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.

Definition

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

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

is the notation 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=0}^{\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

Because 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}

,

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}}(b(J_{\alpha \pm 1})_{n}(b))^{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 [J. Schroeder, 1993]. The Fourier–Bessel series expansion has been successfully applied in diversified areas such as Gear fault diagnosis [G. D’Elia et al., 2012], discrimination of odorants in a turbulent ambient [A. Vergaraa at al., 2011], postural stability analysis [Pachori and Hewson, 2011], detection of voice onset time [Pachori and Gangashetty, 2010], glottal closure instants (epoch) detection [Pachori and Gangashetty, 2010], separation of speech formants [Pachori and Sircar, 2010], EEG signal segmentation [Pachori and Sircar, 2008], speech enhancement [Gurgen and Chen, 1990], and speaker identification [K. Gopalan at al., 1999]. The Fourier–Bessel series expansion has also been used to reduce cross terms in the Wigner–Ville distribution [Pachori and Sircar, 2007].

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=0}^{\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

.

References

Smythe, William R. (1968). Static and Dynamic Electricity (3rd ed.). New York: McGraw-Hill.

Magnus, Wilhelm; Oberhettinger, Fritz; Soni, Raj Pal (1966). Formulas and Theorems for Special Functions of Mathematical Physics. Berlin: Springer.

J. Schroeder, Signal processing via Fourier–Bessel series expansion, Digital Signal Process. 3 (1993), 112–124.

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.

R.B. Pachori and S.V. Gangashetty, AM-FM model based approach for detection of glottal closure instants, Proceedings IEEE International Conference on Information Science, Signal Processing and their Applications, pp. 266-269, 10-13 May, 2010, Kuala Lumpur, Malaysia.

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.

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.

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.

R.B. Pachori and D. Hewson, Assessment of the e�ects of sensory perturbations using Fourier-Bessel expansion method for postural stability analysis, Journal of Intelligent Systems, vol. 20, issue 2, pp. 167-186, August 2011.

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 S.V. Gangashetty, Detection of voice onset time using FB expansion and AM-FM model, Proceedings IEEE International Conference on Information Science, Signal Processing and their Applications, 149-152, 10-13 May, 2010, Kuala Lumpur, Malaysia.

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.

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.

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