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For an example analysis, I’ve picked up the low-shelf filter in Robert Bristow-Johnson’s Audio EQ Cookbook. In the book, the transfer function is given as; $$H(s) = A\frac{s^2 + \frac{\sqrt{A}}{Q}s + A}{As^2 + \frac{\sqrt{A}}{Q}s + 1}$$ Since the analysis is going to be done by hand, the asymptotic approximation method of Bode plot analysis can be followed. ...

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The resonant frequency is related to the "significant frequency" (which is the shelf midpoint frequency) by a factor of $\frac{1}{\sqrt{A}}$ for the lowShelf and the reciprocal of that for the highShelf.

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You can't implement an ideal Hilbert transformer: its impulse response is non causal and infinite in time. So you can only implement an "approximate" one and the best way to do this depends on the specific requirement of your application: what's your frequency range of interest, how much magnitude and/or phase deviation can you tolerate, are you ...

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Your signal has a massive DC bias so your output is dominated by the step response of the band pass filter. It will eventually get there but it's going to take a really long time. Initialize your state with zi = -26040*sosfilt_zi(). See my answer to your other question today. EDIT On second thought: while you can fix some of this in software, you probably ...

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But I don't understand what it does, and how it determines initial conditions from the sos argument and not from the actual signal to filter. From the documentation Compute an initial state zi for the sosfilt function that corresponds to the steady state of the step response. It assumes that the input signal is a unit step. That's useful if you input ...

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