The General Difference Equation for LTI filters

The general difference equation for LTI filters includes feedback terms, and is given by

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

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

This can be implemented in Matlab using the filter function:

B = ...; % feedforward coefficients
A = ...; % feedback coefficients
y = filter(B, A, x);

Matlab specifies coefficients according to the filter transfer function and NOT the difference equation:

• all feedback coefficients (except the first) have a sign opposite to that in the difference equation;
• this is explained by moving the $y$ terms in the difference equation to the left of the equal sign (a step in arriving at the filter transfer function):

  $$y(n) + a_1y(n-1) + \cdots = b_0x(n) + b_1x(n-1) + \cdots$$