# 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.

## Z-transform derivation

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

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

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

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

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

## 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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