Generalized FIR filter

Several different (nonrecursive) filters can be made by changing the delay and the coefficients of the filter terms,

$\displaystyle y(n)$	$\displaystyle =$	$\displaystyle b_0x(n) + b_1x(n-1) + ...$
		$\displaystyle b_2x(n-2) + ... + b_Mx(n-M),$

where

is the maximum delay and thus the order of the filter.

A filter can be defined simply by a set of coefficients. For example if

$\displaystyle b_k = \{1, 3, 3, 1\},$

the filter is third order (has a maximum delay of

), and can be expanded into the difference equation

$\displaystyle y(n) = x(n) + 3x(n-1) + 3x(n-2) + x(n-3)$

``Music 206: Delay and Digital Filters I'' by Tamara Smyth, Computing Science, Simon Fraser University.