Bessel Filters

The Bessel Filter's distinguishing characteristic is the near constant group delay throughout the pass band of the low pass filter.

FilterSolutions normalizes the Bessel filter such that the prototype high frequency attenuation matches the Butterworth filter. The pass band attenuation of the Bessel filter increases with the order of the filter when this normalization is applied. However, FilterSolutions allows the user the option of selecting the desired pass band attenuation in dB's.

 

Derivation:

The continuous transfer function of the prototype Bessel filter is K/F(S) where F(S) is the recursive Bessel relation:

 

Linear Phase Filters with Equiripple Group Delay

Select Equiripple Delay to generate Linear Phase filters with an equiripple group delay. Linear Phase filters are a group of filters that includes Bessel Filters. The synthesis process for equiripple group delay is identical to Delay filters, except the numerator of the prototype transfer function is a constant. The low pass group delay of Linear Phase equiripple filters remains nearly constant at higher frequencies than the Bessel Filters.

The group delay is designed to be accurate for frequencies up to approximately (N-1) /(T*PI) Hz or 2*(N-1)/T Rad/Sec where N is the order of the filter, and T is the design transport delay. This figures varies as the ripple period varies.

 

Ripple Period

FilterSolutions allows one to fine tune the Bessel Filter by changing the ripple period. With an equiripple period of 2.0, the group delay of the filter is set to a forced value approximately every R/(T*2*PI) Hz or R/T Rad/Sec where R is the ripple period and T is the transport delay. Increasing the ripple period has the effect of extending the upper frequency limit of the group delay at the cost of increasing the error of the group delay. The period of each individual ripple is modified slightly to obtain the equiripple effect of the group delay.

The allowable values of the ripple period are 0 to 2.8. Setting the ripple period to zero produces a classical Bessel group delay response.

 

Ripple Magnitude

Several precomputed group delay ripple magnitudes may be selected in lieu of selecting the ripple period. The advantage of using a precomputed filter is that the group delays are far more accurate equiripples.

 

With Stop Bands

Low pass Linear Phase Filters with stop bands are essentially Matched Filters that have been optimized for frequency response. The prototype is derived directly from the Delay Filter prototype:

 

Synthesis Options

Specific element values of synthesized filters are no unique.  Three different options are provided in the Advanced panel for design flexibility.  The FilterQuick design panel using the option that most likely results in all positive element values.

 

Stop Bands and Transmission Zeros

Bessel and Linear Phase filters may be synthesized with stop bands, both equiripple and single point ripple, using advanced synthesis techniques, and with arbitrarily placed transmission zeros.  Stopband synthesis may lead to undesirable negative element values, especially at small attenuations.  Increasing the stop band attenuation will eventually make all values positive.  The attenuation threshold of negative element values changes with different synthesis option selections, so the first attempt to obtain an all-positive element value solution is to  try different synthesis options.  Element values are more likely to be positive when fewer transmission zeros are used, so creating a stopband manually with transmission zeros placement may also help.  Selecting an attenuation requirement that maximizes stop band attenuation also helps mitigate the negative element value problem.