What is a Bessel Function?

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bhq

Joined: 11 Jun 2008
Posts: 81
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16 Aug 2008 10:57 bessel transfer function

What is a Bessel Function?
What it says, and where it is used espacially in Electronics Engineering

Awarapunshee

Joined: 11 Jul 2007
Posts: 455
Helped: 60

16 Aug 2008 11:49 transfer function bessel function

Bessel's equation arises when finding separable solutions to Laplace's equation and the Helmholtz equation in cylindrical or spherical coordinates. Bessel functions are therefore especially important for many problems of wave propagation, static potentials, and so on. In solving problems in cylindrical coordinate systems.

Bessel functions are associated with a wide range of problems in important areas of mathematical physics. Bessel function theory is applied to problems of acoustics, radio physics, hydrodynamics, and atomic and nuclear physics.

LvW

Joined: 07 May 2008
Posts: 1466
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Location: Germany

16 Aug 2008 12:04 bessel function application

bhq wrote:

What is a Bessel Function?
What it says, and where it is used espacially in Electronics Engineering

I suppose,you are asking in context with lowpass filter transfer functions, don´t you ?
Well, opposite to the well known Butterworth approximation with a maximally flat AMPLITUDE response sometimes one is interested in a maximally flat (that means nearly constant) GROUP DELAY function within a certain frequency region.
The ideal transfer function for such a filter is H(s)=exp(-s*T). Each system having this transfer function would simply delay a signal without any influence on its shape (for example: square wave).
However, to realize such a function with lumped elements you need a polynom in s with positive real coefficients. Therefore, the ideal transfer function is approximated by such polynoms. To compute the coefficients for various orders of such polynoms one makes use of the so called "Bessel polynoms". Bessel was a mathematician born in 1784 and - of course - he never has heard something about filter circuits.
As a result, a lowpass response with a maximum flat group delay (resp. a maximum linear phase response) can be designed. And this is called "Bessel low pass" or sometimes "Thomson low pass" or "Thomson-Bessel low pass" as W.E Thomson was the first to propose in 1949 to use these Bessel polynoms for filter designs.

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LvW

Joined: 07 May 2008
Posts: 1466
Helped: 242
Location: Germany

17 Aug 2008 9:14 maximally flat thomson 1949

Supplement to my reply from august 16th, 13:04 :

In my first reply I only spoke about BESSEL polynomes and theit relation to filter theory.
In addition, there are so called BESSEL functions which play a certain role in modulation techniques.
If you frequency modulate (FM) a sinusoidal carrier with another sine function you get a spectrum consisting of the carrier and several sidebands.
Then, the various amplitudes of theses lines depend on the modulation index and can be mathematically derived from these BESSEL functions.