In electronics and signal processing, a Bessel filter is a type of analog linear filter with a maximally flat group delay (i.e., maximally linear phase response), which preserves the wave shape of filtered signals in the passband. Bessel filters are often used in audio crossover systems.
The filter's name is a reference to German mathematician Friedrich Bessel (1784–1846), who developed the mathematical theory on which the filter is based. The filters are also called Bessel–Thomson filters in recognition of W. E. Thomson, who worked out how to apply Bessel functions to filter design in 1949.
The Bessel filter is very similar to the Gaussian filter, and tends towards the same shape as filter order increases. While the time-domain step response of the Gaussian filter has zero overshoot, the Bessel filter has a small amount of overshoot, but still much less than other common frequency-domain filters, such as Butterworth filters. It has been noted that the impulse response of Bessel–Thomson filters tends towards a Gaussian as the order of the filter is increased.
Compared to finite-order approximations of the Gaussian filter, the Bessel filter has better shaping factor, flatter phase delay, and flatter group delay than a Gaussian of the same order, although the Gaussian has lower time delay and zero overshoot.
The transfer function
A Bessel low-pass filter is characterized by its transfer function:
where is a reverse Bessel polynomial from which the filter gets its name and is a frequency chosen to give the desired cut-off frequency. The filter has a low-frequency group delay of . Since is indeterminate by the definition of reverse Bessel polynomials, but is a removable singularity, it is defined that .
Bessel polynomials
The transfer function of the Bessel filter is a rational function whose denominator is a reverse Bessel polynomial, such as the following:
The reverse Bessel polynomials are given by:
where
Setting the cutoff attenuation
There is no standard set attenuation value for Bessel filters, however, −3.0103 dB is a common choice. Some applications may use a higher or lower attenuation, such as −1 dB or −10 dB. The attenuation is set by frequency scaling the denominator. This can be approximated by interpolating Bessel filter attenuation tables, or calculated precisely with Newton's method.
The denominator may be frequency scaled by using a scaling factor that we will refer to as . The frequency-scaled denominator of the Bessel transfer function may be rewritten for the case above and the example below as follows:
The task for modifying to is to find an that results in the desired attenuation at 1 r/s for the normalized transfer function. Newton's method can be easily summarized to do this by its basic definition:
through successive iterations of until is the frequency that attenuates to the desired attenuation at 1 radian/s.
While the above summary may be concise and easily understood, the mechanics of obtaining an accurate derivative of the magnitude function along the axis may be problematic. Digital techniques may be used, but it is generally better to apply Newton's method with continuous functions, if possible, so as to maximize accuracy. Therefore, it is useful to modify the expression to eliminate excess mathematical functions by making the following alterations:
Multiply by to obtain . This will eliminate complex numerical results when evaluating by removing the odd order terms in the polynomial. In , negate all terms of when is divisible by . That would be , , , and so on. We will call the modified function , and this modification will allow the use of real numbers instead of complex numbers when evaluating the polynomial and its derivative. That is, we can now use the real in place of the complex Convert the desired attenuation in dB, , to an arithmetic attenuation value, , using . For example, −3.010 dB is 0.7071, −1 dB is 0.8913 and so on. This simplifies the derivative evaluation. Square the resulting arithmetic attenuation to correlate with the squaring of the function. Calculate the modified in Newton's method using the real value, . Always take the absolute value. Calculate the derivative the modified with respect to the real value, . Negate the derivative computation to account for the effects due to the modifications made to create . DO NOT take the absolute value of the derivative.
When steps 1) through 4) are complete, the expression involving Newton's method may be written: