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# Finite impulse response

(Redirected from FIR filter)

A finite impulse response (FIR) filter is a type of a digital filter, that is normally implemented through digital electronic computation. The Z-transform of an FIR filter has only zeros and no poles. The number of coefficients in an FIR filter is its order (sometimes referred to as "taps").

## Z-transform derivation

Given a time-invariant input signal x(n) and a Pth-order FIR filter h(n), the convolution of x with h is defined as follows:

$y(n) = \sum_{k=0}^{P-1} h(k) x(n-k)$

The z-transform of h(n), denoted H(z) is defined as follows:

$H(z) = \sum_{k=0}^{P-1} h(k) z^{-k} = h(0) + h(1) z^{-1} + \cdots + h({P-1})z^{-(P-1)}$

The z-transform of y(n) is then Y(z) = H(z)X(z).

## Properties

A FIR filter has a number of useful properties which sometimes make it preferable to an infinite impulse response filter:

• FIR filters are inherently stable
• Require no feedback
• Can have linear phase

An FIR filter has linear phase if and only if its coefficients are symmetric about the center coefficient.