# I am looking for an analytic description of a continuous-time Butteworth High-pass filter in the time domain (=impulse response)

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. . .

$$\frac{s^n + b_{n-1}s^{n-1}+\cdots+b_0}{s^n + a_{n-1}s^{n-1}+\cdots+a_0} = 1 + \frac{(b_{n-1}-a_{n-1})s^{n-1}+\cdots+b_0 - a_0}{s^n + a_{n-1}s^{n-1}+\cdots+a_0}$$, and the inverse Laplace transform of 1 is $$\delta(t)$$.