Bessel Filter - Example

Example

The transfer function for a third-order (three-pole) Bessel low-pass filter, normalized to have unit group delay, is

The roots of the denominator polynomial, the filter's poles, include a real pole at s = −2.3222, and a complex-conjugate pair of poles at s = −1.8389 ± j1.7544, plotted above. The numerator 15 is chosen to give a gain of 1 at DC (at s = 0).

The gain is then

The phase is

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

The group delay is

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

The Taylor series expansion of the group delay is

Note that the two terms in ω2 and ω4 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 ω = ∞, 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.

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