Types of Digital Filters

There are two types of digital filters

1. Recursive (Finite Impulse Response)
   
2. non-recursive (Infinite Impulse Response)

1. Recursive (Finite Impulse Response):
   

   A recursive filter is one which in addition to input values also uses previous output values. These, like the previous input values, are stored in the processor's memory.

   The word recursive literally means "running back", and refers to the fact that previously-calculated output values go back into the calculation of the latest output. The expression for a recursive filter therefore contains not only terms involving the input values (x_n, x_n-1, x_n-2, ...) but also terms in y_n-1, y_n-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 is usually the case. To achieve a given frequency response characteristic using a recursive filter generally requires a much lower order filter, and therefore fewer terms to be evaluated by the processor, than the equivalent non-recursive filter.
2. Non-Recursive: he current output (y_n) is calculated solely from the current and previous input values (x_n, x_n-1, x_n-2, ...). This type of filter is said to be non-recursive.

Note: FIR and IIR filters

Some people prefer an alternative terminology in which a non-recursive filter is known as an FIR (or Finite Impulse Response) filter, and a recursive filter as an IIR (or Infinite Impulse Response) filter. These terms refer to the differing "impulse responses" of the two types of filter

The impulse response of a digital filter is the output sequence from the filter when a unit impulse is applied at its input. (A unit impulse is a very simple input sequence consisting of a single value of 1 at time t = 0, followed by zeros at all subsequent sampling instants). An FIR filter is one whose impulse response is of finite duration. An IIR filter is one whose impulse response (theoretically) continues for ever, because the recursive (previous output) terms feed back energy into the filter input and keep it going. The term IIR is not very accurate, because the actual impulse responses of nearly all IIR filters reduce virtually to zero in a finite time. Nevertheless, these two terms are widely used.