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

Bessel function for (i) $\alpha =0$ and (ii) $\alpha =1$.

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

(i) Speech signal (mtlb.mat from Matlab toolbox), (ii) FBSE coefficients of speech signal, and (iii) magnitude of FBSE coefficients of speech signal.

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

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 ($F_{n}$)

(i) Plot of an arbitrary Bessel basis function (ii) Plot of FFT of the considered Bessel basis function (iii) Plot of FBSE coefficients of the considered Bessel basis function.

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 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 based Empirical Wavelet Transform

The Empirical wavelet transform (EWT) is a multi-scale signal processing approach for the decomposition of multi-component signal into intrinsic mode functions (IMFs). The EWT is based on the design of empirical wavelet based filter bank based on the segregation of Fourier spectrum of the multi-component signals. The segregation of Fourier spectrum of multi-component signal is performed using the detection of peaks and then the evaluation of boundary points. For non-stationary signals, the Fourier Bessel Series Expansion (FBSE) is the natural choice as it uses Bessel function as basis for analysis and synthesis of the signal. The FBSE spectrum has produced the number of frequency bins same as the length of the signal in the frequency range [0, ${\frac {F_{s}}{2}}$]. Therefore, in FBSE-EWT, the boundary points are detected using the FBSE based spectrum of the non-stationary signal. Once, the boundary points are obtained, the empirical wavelet based filter-bank is designed in the Fourier domain of the multi-component signal to evaluate IMFs. The FBSE based method used in FBSE-EWT has produced higher number of boundary points as compared to FFT part in EWT based method. The features extracted from the IMFs of EEG and ECG signals obtained using FBSE-EWT based approach have shown better performance for the automated detection of Neurological and cardiac ailments.

FBSE-EWT is extended for images and multivariates signals. Both the extended version uses the concept of mean spectrum magnitude of FBSE, i.e., FBSE spectrum is computed for each channel, and then averaging of the spectrum is performed to obtain the mean spectrum. To the mean spectrum, boundaries are extracted, and on the bases of extracted boundaries filterbanks are designed. Using obtained filterbank, each channel is decomposed into IMFs. In case of 2D-FBSE-EWT, same concept is used for both row and column of the image.

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 :

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

Fourier–Bessel expansion-based discrete energy separation algorithm

For multicomponent amplitude and frequency modulated (AM-FM) signals, the discrete energy separation algorithm (DESA) together with the Gabor's filtering is a traditional approach to estimate the amplitude envelope (AE) and the instantaneous frequency (IF) functions. It has been observed that the filtering operation distorts the amplitude and phase modulations in the separated monocomponent signals. In the work, the Fourier–Bessel expansion-based discrete energy separation algorithm (FB-DESA) for component separation and estimation of the AE and IF functions of a multicomponent AM-FM signal is proposed. The FB-DESA method gives accurate estimations of the AE and IF functions.

Fourier-Bessel Series Expansion based Time-Varying Autoregressive Modeling for EEG signals

The study suggests a Time-Varying Autoregressive Modeling (TVAR) method of second order for the parametric representation of the electroencephalogram (EEG) signals. A feature vector for the segmentation of the EEG signal has been created using the coefficients of the Fourier–Bessel series expansion (FBSE). The method is unique in that it identifies a straight forward model for the parametric representation of the EEG signals by choosing an appropriate data length. It presents the entire procedure for model parameter estimation.

The Fourier-Bessel dyadic decomposition method and Fourier-Bessel Series Expansion-based decomposition

The Fourier-Bessel dyadic decomposition method (FBD) and Fourier-Bessel Series Expansion-based decomposition (FBSED) are implementations of discrete wavelet transform and wavelet packet transform using FBSE. The advantage of using FBSE is that FBSE provides unique FBSE coefficients of length same as the signal's length, whereas Fourier transform provides unique coefficients of length half of the length of the signal. So, better frequency resolution can be obtained, and so FBSED and FBD provide one extra level of decomposition compared to discrete wavelet transform and wavelet packet transform. Another advantage is that FBD and FBSED method uses a grouping of coefficients operation instead of a filtering operation, so any level of decomposition can be obtained in a single step.

Fourier-Bessel series expansion based flexible analytic wavelet transform

Convention wavelet transforms like discrete wavelet transform and wavelet packet transforms are dyadic decomposition methods, i.e., these transform generates filterbanks of fixed Q-factor and are non-adaptive in nature. Fourier-Bessel series expansion based flexible analytic wavelet transform (FBSE-FAWT) generates the filterbanks of desired Q-factor, redundancy, and dilation factor. FBSE-FAWT is FBSE version of  FAWT which uses only positive frequencies for filtering operation. In contrast, conventional FAWT uses both the positive frequency part and the negative frequency part while filtering operations. So the implementation of FBSE-EWT becomes easier than FAWT.

FBSE-FAWT is also extended for image decomposition. The implementation of 2D-FBSE-FAWT is done by performing filtering operation row-wise, followed by column-wise (same 2D discrete wavelet transform).

Fourier-Bessel decomposition method

The Fourier-Bessel decomposition method (FBDM) is the implementation of the Fourier decomposition method using FBSE, as FBSE is considered more suitable for representing non-stationary signals. FBDM decomposes the signal into a finite number of Fourier-Bessel intrinsic functions (FBIBFs). FBIBFs should follow the following properties:

• FBIBFs have zero mean.
• FBIBFs are orthogonal to each other.
• The amplitude envelops, and instantaneous frequency of analytic FBIBF (AFBIBF) should be more than zero.

Fourier-Bessel-dictionary-based spatiotemporal sparse Bayesian learning algorithm with expectation maximization

This Fourier-Bessel-dictionary-based spatiotemporal sparse Bayesian learning algorithm with expectation maximization (FB-SSBL-EM) is based on the existing spatiotemporal sparse Bayesian learning algorithm with expectation maximization (SSBL-EM) method in which discrete cosine transform (DCT) dictionary is used for efficient reconstruction. In FB-SSBL-EM, the FB dictionary is used instead of the DCT dictionary as the Bessel functions have non-stationary characteristics and provide good compression for signals like the EEG.

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 EEG rhythms separation , determination of respiratory rate , parametric modelling of speech , human emotion detection , and epileptic seizure detection.

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

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
• FBSE based signal analysis.pdf