Bandpass Filter

Changing coefficient of $x(n-2)$ to -1:

$$y(n) = x(n) - x(n-2),$$

yields output

• at 0 Hz:
  

  $$y(n) = [A, A, A, ....] - [0, 0, A, A, ...]$$

  $$= [A, A, 0, 0, ...] \approx 0x(n).$$

  
  

• at $f_s/2$ Hz:
  

  $$y(n) = [A, -A, A, -A, ....] - [0, 0, A, -A, A...]$$

  $$= [A, -A, 0, 0, ...] \approx 0x(n).$$

  
  

• at $f_s/4$ Hz:
  

  $$y(n) = [A, 0, -A, 0 A, ....] - [0, 0, A, 0, -A, 0, ...]$$

  $$= [A, 0, -2A, 0, 2A, 0, -2A, 0, ...] \approx 2x(n).$$

  
  

Attenuation is at 0 and $f_s/2$ Hz, and boosts $f_s/4$