Example 9.19.  Solve     with  .  

Solution 9.19.

Take the z-transforms of each term and get

Solve for    and get  

Calculate the residues for    at the poles    and  

        

Similarly,  .  Therefore, the solution is

        ,
which can be written as

          

        .  

Remark.  The solution can also be obtained by applying the z-transform identity with that was proven in example 5 of the first section of this chapter to get  

        ,  
then we have
        .

(c) 2006 John H. Mathews, Russell W. Howell