Recursive and non-recursive filters

For all the examples of digital filters discussed so far, the current output (yn) is calculated solely from thecurrent and previous input values (xn, xn-1, xn-2, ...). This type of filter is said to be non-recursive.A recursive filter is one which in addition to input values also uses previous output values. These, like theprevious input values, are stored in the processor's memory.The word recursive literally means "running back", and refers to the fact that previously-calculated outputvalues go back into the calculation of the latest output.
The expression for a recursive filter therefore containsnot only terms involving the input values (xn, xn-1, xn-2, ...) but also terms in yn-1, yn-2, ...From this explanation, it might seem as though recursive filters require more calculations to be performed,since there are previous output terms in the filter expression as well as input terms.

In fact, the reverse isusually the case:
to achieve a given frequency response characteristic using a recursive filter generally requiresa much lower order filter (and therefore fewer terms to be evaluated by the processor) than the equivalent nonrecursivefilter. This will be demonstrated later.