Discrete Systems and Digital Signal Processing with MATLAB, 2nd Edition

2.12  Nonhomogeneous Difference Equations and Their Solutions

The solution of the nonhomogeneous difference equation

$y(n)-{\displaystyle \sum _{k=1}^N{a}_{k}y(n-k)={\displaystyle \sum _{k=0}^M{b}_{x}x(n-k)}}$

can be obtained by finding the homogeneous solution of

$y(n)-{\displaystyle \sum _{k=1}^N{a}_{k}y(n-k)=0}$

with the initial conditions

$y(-1),y(-2),\cdots ,y(-N)$

then adding it to the particular solution of

$y(n)-{\displaystyle \sum _{k=1}^N{a}_{k}y(n-k)={\displaystyle \sum _{k=0}^M{b}_{x}x(n-k)}}$

So we write the total solution as

$y(n)=y_h(n)+y_p(n)$

Note that if we are given a nonhomogeneous difference equation with initial conditions, we first find y_h(n) and do not evaluate the constants associated with y_h(n) because these initial conditions are given for the total solution y(n). The following example will illustrate this point.

Example 2.19

Consider the system described by the difference ...