Impulse sampling, Electroencephalogram signal, Frequency spectrum

Abstract

The Discrete Fourier Transform (DFT) allows the digital signals representation in the frequency domain by decomposing discrete signals into finite sinusoidal components. DFT works on discrete data with a certain number of samples relevant for modern digital signal processing. In the context of electroencephalogram (EEG), DFT plays an important role in identifying brain frequency spectra, such as alpha, beta, theta, and delta waves, each of which has physiological and clinical significance. This article analyzes the Discrete Fourier Transform (DFT) in the context of finite-dimensional Hilbert space and its application to analyze EEG signals. The objectives of the study are to prove that the DFT is in Hilbert space by showing its mathematical properties and using DFT to analyze the frequency characteristics of EEG signals in finite-dimensional Hilbert space. The method used involves a mathematical approach by establishing a mathematical framework based on finite-dimensional Hilbert space, analyzing the inverse DFT by recasting the signal from the frequency domain to the time domain, identifying the relevant mathematical characteristics of the DFT. Discretization of continuous signals into discrete ones through impulse sampling techniques, and application of DFT for transformation to the frequency domain for analyzing EEG spectral. The results show that DFT satisfies the definition and axioms of complex vector spaces and finite-dimensional Hilbert spaces, having a provable inverse through the properties of unitary operators. The application of DFT to the modeled EEG signal successfully produces a frequency spectrum that matches the frequency components of the initial model, both in magnitude and energy.

Downloads

Download data is not yet available.

References

[1] Gallagher, T. A., Nemeth, A. J., & Hacein-Bey, L. (2008). An introduction to the Fourier transform: relationship to MRI. American journal of roentgenology, 190(5), 1396-1405. https://doi.org/10.2214/AJR.07.2874

[2] Gunawan, G., Harahap, E., & Suwanda. (2019, November). Transformation of the mean value of integral on Fourier series expansion. In Journal of Physics: Conference Series (Vol. 1366, No. 1, p. 012068). IOP Publishing. doi:10.1088/1742-6596/1366/1/012068.

[3] Serov, V. (2017). Fourier series, Fourier transform, and their applications to mathematical physics (Vol. 197). Cham: Springer.

[4] Gray, R. M., & Goodman, J. W. (2012). Fourier Transforms: An Introduction for Engineers (Vol. 322). Springer Science & Business Media.

[5] Sourina, O., & Liu, Y. (2011, January). A fractal-based algorithm of emotion recognition from EEG using arousal-valence model. In International conference on bio-inspired systems and signal processing (Vol. 2, pp. 209-214). Scitepress.

[6] Kroupi, E., Yazdani, A., & Ebrahimi, T. (2011, October). EEG correlates of different emotional states elicited during watching music videos. In International Conference on Affective Computing and Intelligent Interaction (pp. 457-466). Berlin, Heidelberg: Springer Berlin Heidelberg.

[7] Fu, K., Qu, J., Chai, Y., & Zou, T. (2015). Hilbert marginal spectrum analysis for automatic seizure detection in EEG signals. Biomedical Signal Processing and Control, 18, 179-185. https://doi.org/10.1016/j.bspc.2015.01.002

[8] Vempati, R., & Sharma, L. D. (2023). A systematic review on automated human emotion recognition using electroencephalogram signals and artificial intelligence. Results in Engineering, 18, 101027. https://doi.org/10.1016/j.rineng.2023.101027.

[9] Sałabun, W. (2014). Processing and spectral analysis of the raw EEG signal from the MindWave. Przeglad Elektrotechniczny, 90(2), 169-174. doi:10.12915/pe.2014.02.44

[10] Nordin, N. F., & Alias, N. (2022). The classification of human emotions based on the electroencephalogram (EEG) of brain waves. Proc. Sci. Math, 9, 286-297.

[11] Da Silva, F. L. (1999). EEG analysis: theory and practice. Electroencephalography: basic principles, clinical applications and related fields, 1125-1159.

[12] "Your 5 Brainwaves: Delta, Theta, Alpha, Beta, and Gamma," Lucid, 16 June 2016. [Online]. Available: https://lucid.me/blog/5-brainwaves-delta-theta-alpha-beta-gamma/.

[13] Morrison, N. (1994). Introduction to Fourier analysis. John Wiley & Sons.

[14] Norrbo, D. (2025). Norm estimates for the Hilbert matrix operator on weighted Bergman spaces. Journal of Mathematical Analysis and Applications, 548(2), 129408. https://doi.org/10.1016/j.jmaa.2025.129408

[15] Gao, Q., Omran, A. H., Baghersad, Y., Mohammadi, O., Alkhafaji, M. A., Al-Azzawi, A. K. J., ... & Golkar, M. J. (2023). Electroencephalogram signal classification based on Fourier transform and Pattern Recognition Network for epilepsy diagnosis. Engineering Applications of Artificial Intelligence, 123, 106479. https://doi.org/10.1016/j.engappai.2023.106479

[16] Zhuang, N., Zeng, Y., Tong, L., Zhang, C., Zhang, H., & Yan, B. (2017). Emotion recognition from EEG signals using multidimensional information in EMD domain. BioMed research international, 2017(1), 8317357. https://doi.org/10.1155/2017/8317357

[17] Bin Heyat, M. B., Akhtar, F., Khan, A., Noor, A., Benjdira, B., Qamar, Y., ... & Lai, D. (2020). A novel hybrid machine learning classification for the detection of bruxism patients using physiological signals. Applied Sciences, 10(21), 7410, https://doi.org/10.3390/app10217410

[18] Castellano, G., Villalba, S. D., & Camurri, A. (2007, September). Recognising human emotions from body movement and gesture dynamics. In International conference on affective computing and intelligent interaction (pp. 71-82). Berlin, Heidelberg: Springer Berlin Heidelberg.