Exercise 16 (a).   Construct a filter using the zeros    for attenuating signals near   .  

Solution 16 (a).

See text and/or instructor's solution manual.

Answer.   Compute the product  

                    .  

The desired filter is  

                    .  

Solution.   Use the conjugate pair of zeros      and   ,  

and the "attenuating factors"      and    .  

Then calculate  

                      

This is a basic filter and we can use Property (ii) Attenuating Factors.  

The transfer function for part (a) is known to have the form  

                      

and corresponds to the filter

                    .  

        For this exercise, we use      in these equations to get the desired recursive formula  

                    .  

Therefore, the desired filter for part (a) is

                    .  

We are done.   

Aside.  We can let Mathematica double check our work.

Aside.  The Maple commands are similar  

  

                                                              


  

                                                              

We are really done.   

Aside.  We can graph the amplitude response for the filter   .  

                    Amplitude response   ,   

                    and zero-pole plot of   ,  

                    for the filter   .

                    We can see that some of the high-range frequencies are slightly attenuated.

We are really really done.   

Aside.  For illustration, we can graph the causal input sequence   ,  

and the corresponding causal output sequence   .  

The signal component    will be amplified  by the factor  ,  and  

The signal component    will be slightly attenuated by the factor  .  

                    The causal input sequence      and the corresponding causal output sequence.

                    Here we have     and   .  
                    
                    This filter reduces the proportion of the signal component      by a factor of   .

This solution is complements of the authors.

(c) 20098 John H. Mathews, Russell W. Howell