Recursive filter

In signal processing, a recursive filter (also called an infinite impulse response filter) is a type of filter which reuses one or more of its outputs as an input. They allow a system to respond over a long period of time to a brief input signal, without needing to perform complex calculations on every past input.^[1] This feedback typically results in an unending impulse response, characterized by either exponentially growing, decaying, or sinusoidal signal output components.

However, a recursive filter does not always have an infinite impulse response. Some implementations of moving average filter are recursive filters but with a finite impulse response.

Non-recursive Filter Example: y[n] = 0.5x[n − 1] + 0.5x[n].

Recursive Filter Example: y[n] = 0.5y[n − 1] + 0.5x[n].