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] satisfying f(b) = 0 $$f:[0,b]\to \mathbb {R}$$ 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

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

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:

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

where the plus or minus sign is equally valid.

One-to-one relation between order index (n) and continuous frequency ($F_{n}$)

Fourier–Bessel series coefficients are unique for a given signal, and there is one-to-one mapping between continuous frequency ($F_{n}$) and order index $(n)$ which can be expressed as follows:

$u_{n}={\frac {2\pi F_{n}L}{F_{s}}}$

Since, $u_{n}=u_{n-1}+\pi \approx n\pi$. So above equation can be rewritten as follows:

$F_{n}={\frac {F_{s}n}{2L}}$

where $L$ is the length of the signal and $F_{s}$ is the sampling frequency of the signal.

2-D- Fourier-Bessel series expansion

For an image $f(x,y)$ of size M×N, the synthesis equations for order-0 2D-Fourier–Bessel series expansion is as follows:

$f(x,y)=\sum _{m=1}^{M}\sum _{n=1}^{N}F(m,n)J_{0}{\bigg (}{\frac {u_{0,n}y}{N}}{\bigg )}J_{0}{\bigg (}{\frac {u_{0,n}x}{M}}{\bigg )}$

Where $F(m,n)$ is 2D-Fourier–Bessel series expansion coefficients whose mathematical expressions are as follows:

$F(m,n)={\frac {4}{\alpha _{1}}}\sum _{x=0}^{M-1}\sum _{y=0}^{N-1}xyf(x,y)J_{0}{\bigg (}{\frac {u_{0,n}y}{N}}{\bigg )}J_{0}{\bigg (}{\frac {u_{0,m}x}{M}}{\bigg )}$

where, $\alpha _{1}=(NM)^{2}(J_{1}(u_{0,m})J_{1}(u_{0,n}))^{2}$

Fourier-Bessel series expansion based entropies

For a signal of length $b$, Fourier-Bessel based spectral entropy such as Shannon spectral entropy ($H_{\text{SSE}}$), log energy entropy ($H_{\text{LLE}}$), and Wiener entropy ($H_{\text{WE}}$) are defined as follows:

$H_{\text{SSE}}=-\sum _{n=1}^{b}P(n)~{\text{log}}_{2}\left(P(n)\right)$

$H_{\text{WE}}=b{\frac {\sqrt {\displaystyle \prod _{n=1}^{b}E_{n}}}{\displaystyle \sum _{n=1}^{b}E_{n}}}$

$H_{\text{LE}}=-\sum _{n=1}^{b}~{\text{log}}_{2}\left(P(n)\right)$

where $P_{n}$ is is the normalized energy distribution which is mathematically defined as follows:

$P(n)={\frac {E_{n}}{\displaystyle \sum _{n=1}^{b}E_{n}}}$

$E_{n}$ is energy spectrum which is mathematically defined as follows:

$E_{n}={\frac {c_{n}^{2}b^{2}[J_{1}(u_{1,n})]^{2}}{2}}$

Fourier-Bessel Series Expansion Domain Discrete Stockwell Transform

For a discrete time signal, x(n), the FBSE domain discrete Stockwell transform (FBSE-DST) is evaluated as follows:$$T(n,l)=\sum _{m=1}^{L}Y{\Big (}m+l{\Big )}g(m,l)J_{0}{\Big (}{\frac {\lambda _{l}}{N}}n{\Big )}$$where Y(l) are the FBSE coefficients and these coefficients are calculated using the following expression as

$Y(l)={\frac {2}{N^{2}[J_{1}(\lambda _{l})]^{2}}}\sum _{n=0}^{N-1}nx(n)J_{0}{\Big (}{\frac {\lambda _{l}}{N}}n{\Big )}$

The $\lambda _{l}$is termed as the $l^{th}$ root of the Bessel function, and it is evaluated in an iterative manner based on the solution of $J_{0}(\lambda _{l})=0$using the Newton-Rapson method. Similarly, the g(m,l) is the FBSE domain Gaussian window and it is given as follows :

$g(m,l)={\text{e}}^{-{\frac {2\pi ^{2}\lambda _{m}^{2}}{\lambda _{l}^{2}}}},~{\{l,m=1,2,...L}\}$

Advantages

The Fourier–Bessel series expansion does not require use of window function in order to obtain spectrum of the signal. It represents real signal in terms of real Bessel basis functions. It provides representation of real signals it terms of positive frequencies. The basis functions used are aperiodic in nature and converge. The basis functions include amplitude modulation in the representation. The Fourier–Bessel series expansion spectrum provides frequency points equal to the signal length.

Applications

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 n-th 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
• Kapteyn series
• Neumann polynomial
• Schlömilch's series

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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