Fun with Bessel Functions

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Well, I certainly forget things faster than I learn them. Today is a quick review of Bessel functions and their applications to signal processing.

The Bessel functions appear in lots of situations (think wave propagation and static potentials), particularly those that involve cylindrical symmetry. While special types of what would later be known as Bessel functions were studied by Euler, Lagrange, and the Bernoullis, the Bessel functions were first used by F. W. Bessel to describe three body motion, with the Bessel functions appearing in the series expansion on planetary perturbation

First, I think they should be called Bernoulli-Bessel functions both because that sounds more pompous and because they were discovered by Daniel Bernoulli and generalized by Friedrich Bessel. While they sound (and can be) complicated, they are the canonical solutions [latex]y(x)[/latex] of Bessel’s differential equation:

$$ x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 – \alpha^2)y = 0 $$

for an arbitrary complex number [latex]\alpha[/latex] (where [latex]\alpha[/latex] denotes the order of the Bessel function). The most important cases are for [latex]\alpha[/latex] as an integer or half-integer. Since all math ties together, I find it pretty cool that Bessel functions appear in the solution to Laplace’s equation in cylindrical coordinates.

Although [latex]\alpha[/latex] and −[latex]\alpha[/latex] produce the same differential equation that a real [latex]\alpha[/latex] does, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of [latex]\alpha[/latex].

Bessel functions of the first kind: [latex]J_\alpha[/latex]

Bessel functions of the first kind, known as [latex]J_\alpha (x)[/latex], are solutions of Bessel’s differential equation that are finite at the origin ([latex]x = 0[/latex]) for integer or positive [latex]\alpha[/latex], and diverge as x approaches zero for negative non-integer [latex]\alpha[/latex]. It is possible to define the function by its Taylor series expansion around [latex]x = 0[/latex].

$$ J_\alpha(x) = \sum_{m=0}^\infty \frac{(-1)^m}{m! \, \Gamma(m+\alpha+1)} {\left(\frac{x}{2}\right)}^{2m+\alpha} $$

where [latex]\Gamma(z)[/latex] is the gamma function, a shifted generalization of the factorial function to non-integer values.

Bessel functions of the second kind : [latex]Y_\alpha[/latex]

The Bessel functions of the second kind, denoted by [latex] Y_\alpha (x) [/latex] are solutions of the Bessel differential equation that have a singularity at the origin.

For non-integer [latex]\alpha[/latex], [latex]Y_\alpha[/latex] is related to [latex]J_\alpha[/latex] by:

$$ Y_\alpha(x) = \frac{J_\alpha(x) \cos(\alpha\pi) – J_{-\alpha}(x)}{\sin(\alpha\pi)} $$

In the case of integer order n, the function is defined by taking the limit as a non-integer [latex]\alpha[/latex] tends to [latex]n[/latex]:

$$ Y_n(x) = \lim_{\alpha \to n} Y_\alpha(x). $$

There is also a corresponding integral formula (for Re(x) > 0),

$$ Y_n(x) =\frac{1}{\pi} \int_0^\pi \sin(x \sin\theta – n\theta) \, d\theta – \frac{1}{\pi} \int_0^\infty \left[ e^{n t} + (-1)^n e^{-n t} \right] e^{-x \sinh t} \, dt.$$

[latex]Y_\alpha(x)[/latex] is necessary as the second linearly independent solution of the Bessel’s equation when [latex]\alpha[/latex] is an integer. But [latex]Y_\alpha (x)[/latex] can be considered as a ‘natural’ partner of [latex]J_\alpha (x)[/latex].

When [latex]\alpha[/latex] is an integer, moreover, as was similarly the case for the functions of the first kind, the following relationship is valid:

$$ Y_{-n}(x) = (-1)^n Y_n(x).\,$$

Both [latex]J_\alpha (x)[/latex] and [latex]Y_\alpha (x)[/latex] are holomorphic functions of x on the complex plane cut along the negative real axis. When [latex]\alpha[/latex] is an integer, the Bessel functions [latex]J[/latex] are entire functions of x. If x is held fixed, then the Bessel functions are entire functions of [latex]\alpha[/latex].

Bessel Filters

In electronics and signal processing, a Bessel filter is a type of linear filter with a maximally flat group delay. The Bessel filter is used because a low pass filter is characterized by transfer function. The denominator of the Bessel filter is a reverse Bessel polynomial. Bessel filters are often used in audio crossover systems. Analog Bessel filters are characterized by almost constant group delay across the entire passband, thus preserving the wave shape of filtered signals in the passband.

A low pass active filter with a Bessel response is used when the filter needs to exhibit minimum differential delay between the various frequency components of interest contained within the input signal being filtered. In essence this means that the fundamental frequency of say an applied squarewave experiences the same input-to-output delay as the other harmonics within the filter’s pass-band. This results in a high degree of fidelity of the output signal relative to the input signal.

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