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Adjustable Gauss Transitional Filter       Home    

The Adjustable Gauss Polynomial is a new polynomial for filter design that can generate a response anywhere between a Gaussian and Butterworth response.

Transitional filters are not new. According to the Electronics Filter Design Handbook by Williams and Taylor, they are typically "generated using mathematical techniques which involve interpolation of pole locations". Humpherys also describes this method in The Analysis, Design, and Synthesis of Electrical Filters. He then goes a step further and describes a method that involves the simultaneous solution of n non linear equations. Invariably, these discussions end with, "the interested reader is referred to the literature".

The Adjustable Gauss Filter is a new filter in the sense that we have a closed form solution for the pole locations. We start with the Gauss polynomial, which is defined as:

Equation1.png

We modify this by taking the coefficients to the power of Gamma, -2 <= Γ <= 1, as shown here:

Equation2.png

See  Adjustable Gauss Algorithm for more details.

 

Adjustable Gauss Filter Responses

Except for the Bessel, there is no middle ground between a Gauss and Butterworth response. The Gauss has a step response with no overshoot and constant group delay, but its stop band is almost non existent. The Butterworth on the other hand, has good attenuation characteristics, but can distort a signal badly.

Here we show the filter responses generated by the Adjustable Gauss as we vary Gamma. It transitions smoothly from the Gauss, to the Bessel, to the Butterworth.

10 Pole Adjustable Gauss Gain and GD Plots.png

10 Pole Adjustable Gauss Step Response Plots.png

 

The following plots compare the response of this polynomial to a Gauss, Bessel, and Butterworth.

Adj Gauss and Gauss.png

Adj Gauss and Bessel.png

Adj Gauss and Butterworth.png

It should be noted here that little effort was made to get this polynomial to replicate a Butterworth, Gauss, or Bessel, nor is it necessary. It's purpose is to provide a means to transition between these various responses in a smooth way. If a true Gauss response is needed, then the Gauss polynomial should be used. Likewise for the Butterworth and Bessel.

This plot shows how the poles move across the s plane as Gamma is varied from -1 to 1. These are frequency corrected pole locations, which is to say, these pole locations give a 3 dB corner at Omega = 1.

Adjustable Gauss Pole Trajectories.png

Adjustable Gauss Algorithm

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