Finite impulse response

A finite impulse response (FIR) filter is a type of a digital filter in discrete time, 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.

Given an input signal <math>x_n<math> and a <math>P<math>^th-order FIR filter <math>h_n<math>, the convolution of <math>x<math> with <math>h<math> is defined as follows:

<math>y_n = \sum_{k=0}^{P-1} h_k x_{n-k}<math>

The z-transform of <math>h_n<math>, denoted <math>H(z)<math> is defined as follows:

<math>H(z) = \sum_{k=0}^{P-1} h_k z^{-k} = h_0 + h_1 z^{-1} + \cdots + h_{P-1}z^{-(P-1)}<math>

The z-transform of <math>y_n<math> is then <math>Y(z) = H(z) X(z)<math>.

A finite impulse response 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, and can have linear phase (i.e. the phase response of the filter is a linear function of frequency, excluding the possibility of wraps at <math>\pm\pi<math>). An FIR filter has linear phase if and only if its coefficients are symmetric about the center coefficient.

See also

digital filter, infinite impulse response, filter (signal processing)