# Transform lowpass filter to highpass filter, what happened to unit sample response and difference equation?

I know that multiple a lowpass filter signal by $$(-1)^n$$ can transform it to a highpass filter. $$s_{hp}[n]=(-1)^n s_{lp}[n]$$

But exactly what happened to the unit sample response and difference equation? Assume that the original lowpass filter signal has a difference equation. Can someone explain to me?? Thanks!!!

Depends a bit what you mean by that. If you assume that this will turn 1 kHz lowpass signal into a 1kHz high pass signal, your assumption is plain wrong. What happens here is that SHIFT your signal frequency spectrum by half the sample rate, $$f_s$$. Due to the periodicity of the spectrum, this looks like the spectrum has been mirrored at $$fs_/4$$ (at least for real valued signals), i.e. DC become Nyquist, Nyquist becomes DC.