having derived the Butterworth Lowpass Time domain response, I am now struggling to find a similar function for a Butterworth Highpass filter. I understand you need to replace s by 1/s. But this leads to a transfer function with as many zeros as there are poles. So you can no longer use the Heaviside 'cover-up' method to decompose the individual exponential functions, to arrive at the time domain response.
Is there perhaps someone who stumbled on the same problem and arrived at a solution ?
The ultimate goal is to arrive at a Butterworth bandpass filter whereby the lowpass and high pass sides can have a different order (slope). So I still need to convolve the LP- with the HP-filter. . .