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Given a an IIR lowpass-filter in z-space:

$ H(z) = \frac{\sum_{i=0}^P b_{i} z^{-i}}{1+\sum_{j=1}^Q a_{j} z^{-j}} $

How to calculate it's 3dB cut-off frequency?

I about evaluating it's fourier transform (as long it exits) and take it's magnitude:

$\frac{1}{\sqrt{2}} = \frac{|H(e^{j\omega_c)}|}{|H(e^{j0})|}$

This is then transformed to calculate $\omega_c$. Is this correct? Are there other ways to do that?

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This will work but only if the it's actually a low pass filter. Your formula describes any possible IIR filter so it could really be anything.

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