Order of a recursive (IIR) digital filter

The order of a digital filter was defined earlier as the number of previous inputs which have to be stored in order to generate a given output. This definition is appropriate for non-recursive (FIR) filters, which use only the current and previous inputs to compute the current output. In the case of recursive filters, the definition can be extended as follows:

The order of a recursive filter is the largest number of previous input or output values required to compute the current output.

This definition can be regarded as being quite general: it applies both to FIR and IIR filters.

For example, the recursive filter discussed above, given by the expression

y_n = x_n + y_n-1

is classed as being of first order, because it uses one previous output value (y_n-1), even though no previous inputs are required.

In practice, recursive filters usually require the same number of previous inputs and outputs. Thus, a first-order recursive filter generally requires one previous input (x_n-1) and one previous output (y_n-1), while a second-order recursive filter makes use of two previous inputs (x_n-1 and x_n-2) and two previous outputs (y_n-1 and y_n-2); and so on, for higher orders.

Note that a recursive (IIR) filter must, by definition, be of at least first order; a zero-order recursive filter is an impossibility