Notch Filter

Changing the delay of the second term (and adding):

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

has output

• at 0 Hz ( $x_1(n) = [A, A, A, ....]$):
  

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

  $$= [A, A, 2A, 2A, ...] \approx 2x_1(n).$$

  
  

• at $f_s/2$ Hz ( $x_2(n) = [A, -A, A, -A, ...]$):
  

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

  $$= [A, -A, 2A, -2A, ...] \approx 2x_2(n)$$

  
  

This filter boosts both low and high frequencies!

Output at $f_s/4$ Hz ( $x_3(n) = [A, 0, -A, 0, A, 0, ...]$):

$$y(n) = x_3(n) + x_3(n-2)$$

$$= [A, 0, -A, 0, A, ....] + [0, 0, A, 0, -A, 0, ...]$$

$$= [A, 0, 0, 0, ...] \approx 0x_3(n)$$