Exercise 4.  Given the recursion formula   .  

4 (a).  Calculate the amplitude response   ,   ,   ,   and   .  

4 (b).  Discuss what happens to the the filtered signal for the input   .  

Solution 4.

See text and/or instructor's solution manual.

Answer.   ,   ,   ,   ,   .  

The signal component      is slightly attenuated by the factor   .  

The signal component      is attenuated quite a bit by the factor   .  

Solution 4 (a).  Given the filter   ,  

the transfer function (9-27) is   

                    ,  

and formula (9-28) for the amplitude response is

                    .  

Now calculate  

                    ,  

                    ,  

                    ,  

                    .  

Solution 4 (b).  From the above calculations we expect that component      of the signal is slightly attenuated by the factor   

and the component      attenuated quite a bit by the factor   .  

Hence the filter almost eliminates the signal component      which is close to the "zero-out" frequency   .  

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   .  

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

                    The lower frequencies are slightly attenuated and      when   .  

                    The higher frequencies are amplified and      when   .  

We are really really done.   

Remark.   The transfer function is  

                    ,  

and has conjugate zeros at   .  

The argument of      is   ,   and there is a zero amplitude response at   .  

Since  ,   we can expect that   .  

Since  ,   we can expect that   .  

Also, the amplitude response increases for values of   in the interval  .

We are really really really done.   

Aside.  We can graph the causal input sequence and the corresponding causal output sequence.  

                    The input signal      and output signal   .  

                    Here we have   .  

                    We can see that the signal component    is mostly eliminated.

Remark.  In Exercise 2 we saw what happens when we change the sign of the term   .  

This solution is complements of the authors.

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