Could not load that page.

Fourier–Bessel series

☆ Save On Wikipedia ↗

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) α = 0 {\displaystyle \alpha =0} {\displaystyle \alpha =0} and (ii) α = 1 {\displaystyle \alpha =1} {\displaystyle \alpha =1}.

f : [ 0 , b ] → R {\displaystyle f:[0,b]\to \mathbb {R} } {\displaystyle 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 Jα, where the argument to each version n is differently scaled, according to [1][2] ( J α ) n ( x ) := J α ( u α , n b x ) {\displaystyle (J_{\alpha })_{n}(x):=J_{\alpha }\left({\frac {u_{\alpha ,n}}{b}}x\right)} {\displaystyle (J_{\alpha })_{n}(x):=J_{\alpha }\left({\frac {u_{\alpha ,n}}{b}}x\right)} where uα,n is a root, numbered n associated with the Bessel function Jα and cn are the assigned coefficients:[3] f ( x ) ∼ ∑ n = 1 ∞ c n J α ( u α , n b x ) . {\displaystyle f(x)\sim \sum _{n=1}^{\infty }c_{n}J_{\alpha }\left({\frac {u_{\alpha ,n}}{b}}x\right).} {\displaystyle 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.

⟨ f , g ⟩ = ∫ 0 b x f ( x ) g ( x ) d x {\displaystyle \langle f,g\rangle =\int _{0}^{b}xf(x)g(x)\,dx} {\displaystyle \langle f,g\rangle =\int _{0}^{b}xf(x)g(x)\,dx}

according to

∫ 0 b x J α ( x u α , n b ) J α ( x u α , m b ) d x = b 2 2 δ m n [ J α + 1 ( u α , n ) ] 2 , {\displaystyle \int _{0}^{b}xJ_{\alpha }\left({\frac {xu_{\alpha ,n}}{b}}\right)\,J_{\alpha }\left({\frac {xu_{\alpha ,m}}{b}}\right)\,dx={\frac {b^{2}}{2}}\delta _{mn}[J_{\alpha +1}(u_{\alpha ,n})]^{2},} {\displaystyle \int _{0}^{b}xJ_{\alpha }\left({\frac {xu_{\alpha ,n}}{b}}\right)\,J_{\alpha }\left({\frac {xu_{\alpha ,m}}{b}}\right)\,dx={\frac {b^{2}}{2}}\delta _{mn}[J_{\alpha +1}(u_{\alpha ,n})]^{2},}

(where: δ m n {\displaystyle \delta _{mn}} {\displaystyle \delta _{mn}} is the Kronecker delta). The coefficients can be obtained from projecting the function f(x) onto the respective Bessel functions:

c n = ⟨ f , ( J α ) n ⟩ ⟨ ( J α ) n , ( J α ) n ⟩ = ∫ 0 b x f ( x ) ( J α ) n ( x ) d x 1 2 ( b J α ± 1 ( u α , n ) ) 2 {\displaystyle 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}}}} {\displaystyle 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.

For the inverse transform, one makes use of the following representation of the Dirac delta function[4]

2 x α y 1 − α b 2 ∑ k = 1 ∞ J α ( x u α , k b ) J α ( y u α , k b ) J α + 1 2 ( u α , k ) = δ ( x − y ) . {\displaystyle {\frac {2x^{\alpha }y^{1-\alpha }}{b^{2}}}\sum _{k=1}^{\infty }{\frac {J_{\alpha }\left({\frac {xu_{\alpha ,k}}{b}}\right)\,J_{\alpha }\left({\frac {yu_{\alpha ,k}}{b}}\right)}{J_{\alpha +1}^{2}(u_{\alpha ,k})}}=\delta (x-y).} {\displaystyle {\frac {2x^{\alpha }y^{1-\alpha }}{b^{2}}}\sum _{k=1}^{\infty }{\frac {J_{\alpha }\left({\frac {xu_{\alpha ,k}}{b}}\right)\,J_{\alpha }\left({\frac {yu_{\alpha ,k}}{b}}\right)}{J_{\alpha +1}^{2}(u_{\alpha ,k})}}=\delta (x-y).}

