Fourier–Bessel series

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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 Jα, 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 uα,n is a root, numbered n associated with the Bessel-Function Jα and cn 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

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

$\langle f,g \rangle = \int_0^b x f(x) g(x) \mathrm{d}x$

according to

$\int_0^1 x J_\alpha(x u_{\alpha,n})\,J_\alpha(x u_{\alpha,m})\,dx = \frac{\delta_{mn}}{2} [J_{\alpha+1}(u_{\alpha,n})]^2$,

the coefficients can be obtained from projecting the function f(x) onto the respective Bessel functions:

$c_n = \frac{ \langle f,(J_\alpha)_n \rangle }{ \langle (J_\alpha)_n,(J_\alpha)_n \rangle } = \frac{ \int_0^b x f(x) (J_\alpha)_n(x) \mathrm{d}x }{ \frac12 (b(J_{\alpha\pm1})_n(b))^2}$

where the plus or minus sign is equally valid.

Application

The Fourier–Bessel series expansion employs aperiodic and decaying Bessel functions as the basis. 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

$b f'(b) + c f(b) = 0$, 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 $x J'_\alpha(x)+cJ_\alpha(x)$.

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.
• 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.