Exercise 8.  Use the recursion formula      in Exercise 7 (a).  

8 (a).  Start with   ,   ,   and show by induction that   .  

8 (b).  Use the transfer function      and find the unit-sample response   .  

8 (c).  Verify that the general term in part (a) is given by the convolution formula   .  

Solution 8.

See text and/or instructor's solution manual.

8 (a).   Start with

                      

                    and  

                      

For fun, calculate the two term

                      

                    and  

                      

By induction we have

                    

8 (b).   The transfer function is    and unit-sample response is  

                    .

8 (c).   The solution has the form of the convolution sum

                    .  

Remark.  Note that this filter has an "infinite memory",

i.e. it's output uses all previous inputs but weights them in a decreasing exponential fashion back to the term  .

This is the motivation for using the terminology "infinite impulse response" filter or IIR filter.

We are done.   

Aside.  We can let Mathematica double check our work.

Aside.  The Maple command is similar  

  

                                                            

This solution is complements of the authors.

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