Fourier-Bessel

Three-dimensional sound can be described, from a physical point of view, by an acoustic field, which is defined for each point (x,y,z) in space and for each instant t using the pressure field p(x,y,z,t). Nevertheless, manipulating an acoustic field using its primary representation p(x,y,z,t) is not easy because it would be necessary to know it for each value of (x,y,z,t). Therefore, an acoustic field is decomposed, in spherical coordinates, into its Fourier-Bessel expansion, offering a much convenient and compact representation. From the four dimensional continuous function p(r,,,t), the Fourier-Bessel decomposition gives a set of signals called Fourier-Bessel coefficients of the acoustic field, denoted pl,m(t), where l and m are integers that satisfy l = 0 and -l = m = l. In the Fourier-Bessel formalism, l is called the order. In the frequency domain, P(r,,,f) and Pl,m(f) are the Fourier transforms of P(r,,,f) and pl,m(t) respectively. This decomposition is given by the following expression:

where k = 2_f=c and c is the speed of sound, approximately 340 m/s. The Fourier-Bessel expansion is generally truncated at some order L. This order determines the resolution of the acoustic field representation. The higher the order, the higher the acoustic field representation fidelity will be, but the more computation power and signals will be required.

They are composed of two parts:

sperical harmonics 

and

sperical Bessel functions 

Spherical Bessel functions give the radial behavior of Fourier-Bessel functions whereas spherical harmonics give their angular behavior.
The variable l is called order.
The Fourier-Bessel functions are defined for each l > 0 and for each mverifying -l < m < l.

The pressure field corresponding to each elementary Fourier-Bessel function is represented on the following figure, where each row corresponds to a value ofl (from 0 to 2) and each column corresponds to a value of m (from -2 to 2).
The indication tells in which plane the field is represented (for example, X,Yindicates the horizontal plane).

Spherical harmonics are already used by Ambisonics, but most of the time only at orders 0 and 1.
Using higher order Ambisonics is indeed difficult because of the lack of good high order directivity microphones.
Spherical harmonics are defined for each l > 0 and for each m verifying -l < m < l.
They are bi-dimensional objects that allow to model the direction of arrival of sound.
Their usual representation is obtained by Spherical Fourier Transform of the Fourier-Bessel coefficients.

The first spherical harmonics are represented on the figure below, where each row corresponds to a value of l
(from 0 to 3) and each column corresponds to a value of m (from -3 to 3).

However, considering spherical harmonics alone is not sufficient to manage acoustic fields (record, manipulate or reproduce).

Our research works are addressing the complete Fourier-Bessel representation, including its radial part, and we developed a unique knowledge to manipulate these complex mathematical objects.