Gaussian Filters

The Gaussian Filter is the filter type that results in the most gradual pass band roll-off and the lowest group delay. As the name states, the Gaussian Filter is derived from the same basic equations used to derive the Gaussian Distribution. The significant characteristic of the Gaussian Filter is that the step response contains no overshoot at all.

FilterSolutions normalizes the Gaussian filter such that the prototype high frequency attenuation matches the Butterworth filter. The pass band attenuation of the Gaussian 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 transfer function of the prototype Gaussian filter is one over the square root of the Maclaurin series expansion of using the same number of terms as the order of the filter and using left half plane poles only for the square root.

The prototype transfer function is normalized so that the high frequency attenuation matches the Butterworth filter.

 

Transitional Filters

A transition option is available for Gaussian filters of order 3 or more. If a transition is selected, the attenuation of the Gaussian filter greatly increases for frequencies higher than the frequency that exists at the transition dB selection point. Gaussian Transition Filters are created with numerical methods that minimize the magnitude RMS error between the realizable Gaussian Transition filter and the ideal Gaussian Transition filter.

All Gaussian Transitional filters use -3 dB as the attenuation cutoff frequency. The steep roll off frequency is selectable by the user.

 

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

Gaussian and Transitional 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.