Chemistry - Finite impulse response

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.

Finite impulse response

Z-transform derivation

Properties

See also