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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 P^th-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.

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