Recursive Filters

Using FIR filters often required significant computation and coefficients to reproduce a desired frequency response.
It is often possible to reduce the number of feedforward coefficients needed to obtain a frequency response by introducing feedback coefficients.
A simple example of a first order recursive low-pass filter is given by

$\displaystyle y(n) = b_0x(n) + a_1y(n-1)$
The general difference equation for LTI filter therefore, is given by

$\displaystyle y(n) = b_0x(n)$ $\displaystyle +$ $\displaystyle b_1x(n-1) + \cdots + b_Mx(n-m)$

$\displaystyle -$ $\displaystyle a_1y(n-1) - \cdots - a_Ny(n-N)$

``Mus 270a: Introduction to Digital Filters'' by Tamara Smyth, Department of Music, University of California, San Diego.

$\displaystyle y(n) = b_0x(n)$	$\displaystyle +$	$\displaystyle b_1x(n-1) + \cdots + b_Mx(n-m)$
	$\displaystyle -$	$\displaystyle a_1y(n-1) - \cdots - a_Ny(n-N)$