I have a signal that has a spectral shape very similar to the transfer function of a second-order lowpass filter (butterworth design).
I would like to whiten the spectrum of this spectrum in the time domain. For that, I need a filter that reverses the effect of the second-order lowpass filter.
I can calculate the inverse filter of the second-order lowpass by exchanging the A- and B-coefficients, but this will amplify high frequency components to infinity.
How do I design a filter that has the inverse transfer function of a second-order butterworth design but does not amplify to infinity at $f_s/2$? The transfer function should have the same corner frequency, but smoothly rise with 12 dB per octave.
Is this even possible? Any pointers as to how to design such a filter would be helpful!