Exercise 14 (a).   Construct a filter using the zeros  .  What signals are "zeroed out" ?

Solution 14 (a).

See text and/or instructor's solution manual.

Answer.   Compute the product  

                    .  

The desired filter is  

                    .  

Solution.   Use the conjugate pairs of zeros      and   ,  

and the "zero out factors"      and   .   

Then calculate

                       

        Now use the General Filter Equation (9-29) and the corresponding Transfer Function (9-34) in the case  .     

Get the new fact that the transfer function  

                    ,  

corresponds to the filter  

                    .  

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

                    .  

Therefore, the desired filter for (a) is

                    .  

for "zeroing out" the signals  .

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 the mid-range frequencies are slightly amplified, and      for   .  

Remark.  In Exercise 12 (a) we saw what happens when we change the sign of the term   .  

We are really really done.   

Aside.  Let us investigate how well the filter works to eliminate signals    which are close to the "zero-out" frequencies.

        For illustration purposes we will explore the casual input signal   .

The signal component    will be slightly reduced by the factor  .  

The signal component    will be attenuated by the factor  ,  and

The signal component    will be attenuated by the factor  .  

                    The causal input sequence      and the corresponding causal output sequence.

                    Since we have     and   ,
                    
                    this filter reduces the proportion of the signal component      by a factor of   .
                    
                    Since we have     and   ,
                    
                    this filter reduces the proportion of the signal component      by a factor of   .

We are really really really done.   

Aside.  Let us investigate how well the filter works to eliminate signals   which are close to the "zero-out" frequencies,

and the filter retains the signal   .  

Remark.  In part (b) we will see how to eliminate the latter.

        For illustration purposes we will explore the casual input signal   .

The signal component    will be slightly reduced by the factor  .  

The signal component    will be attenuated by the factor  ,  and

The signal component    will be amplified by the factor  ,  and

The signal component    will be attenuated by the factor  .  

                    The causal input sequence      and the corresponding causal output sequence.

                    Since we have     and   ,
                    
                    this filter reduces the proportion of the signal component      by a factor of   .
                    
                    Since we have     and   ,
                    
                    this filter increases the proportion of the signal component      by a factor of   .
                    
                    Since 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