Two-sided impulse response and its inverse in real life
The analog front end will always contain some sort of low pass filter. This can be chosen to be either linear phase or minimum phase. Linear phase preserves the phase response of the original signal but the time domain response is non-causal, i.e. the impulse response is symmetric around the center tap.

Minimum phase has a one sided impulse response, with the strongest tap up front, but it incurs some phase distortion. Ideally we want both, but we have to pick one.

The same consideration implies to the inverse: the inverse of a linear phase filter is linear phase and the inverse of a minimum filter is minimum phase. However there is an added complexity: not all filters are invertible: If the magnitude of the original transfer function is very small or zero at some frequencies, the inverse is not well defined. You get "divide by zero" or "divide by noise" problems.