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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). The impulse response (that is, the output in response to a Kronecker delta input) of an Nth-order discrete-time FIR filter lasts exactly 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: is the input signal, is the output signal, is the filter order; an th-order filter has terms on the right-hand side is the value of the impulse response at the ith instant for of an -order FIR filter. If the filter is a direct form FIR filter then is also a coefficient of the filter. This computation is also known as discrete convolution. The in these terms are commonly referred to as s, 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: If an FIR filter is non-causal, the range of nonzero values in its impulse response can start before , with the defining formula appropriately generalized. An FIR filter has a number of useful properties which sometimes make it preferable to an infinite impulse response (IIR) filter. FIR filters: Require no feedback. This means that any rounding errors are not compounded by summed iterations. The same relative error occurs in each calculation. This also makes implementation simpler.

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