Fourier–Bessel series

Infinite series of Bessel functions

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.

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.

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. It has been observed that the filtering operation distorts the amplitude and phase modulations in the separated monocomponent signals.

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.

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, speech enhancement, and speaker identification. The Fourier–Bessel series expansion has also been used to reduce cross terms in the Wigner–Ville distribution.