Bilinear Approximation to nth order Low Pass Elliptic Filter.

Cuthbert Nyack
Frequency Response, Transfer function and pole locations are shown by the applet below.
Fn = 0 shows the frequency response.
Fn = 1 shows the transfer function represented as a product of quadratic factors.
Fn = 2 shows the pole and zero locations for the continuous Elliptic filter and the discrete Bilinear approximation.

To design a filter which has a gain of ~ -0.5dB at 0.12ws, ~ -85dB at 0.18ws and a passband ripple of 0.15dB, proceed as follows:-

Set BL = 90, WL to 135 and R = 0.15 to locate the frequency markers and set the pass band ripple.
Adjust N, fc/fs until the conditions are met.
In this case N = 8, fc/fs = 0.119 and ws/wp = 1.639(ratio of stop band frequency to pass band frequency).

The last entry in the first line of white text shows the gain at 0.18ws as -88.46dB and the second line shows the attenuation at 0.12ws is -0.421dB.
In this case, the minimum stop band attenuation is 96.34dB.
Changing Fn to 1 shows the quadratic factors needed to realize the filter and Fn = 2 show the zeros and poles.