where uα,n is a
root, numbered n associated with the Bessel function Jα and cn are the assigned coefficients:[3]
Interpretation
The FourierâBessel series may be thought of as a Fourier expansion in the Ï 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 FourierâBessel 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.
For the inverse transform, one makes use of the following representation of the
Dirac delta function[4]
One-to-one relation between order index (n) and continuous frequency ()
FourierâBessel series coefficients are unique for a given signal, and there is one-to-one mapping between continuous frequency () and order index which can be expressed as follows:
Since, . So above equation can be rewritten as follows:
where is the length of the signal and is the sampling frequency of the signal.
2-D- Fourier-Bessel series expansion
For an image of size MĂN, the synthesis equations for order-0 2D-FourierâBessel series expansion is as follows:
Where is 2D-FourierâBessel series expansion coefficients whose mathematical expressions are as follows:
where,
Fourier-Bessel series expansion based entropies
For a signal of length , Fourier-Bessel based spectral entropy such as Shannon spectral entropy (), log energy entropy (), and Wiener entropy () are defined as follows:
where is the normalized energy distribution which is mathematically defined as follows:
is energy spectrum which is mathematically defined as follows:
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).[5] 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.[5] 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, ]. 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.
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:
where Y(l) are the FBSE coefficients and these coefficients are calculated using the following expression as
The is termed as the root of the Bessel function, and it is evaluated in an iterative manner based on the solution of using the Newton-Rapson method. Similarly, the g(m,l) is the FBSE domain Gaussian window and it is given as follows :
FourierâBessel 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.[6] It has been observed that the filtering operation distorts the amplitude and phase modulations in the separated monocomponent signals.
Advantages
The FourierâBessel 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 FourierâBessel series expansion spectrum provides frequency points equal to the signal length.
Applications
The FourierâBessel series expansion employs aperiodic and decaying Bessel functions as the basis. The FourierâBessel series expansion has been successfully applied in diversified areas such as Gear fault diagnosis,[7] discrimination of odorants in a turbulent ambient,[8] postural stability analysis, detection of voice onset time, glottal closure instants (epoch) detection, separation of speech formants, speech enhancement,[9] and speaker identification.[10] The FourierâBessel series expansion has also been used to reduce cross terms in the WignerâVille distribution.
Dini series
A second FourierâBessel 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