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,[5] discrimination of odorants in a turbulent ambient,[6] postural stability analysis, detection of voice onset time, glottal closure instants (epoch) detection, separation of speech formants, speech enhancement,[7] and speaker identification.[8] 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 b f ′ ( b ) + c f ( b ) = 0 , {\displaystyle bf'(b)+cf(b)=0,} {\displaystyle bf'(b)+cf(b)=0,} where c {\displaystyle c} {\displaystyle c} is an arbitrary constant. The Dini series can be defined by f ( x ) ∼ ∑ n = 1 ∞ b n J α ( γ n x / b ) , {\displaystyle f(x)\sim \sum _{n=1}^{\infty }b_{n}J_{\alpha }(\gamma _{n}x/b),} {\displaystyle f(x)\sim \sum _{n=1}^{\infty }b_{n}J_{\alpha }(\gamma _{n}x/b),}

where γ n {\displaystyle \gamma _{n}} {\displaystyle \gamma _{n}} is the n-th zero of x J α ′ ( x ) + c J α ( x ) {\displaystyle xJ'_{\alpha }(x)+cJ_{\alpha }(x)} {\displaystyle xJ'_{\alpha }(x)+cJ_{\alpha }(x)}.

The coefficients b n {\displaystyle b_{n}} {\displaystyle b_{n}} are given by b n = 2 γ n 2 b 2 ( c 2 + γ n 2 − α 2 ) J α 2 ( γ n ) ∫ 0 b J α ( γ n x / b ) f ( x ) x d x . {\displaystyle 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.} {\displaystyle 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

References

  1. Magnus, Wilhelm; Oberhettinger, Fritz; Soni, Raj Pal (1966). Formulas and Theorems for the Special Functions of Mathematical Physics. doi:10.1007/978-3-662-11761-3. ISBN 978-3-662-11763-7.
  2. R., Smythe, William (1968). Static and dynamic electricity. - 3rd ed. McGraw-Hill. OCLC 878854927.{{cite book}}: CS1 maint: multiple names: authors list (link)
  3. Schroeder, Jim (April 1993). "Signal Processing via Fourier-Bessel Series Expansion". Digital Signal Processing. 3 (2): 112–124. Bibcode:1993DSP.....3..112S. doi:10.1006/dspr.1993.1016. ISSN 1051-2004.
  4. Cahill, Kevin (2019). Physical Mathematics. Cambridge University Press. p. 385. ISBN 9781108470032. Retrieved 9 March 2023.
  5. D’Elia, Gianluca; Delvecchio, Simone; Dalpiaz, Giorgio (2012), "On the Use of Fourier-Bessel Series Expansion for Gear Diagnostics", Condition Monitoring of Machinery in Non-Stationary Operations, Berlin, Heidelberg: Springer Berlin Heidelberg, pp. 267–275, doi:10.1007/978-3-642-28768-8_28, ISBN 978-3-642-28767-1, retrieved 2022-10-22
  6. Vergaraa, A.; Martinelli, E.; Huerta, R.; D’Amico, A.; Di Natale, C. (2011). "Orthogonal Decomposition of Chemo-Sensory Signals: Discriminating Odorants in a Turbulent Ambient". Procedia Engineering. 25: 491–494. doi:10.1016/j.proeng.2011.12.122. ISSN 1877-7058.
  7. Gurgen, F.S.; Chen, C.S. (1990). "Speech enhancement by fourier–bessel coefficients of speech and noise". IEE Proceedings I - Communications, Speech and Vision. 137 (5): 290. doi:10.1049/ip-i-2.1990.0040. ISSN 0956-3776.
  8. Gopalan, K.; Anderson, T.R.; Cupples, E.J. (May 1999). "A comparison of speaker identification results using features based on cepstrum and Fourier-Bessel expansion". IEEE Transactions on Speech and Audio Processing. 7 (3): 289–294. doi:10.1109/89.759036. ISSN 1063-6676.