Saturday, October 17, 2009

Bessel filter

In electronics and signal processing, a Bessel filter is a type of linear filter with a maximally flat group delay (maximally 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's name is a reference to Friedrich Bessel, a German mathematician (1784–1846), who developed the mathematical theory on which the filter is based.

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:[1]
H(s) = \frac{\theta_n(0)}{\theta_n(s/\omega_0)}\,
where θn(s) is a reverse Bessel polynomials from which the filter gets its name and ω0 is a frequency chosen to give the desired cut-off frequency. The filter has a low-frequency group delay of 1 / ω0.

Bessel polynomials

The roots of the third-order Bessel polynomial are the poles of filter transfer function in the s plane, here plotted as crosses.
The transfer function of the Bessel filter is a rational function whose denominator is a reverse Bessel polynomial, such as the following:
n=1; \quad s+1
n=2; \quad s^2+3s+3
n=3; \quad s^3+6s^2+15s+15
The reverse Bessel polynomials are given by:[1]
P(s)=\sum_{k=0}^N a_ks^k where
a_k=\frac{(2N-k)!}{2^{N-k}k!(N-k)!} \quad k=0,1,...,N

Example

Gain plot of the third-order Bessel filter, versus normalized frequency
Group delay plot of the third-order Bessel filter, illustrating flat unit delay in the passband
The transfer function for a third-order (three-pole) Bessel low-pass filter, normalized to have unit group delay, is
H(s)=\frac{15}{s^3+6s^2+15s+15}\,
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 \plusmn i 1.7544\quad, plotted above. The numerator 15 is chosen to give unity gain at DC (at s = 0).
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
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 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 \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
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