In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Fourier–Bessel series is a particular kind of
generalized Fourier series
A generalized Fourier series is the expansion of a square integrable function into a sum of square integrable orthogonal basis functions. The standard Fourier series uses an orthonormal basis of trigonometric functions, and the series expansion ...
(an
infinite series
In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathemati ...
expansion on a finite interval) based on
Bessel function
Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary complex ...
s.
Fourier–Bessel series are used in the solution to
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives.
The function is often thought of as an "unknown" that solves the equation, similar to ho ...
s, particularly in
cylindrical coordinate
A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions around a main axis (a chosen directed line) and an auxiliary axis (a reference ray). The three cylindrical coordinates are: the point perpen ...
systems.
Definition
The Fourier–Bessel series of a function with a
domain
A domain is a geographic area controlled by a single person or organization. Domain may also refer to:
Law and human geography
* Demesne, in English common law and other Medieval European contexts, lands directly managed by their holder rather ...
of satisfying
is the representation of that function as a
linear combination
In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...
of many
orthogonal
In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ...
versions of the same
Bessel function of the first kind
Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary complex ...
''J''
''α'', where the argument to each version ''n'' is differently scaled, according to
where ''u''
''α'',''n'' is a
root
In vascular plants, the roots are the plant organ, organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often bel ...
, numbered ''n'' associated with the Bessel function ''J''
''α'' and ''c''
''n'' are the assigned coefficients:
Interpretation
The Fourier–Bessel series may be thought of as a Fourier expansion in the ρ coordinate of
cylindrical coordinates
A cylinder () has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
A cylinder may also be defined as an infinite ...
. Just as the
Fourier series
A Fourier series () is an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems ...
is defined for a finite interval and has a counterpart, the
continuous Fourier transform
In mathematics, the Fourier transform (FT) is an integral transform that takes a function (mathematics), function as input then outputs another function that describes the extent to which various Frequency, frequencies are present in the origin ...
over an infinite interval, so the Fourier–Bessel series has a counterpart over an infinite interval, namely the
Hankel transform
In mathematics, the Hankel transform expresses any given function ''f''(''r'') as the weighted sum of an infinite number of Bessel functions of the first kind . The Bessel functions in the sum are all of the same order ν, but differ in a scalin ...
.
Calculating the coefficients
As said, differently scaled Bessel Functions are orthogonal with respect to the
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, ofte ...
according to
(where:
is the Kronecker delta). The coefficients can be obtained from
projecting the function 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
In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
One-to-one relation between order index (''n'') and continuous frequency ()

Fourier–Bessel 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 M×N, the synthesis equations for order-0 2D-Fourier–Bessel series expansion is as follows:
Where
is 2D-Fourier–Bessel 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).
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
\frac">, 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-Raphson method. Similarly, the g(m,l) is the FBSE domain Gaussian window and it is given as follows :
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.
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, 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
In mathematics, the Robin boundary condition ( , ), or third type boundary condition, is a type of boundary condition, named after Victor Gustave Robin (1855–1897). When imposed on an ordinary or a partial differential equation, it is ...
where
is an arbitrary constant.
The Dini series can be defined by
where
is the ''n''-th zero of
.
The coefficients
are given by
See also
*
Orthogonality
In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendicular'' is more specifically ...
*
Generalized Fourier series
A generalized Fourier series is the expansion of a square integrable function into a sum of square integrable orthogonal basis functions. The standard Fourier series uses an orthonormal basis of trigonometric functions, and the series expansion ...
*
Hankel transform
In mathematics, the Hankel transform expresses any given function ''f''(''r'') as the weighted sum of an infinite number of Bessel functions of the first kind . The Bessel functions in the sum are all of the same order ν, but differ in a scalin ...
*
Kapteyn series Kapteyn may refer to:
* Jacobus Kapteyn - Astronomer
** Parallactic instrument of Kapteyn - the instrument used by Kapteyn to analyze photographic plates
** Jacobus Kapteyn Telescope - telescope named after Jacobus Kapteyn
** Kapteyn's Star - s ...
*
Neumann polynomial In mathematics, the Neumann polynomials, introduced by Carl Neumann for the special case \alpha=0, are a sequence of polynomials in 1/t used to expand functions in term of Bessel functions.
The first few polynomials are
:O_0^(t)=\frac 1 t,
:O_1^(t) ...
*
Schlömilch's series
References
External links
*
*
* Fourier–Bessel series applied to Acoustic Field analysis o
Trinnov Audio's research page
{{DEFAULTSORT:Fourier-Bessel series
Fourier series