Bessel filter

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Linear analog electronic filters
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In electronics and signal processing, a Bessel filter is a variety of linear filter with a maximally flat group delay (linear phase response). 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. The filter is named in honor of Friedrich Bessel, a German mathematician (1784–1846).

1 The transfer function
2 A simple example
3 See also
4 External links

[edit] The transfer function

A plot of the gain and group delay for a fourth-order low pass Bessel filter. Note that the transition from the pass band to the stop band is much slower than for other filters, but the group delay is practically constant in the passband. The Bessel filter maximizes the flatness of the group delay curve at zero frequency.

A Bessel low-pass filter is characterized by its transfer function:

$H(s) = \frac{\theta_n(0)}{\theta_n(s/\omega_0)}\,$

where $θ n (s)$ is a reverse Bessel polynomial from which the filter gets its name and $ω 0$ is a frequency chosen to give the desired cut-off frequency.

[edit] A simple example

The transfer function for a third order Bessel low pass filter is

$H(s)=\frac{15}{s^3+6s^2+15s+15}\,$

The gain is then

$G(\omega) = |H(j\omega)| = \frac{15}{\sqrt{\omega^6+6\omega^4+45\omega^2+225}}$

The phase is

$\phi(\omega)=-\mathrm{arg}(H(j\omega))= -\mathrm{arctan}\left(\frac{15\omega-\omega^3}{15-6\omega^2}\right)\,$

The group delay is then

$D(\omega)=-\frac{d\phi}{d\omega} = \frac{6 \omega^4+ 45 \omega^2+225}{\omega^6+6\omega^4+45\omega^2+225}$

The Taylor series expansion of the group delay is

$D(\omega) = 1-\frac{\omega^6}{225}+\frac{\omega^8}{1125}+\cdots$

Note that the two terms in $ω 2$ and $ω 4$ are zero, resulting in a very flat group delay at ω=0. This is the greatest number of terms that can be set to zero, since there are a total of four coefficients in the third order Bessel polynomial, requiring four equations in order to be defined. One equation specifies that the gain be unity at $ω = 0$ and a second specifies that the gain be zero at $\omega=\infty$ , leaving two equations to specify two terms in the series expansion to be zero. This is a general property of the group delay for a Bessel filter of order n: the first n-1 terms in the series expansion of the group delay will be zero, thus maximizing the flatness of the group delay at $ω = 0$ .