Pole Zero Analysis

• The transfer function for the recursive LTI digital filter
  $$H(z) = g\frac{1 + \beta_1 z^{-1} + \cdots + \beta_M z^{-M}} {1 + a_1z^{-1} + \cdots + a_Nz^{-N}}$$

• We can factor the numerator and denominator to obtain
  $$H(z) = g\frac{(1-q_1z^{-1})(1-q_2z^{-1})\cdots(1-q_Mz^{-1})} {(1-p_1z^{-1})(1-p_2z^{-1})\cdots(1-p_Nz^{-1})}$$
  
• We may therefore characterize a transfer function, by its
  1. Numerator roots, called the zeros of the filter
  2. Denominator roots, called poles of the filter
  3. Constant gain factor