Exercise 1.  Use direct substitution and trigonometric identities to show the following:  

1 (d).     will "zero-out"      and   .

Solution 1 (d).

See text and/or instructor's solution manual.

Substitute and get  

                      

Substitute and get  

                      

We are done.   

Aside.  We can let Mathematica double check our work.

First.   Substitute  .

Aside.  The Maple commands are similar  

  

                                                            


  

                                                            


  

                                                            

Second.   Substitute  .

Aside.  The Maple commands are similar  

  

                                                            


  

                                                            


  

                                                            

We are really done.   

Remark.   The above computations are straightforward and only use basic facts from trigonometry.  

It is our goal to learn about the transfer function and the formula for the amplitude response.  

Aside.   Given this filter   ,  

use formula (9-27) to calculate the transfer function  

                      

then formula (9-28) will give the amplitude response  

                    .  

                    The amplitude response      and zero-pole plot of   ,  
  
                    for the filter   .  

We are really really done.   

Remark.   The transfer function can be written using "zero-out" factors    

                    ,  

and has conjugate zeros at   .   The argument of      is   ,   

and there is a zero amplitude response at   ,   i. e.   .  

Remark.  In part 1 (e) we will see what happens when we change the sign of the term   .  

This solution is complements of the authors.

(c) 2009 John H. Mathews, Russell W. Howell