This page is a schoolbook exercise in digital filter design. The applet above allows you to explore a digital filter with zero or two poles and zero or two zeros.

The above shows the unit circle in Z space. The input to this filter is complex with a frequency in a specified proportion to the sampling frequency.

The difference equation for the general filter is: 

   y[ n ] = x[ n ] - 2 Cos( theta ) x[ n - 1 ] + x[ n - 2 ]
       + 2 r Cos( phi ) y[ n - 1 ] - r^2 y[ n - 2 ]
where theta is the angle to the zero, and one pole is at the point with radius r and angle phi. The transfer function is: 
  Y(z)   z^2 - 2 Cos( theta ) z^1 + 1
  ---- = ------------------------------
  X(z)   z^2 + 2 r Cos( phi ) z^1 + r^2