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

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

where uα,n is a root, numbered n associated with the Bessel function Jα and cn are the assigned coefficients:

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

according to

,

(where:  is the Kronecker delta). The coefficients can be obtained from projecting the function f(x) onto the respective Bessel functions:

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
, where  is an arbitrary constant.
The Dini series can be defined by
,

where  is the nth zero of .

The coefficients  are given by

See also
Orthogonality
Generalized Fourier series
Hankel transform
Neumann polynomial

References

 
 
 R.B. Pachori and P. Sircar, Non-stationary signal analysis: Methods based on Fourier-Bessel representation, LAP LAMBERT Academic Publishing, Saarbrucken, Germany, 2010, .
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 G. D’Elia, S. Delvecchio and G. Dalpiaz, On the use of Fourier–Bessel series expansion for gear diagnostics, Proc. of the Second Int. Conf. Condition Monitoring of Machinery in Non-Stationary Operations (2012), 267-275.
 A. Vergaraa, E. Martinelli, R. Huerta, A. D’Amico and C. Di Natale, Orthogonal decomposition of chemo-sensory signals: Discriminating odorants in a turbulent ambient, Procedia Engineering 25 (2011), 491–494.
 F.S. Gurgen and C. S. Chen, Speech enhancement by Fourier–Bessel coefficients of speech and noise, IEE Proc. Comm. Speech Vis. 137 (1990), 290–294.
 K. Gopalan, T. R. Anderson and E. J. Cupples, A comparison of speaker identification results using features based on cepstrum and Fourier–Bessel expansion, IEEE Trans. Speech Audio Process. 7 (1999), 289–294.
 R.B. Pachori and P. Sircar, Analysis of multicomponent AM-FM signals using FB-DESA method, Digital Signal Processing, vol. 20, pp. 42-62, January 2010.
 R.B. Pachori and P. Sircar, EEG signal analysis using FB expansion and second-order linear TVAR process, Signal Processing, vol. 88, no. 2, pp. 415-420, February 2008.
 R.B. Pachori and P. Sircar, A new technique to reduce cross terms in the Wigner distribution, Digital Signal Processing, vol. 17, no. 2, pp. 466-474, March 2007.
 A. Bhattacharyya, L. Singh, and R.B. Pachori, Fourier-Bessel series expansion based empirical wavelet transform for analysis of non-stationary signals, Digital Signal Processing, vol. 78, pp. 185-196, July 2018.
 P. Jain and R.B. Pachori, Event-based method for instantaneous fundamental frequency estimation from voiced speech based on eigenvalue decomposition of Hankel matrix, IEEE/ACM Transactions on Audio, Speech and Language Processing, vol. 22. issue 10, pp. 1467-1482, October 2014.
R.K. Tripathy, A. Bhattacharyya, and R.B. Pachori, A novel approach for detection of myocardial infarction from ECG signals of multiple electrodes, IEEE Sensors Journal, DOI: 10.1109/JSEN.2019.2896308, 2019.
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.

External links
 
 
 Fourier–Bessel series applied to Acoustic Field analysis on Trinnov Audio's research page

Category:Fourier series