Exercises for Section 9.3.   Exercises for the Z-Transform Digital Signal Filters

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

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

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

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

Hint.  This is similar to Example 9.21 (a).  

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

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

Solution 1 (a).

Solution 1 (b).

Solution 1 (c).

Solution 1 (d).

Solution 1 (e).

Exercise 2.  Given the recursion formula   .  

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

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

Solution 2.

Exercise 3.  Given the recursion formula   .  

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

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

Hint.  This is similar to Examples 9.21 (b) and 9.21 (c).  

Solution 3.

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.

Exercise 5.  Given the recursion formula   .  

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

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

Solution 5.

Exercise 6.  Given the recursion formula   .  

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

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

Hint.  This is similar to Example 9.22.  

Solution 6.

Exercise 7.  The single-pole low-pass filter is   ,   where constant K is between  .  

7 (a).  Use    and find   ,   ,   ,   ,   and   .  

7 (b).  Use   and find   ,   ,   ,   ,   and   .  

7 (c).  Use   and find   ,   ,   ,   ,   and   .  

Solution 7 (a).

Solution 7 (b).

Solution 7 (c).

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.

Exercise 9.  Show that the moving average filter     

is designed to "zero out"  .  

Hint.  This is similar to Example 9.23 (b).

Solution 9.

Exercise 10.  Use the transfer function     and show that the moving average filter in Exercise 9

has an alternative formula   .

Solution 10.

Exercise 11.  Use the transfer function      and show that the moving average filter in Example 9.24   

has an alternative formula   .

Solution 11.

Exercise 12 (a).  Construct a filter using the zeros   .   What signals are "zeroed out" ?

Exercise 12 (b).  Construct a filter using the zeros   .   What signals are "zeroed out" ?

Solution 12 (a).

Solution 12 (b).

Exercise 13 (a).   Construct a filter using the zeros   .   What signals are "zeroed out" ?

Exercise 13 (b).  Construct a filter using the zeros   .   What signals are "zeroed out" ?

Solution 13 (a).

Solution 13 (b).

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

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

Solution 14 (a).

Solution 14 (b).

Exercise 15 (a).  Construct a filter using the zeros   .   What signals are "zeroed out" ?

Exercise 15 (b).  Construct a filter using the zeros   .   What signals are "zeroed out" ?

Solution 15 (a).

Solution 15 (b).

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

Exercise 16 (b).   Construct a filter using the poles      for boosting up signals near    and    and low frequency signals.  

Hint.  This is similar to Example 9.25 (b).  

Exercise 16 (c).  Construct the combination filter using the zeros       and  poles     

for attenuating high frequencies, and boosting up low frequencies.

Hint.  This is an modification of the filter in Example 9.26.

Solution 16 (a).

Solution 16 (b).

Solution 16 (c).

Exercise 17 (a).   Construct a filter using the zeros     for "zeroing out"   ,  ,  and  .  

Exercise 17 (b).   Construct a filter using the poles      for boosting up signals near    and    and low frequency signals.  

Hint.  This is similar to Example 9.25 (b).  

Exercise 17 (c).   Construct a filter using the zeros and poles in part (a) and (b).  

Solution 17 (a).

Solution 17 (b).

Solution 17 (c).

Exercise 18 (a).   Construct a filter using the zeros      for "zeroing out"   ,  ,  and  .

Exercise 18 (b).   Construct a filter using the poles      for boosting up signals near    and  .

Exercise 18 (c).   Construct a filter using the zeros and poles in part (a) and (b).

Solution 18 (a).

Solution 18 (b).

Solution 18 (c).

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