Operation of Digital Filters

• Digital filters are described by difference equations.
• Let x_n denote the n-th sample of the input and y_n denote the n-th sample of the output of the filter.
• Then, a relationship like the following describes the operation of the filter
  
  $$y_n = a_0 x_n + a_1 x_{n-1} + \ldots a_N x_{n-N} - b_1 y_{n-1} - \ldots b_M y_{n-M}$$
  
  

• That means, the n-th output sample is computed as a weighted average of the N+1 most recent inputs minus a weighted average of the M most recent outputs.
• The numbers a_i and b_i are called filter coefficients.
• We will focus on the case when the coefficients b_i are equal to zero.
• The resulting filters
  
  $$y_n = a_0 x_n + a_1 x_{n-1} + \ldots a_N x_{n-N}$$
  
  
  are called finite impulse response (FIR) filters.