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. Digital filters are often characterised by their order which is determined by the number of delay elements , e.g. a second order filter has two delay elements. Alternatively a digital filter may be characterised by the number of taps, which is equal to the order plus one, e.g. a second order filter has three taps.

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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

See also: Finite impulse response, Convolution, Digital, Digital filter, Electronic filter, Filter (signal processing), Iff, Infinite impulse response, Linear phase, Pole (complex analysis)