Bilinear Approximation to nth order Low Pass Butterworth Filter.

Cuthbert Nyack
Frequency Response, Transfer function and pole locations are shown by the applet below. N is the order of the filter and fc/fs is the cut off frequency normalised to the sampling frequency.
Fn = 0 shows the frequency response.
Fn = 1 shows the transfer function represented as a product of quadratic factors.
Fn = 2 shows the pole locations for the continuous Butterworth filter and the discrete Bilinear approximation.

To design a filter which has a gain of ~ -1dB at 0.1ws and ~ -40dB at 0.15ws, proceed as follows:-

Set BL = 75 and WL to 113 to locate the frequency markers.
Adjust N, fc/fs until the conditions are met.
In this case N = 12 and fc/fs = 0.106.

The last entry in the first line of white text shows the gain at 0.15ws as -40.75dB and the second line shows the attenuation at 0.1ws is -0.8472dB.
Changing Fn to 1 shows the quadratic factors needed to realize the filter and Fn = 2 show the poles.

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COPYRIGHT 2008 Cuthbert Nyack.