Does anyone have any good references for deriving parameters of an IIR Low pass/High Pass filter directly in the digital domain using the magnitude squared at the corner frequency?
I have been able to derive the parameters of a first order Low/High pass filter with $3\textrm{ dB}$ attenuation at the corner frequency i.e. calculating $k$ and $\alpha$ in:
$$H(z) = k\frac{\left(1+z^{-1}\right)}{\left(1-\alpha z^{-1}\right)}$$
My issue is that I distinctly remember deriving the parameters using a $6\textrm{ dB}$ attenuation at the corner frequency in a DSP course I have done previously but I have forgotten the trigonometric identiftes used to finish the derivation.
The general procedure is as follows:
- Let $\omega = 0/\pi$ to calculate the gain term $k$ such that there is a $0\textrm{ dB}$ gain at $0/\pi$
- Calculate the magnitude squared at the corner frequency to obtain a value for $\alpha$ in terms of the corner frequency.
The problem may be that it should be a second order filter or I am recalling the method for a band pass/stop filter but I'm not sure and it appears this method is not used very often except in the case of band pass/stop filters for parametric EQ.
I hope the question is clear and I will try to improve the structure with the responses so it will be useful for others. Any help will be appreciated.