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Sampling frequency in bilinear transform when designing butterworth filter

Bilinear transform has a nonlinear frequency mapping $$ \varOmega = \frac{2}{T}\tan{\frac{\omega}{2}} $$ Substituting $\varOmega = 1$ and $\omega = 0.5\pi$ into it and you can derive that $$ T = 2 $$ ...
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How to design digital Butterworth filter and return its zeros, poles and gain

This can be designed in the following steps: Design a $N$-th order lowpass prototype of the analog Butterworth filter and you get an all-pole filter with poles $$ p_{a, n} = e^{j\pi(2n-1+N)/2N}, \ \ \...
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