Exercise 20.  Consider the system of difference equations

with the initial conditions   .  

20 (b).  Use z-transforms and residues to construct the solution in part (a).

Solution 20 (b).

Method (i)   Take the z-transform of both sides and use the initial conditions   .  

                    ,  

                    and    

                    .  

then get  

                    ,      

                    and    

                    ,  

which can be written as    

                    ,        

                    and    

                    .

Solve for and and get

                       
                    
Then get

                     ,  

                    and    

                    .    
                    

Use Table 9.1 to find the inverse z-transform   

                    ,  

                    and    

                    .  

We are done.   

Method (ii)   Using residues we obtain    

                      

                    and

                      

Thus,

                      

Using residues we obtain    

                      
    
                    and

                      

Thus,

                      

Therefore,   

                       

                    and   

                    .  

We are done.   

Aside.  We can let Mathematica double check our work.

Using limits and residues we obtain    

Using limits and residues we obtain    

          The Maple commands are similar  


  

                                                            


  

                                                            

Using limits and residues we obtain    

  

                                                            


  

                                                            

Using limits and residues we obtain    

  

                                                            


  

                                                            

Aside.  We can use Mathematica's InverseZTransform subroutine.

The following results were obtained using Mathematica 7.0

            

            


            

            


            

            


            

          

          The Maple commands are similar.  However, the inverse Z-transforms are complicated.

Hence, we will verify that the Z-transforms of      and   ,  

are      and   ,   respectively.

We are really really done.   

Aside.  We can use Mathematica's Rsolve subroutine.

The following results were obtained using Mathematica 5.0

            

            

            

            

            

          

          The Maple command is similar 

  

                              

We are really really really done.   

Aside.  We can graph the solution.

                                                                                                                        The sequence   .   

                                                                                                                       The sequence   .   

                                                                                                                        The sequence   .   

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell