Fourier–Bessel series

In mathematics, Fourier–Bessel series is a particular kind of generalized Fourier series (an infinite series expansion on a finite interval) based on Bessel functions.

Fourier–Bessel series are used in the solution to partial differential equations, particularly in cylindrical coordinate systems. The series formed by the Bessel function of the first kind is known as the Schlömilch's series.

Definition

The Fourier–Bessel series of a function f(x) with a domain of [0,b] satisfying f(b)=0

$f:[0,b]\rightarrow \mathbb {R}$

is the representation of that function as a linear combination of many orthogonal versions of the same Bessel function of the first kind J_α, where the argument to each version n is differently scaled, according to

$(J_{\alpha })_{n}(x):=J_{\alpha }\left({\frac {u_{\alpha ,n}}{b}}x\right)$

where u_α,n is a root, numbered n associated with the Bessel function J_α and c_n are the assigned coefficients:

$f(x)\sim \sum _{n=1}^{\infty }c_{n}J_{\alpha }\left({\frac {u_{\alpha ,n}}{b}}x\right).$

Interpretation

The Fourier–Bessel series may be thought of as a Fourier expansion in the ρ coordinate of cylindrical coordinates. Just as the Fourier series is defined for a finite interval and has a counterpart, the continuous Fourier transform over an infinite interval, so the Fourier–Bessel series has a counterpart over an infinite interval, namely the Hankel transform.

Calculating the coefficients

As said, differently scaled Bessel Functions are orthogonal with respect to the inner product

$\langle f,g\rangle =\int _{0}^{b}xf(x)g(x)\mathrm {d} x$

according to

$\int _{0}^{1}xJ_{\alpha }(xu_{\alpha ,n})\,J_{\alpha }(xu_{\alpha ,m})\,dx={\frac {\delta _{mn}}{2}}[J_{\alpha +1}(u_{\alpha ,n})]^{2}$,

(where: ${\delta _{mn}}$ is the Kronecker delta). The coefficients can be obtained from projecting the function f(x) onto the respective Bessel functions:

$c_{n}={\frac {\langle f,(J_{\alpha })_{n}\rangle }{\langle (J_{\alpha })_{n},(J_{\alpha })_{n}\rangle }}={\frac {\int _{0}^{b}xf(x)(J_{\alpha })_{n}(x)\mathrm {d} x}{{\frac {1}{2}}(bJ_{\alpha \pm 1}(u_{\alpha ,n}))^{2}}}$

where the plus or minus sign is equally valid.

Application

The Fourier–Bessel series expansion employs aperiodic and decaying Bessel functions as the basis. The Fourier–Bessel series expansion has been successfully applied in diversified areas such as Gear fault diagnosis, discrimination of odorants in a turbulent ambient, postural stability analysis, detection of voice onset time, glottal closure instants (epoch) detection, separation of speech formants, EEG signal segmentation, speech enhancement, and speaker identification. The Fourier–Bessel series expansion has also been used to reduce cross terms in the Wigner–Ville distribution.

Dini series

A second Fourier–Bessel series, also known as Dini series, is associated with the Robin boundary condition

$bf'(b)+cf(b)=0$, where $c$ is an arbitrary constant.

The Dini series can be defined by

$f(x)\sim \sum _{n=1}^{\infty }b_{n}J_{\alpha }(\gamma _{n}x/b)$,

where $\gamma _{n}$ is the nth zero of $xJ'_{\alpha }(x)+cJ_{\alpha }(x)$.

The coefficients $b_{n}$ are given by

$b_{n}={\frac {2\gamma _{n}^{2}}{b^{2}(c^{2}+\gamma _{n}^{2}-\alpha ^{2})J_{\alpha }^{2}(\gamma _{n})}}\int _{0}^{b}J_{\alpha }(\gamma _{n}x/b)\,f(x)\,x\,dx.$

See also

• Orthogonality
• Generalized Fourier series
• Hankel transform
• Neumann polynomial

References

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• V. Gupta and R.B. Pachori, Epileptic seizure identification using entropy of FBSE based EEG rhythms, Biomedical Signal Processing and Control, DOI: 10.1016/j.bspc.2019.101569, 2019.
• R. Katiyar, V. Gupta, and R.B. Pachori, FBSE-EWT based approach for the determination of respiratory rate from PPG signals, IEEE Sensors Letters, DOI: 10.1109/LSENS.2019.2926834, 2019.
• R.K. Tripathy, A. Bhattacharyya, and R.B. Pachori, Localization of myocardial infarction from multi lead electrocardiogram signals using multiscale convolution neural network, IEEE Sensors Journal, DOI: 10.1109/JSEN.2019.2935552, 2019.
• P. Gajbhiye, R.K. Tripathy, and R.B. Pachori, Elimination of ocular artifacts from single channel EEG signals using FBSE-EWT based rhythms, IEEE Sensors Journal, 2019.
• A.S. Hood, R.B. Pachori, V.K. Reddy, and P. Sircar, Parametric representation of speech employing multi-component AFM signal model, International Journal of Speech Technology, vol. 18, issue 03, pp. 287-303, September 2015.
• A. Anuragi, D. Sisodia, and RB Pachori, Automated alcoholism detection using Fourier-Bessel series expansion based empirical wavelet transform, IEEE Sensors Journal, DOI: 10.1109/JSEN.2020.2966766, 2020.
• T. Siddharth, P. Gajbhiye, R.K. Tripathy, and R.B. Pachori, EEG based detection of focal seizure area using FBSE-EWT rhythm and SAE-SVM network, IEEE Sensors Journal, DOI: 10.1109/JSEN.2020.2995749.
• V. Gupta and R.B. Pachori, Classification of focal EEG signals using FBSE based flexible time-frequency coverage wavelet transform, Biomedical Signal Processing and Control, DOI: https://doi.org/10.1016/j.bspc.2020.102124.
• A. Bhattacharyya, R.K. Tripathy, L. Garg, and R.B. Pachori, A novel multivariate-multiscale approach for computing EEG spectral and temporal complexity for human emotion recognition, IEEE Sensors Journal, DOI: 10.1109/JSEN.2020.3027181.
• P.K. Chaudhary and R.B. Pachori, Automatic diagnosis of glaucoma using two-dimensional Fourier-Bessel series expansion based empirical wavelet transform, Biomedical Signal Processing and Control, DOI: https://doi.org/10.1016/j.bspc.2020.102237.
• V. Gupta and R.B. Pachori, FBDM based time-frequency representation for sleep stages classification using EEG signals, Biomedical Signal Processing and Control, DOI: https://doi.org/10.1016/j.bspc.2020.102265.

External links

• "Fourier-Bessel series", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Weisstein, Eric. W. "Fourier-Bessel Series". From MathWorld--A Wolfram Web Resource.
• Fourier–Bessel series applied to Acoustic Field analysis on Trinnov Audio's research page