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 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).

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 $\pm\pi$ ). An FIR filter has linear phase if and only if its coefficients are symmetric about the center coefficient.

Finite impulse response

See also