Coefficients of recursive (IIR) digital filters

From the above discussion, we can see that a recursive filter is basically like a non-recursive filter, with the addition of extra terms involving previous outputs (y_n-1, y_n-2 etc.).

A first-order recursive filter can be written in the general form

y_n = (a₀x_n + a₁x_n-1 - b₁y_n-1) / b₀

Note the minus sign in front of the "recursive" term b₁y_n-1, and the factor (1/b₀) applied to all the coefficients. The reason for expressing the filter in this way is that it allows us to rewrite the expression in the following symmetrical form:

b₀y_n + b₁y_n-1 = a₀x_n + a₁x_n-1

In the case of a second-order filter, the general form is

y_n = (a₀x_n + a₁x_n-1 + a₂x_n-2 - b₁y_n-1 - b₂y_n-2) / b₀

An alternative "symmetrical" form of this expression is

b₀y_n + b₁y_n-1 + b₂y_n-2 = a₀x_n + a₁x_n-1 + a₂x_n-2

Note the convention that the coefficients of the inputs (the x's) are denoted by a's, while the coefficients of the outputs (the y's) are denoted by b's.