Finite impulse response

In signal processing, a finite impulse response (FIR) filter is a filter whose impulse response (or response to any finite length input) is of finite duration, because it settles to zero in finite time. This is in contrast to infinite impulse response (IIR) filters, which may have internal feedback and may continue to respond indefinitely (usually decaying).^{[citation needed]}

The impulse response (that is, the output in response to a Kronecker delta input) of an N^th-order discrete-time FIR filter lasts exactly ${\displaystyle N+1}$ samples (from first nonzero element through last nonzero element) before it then settles to zero.

FIR filters can be discrete-time or continuous-time, and digital or analog.

For a causal discrete-time FIR filter of order N, each value of the output sequence is a weighted sum of the most recent input values:

where:

This computation is also known as discrete convolution.

The ${\textstyle x[n-i]}$ in these terms are commonly referred to as taps, based on the structure of a tapped delay line that in many implementations or block diagrams provides the delayed inputs to the multiplication operations. One may speak of a 5th order/6-tap filter, for instance.

The impulse response of the filter as defined is nonzero over a finite duration. Including zeros, the impulse response is the infinite sequence: