inverse z transform

1. The inverse z transform The inverse z-transform can be found by one of the following ways Inspection method Partial fraction expansion Power series expansion Each will be explained briefly next 1

2. Inverse z transform by inspection method The inspection method is based on the z transform pair table. In order to find the inverse z transform we compare to one of the standard transform pairs listed in the z transform pairs table   2

3. Inverse z transform by partial fraction If is not in a form listed in the table of z transform pairs we can use the partial fraction method to simplify the function into one of the standard forms listed in the z transform pair table   3

4. Inverse z- transform example 1 Consider a sequence with z transform of Where the ROC is as shown   4

5. Inverse z- transform example 1 S Note that can be rewritten as To find the constants and we use the following Similarly   5

6. Inverse z- transform example 1 Note that Now can be rewritten as The inverse z transform results in the shown below   6

7. Inverse z transform by the partial fraction with -   The partial fraction method can be used to find the inverse z-transform for rational functions with numerator of order and denominator of order The partial fraction can be used only if the numerator order is less than denominator order If the numerator order is greater than or equal the denominator order then we use long division to make the denominator order greater than the numerator order before we can use the partial fraction method   7

8. Partial fraction with -   The long division converts the function in the following form Where is the numerator order, is the denominator order, are constants of the partial fraction and are the roots   8

9. Partial fraction example 2 Find the inverse z transform for the sequence given by If the ROC is as shown 9

10. Partial fraction example 2 As it can be seen from the order of the numerator is equal to the order of the denominator Long division can be used to make the order of the numerator less than the order of the denominator as shown below   10

11. Partial fraction example 2 Now the function can be rewritten as shown below Or Where   11

12. Partial fraction example 2 The constants and can be found as follows can now be written as   12

13. Partial fraction example 2 Recall that from the z-transform pairs table we have Therefore is given by   13

14. Partial fraction with multiple poles and greater than   If the function contains multiple poles and as shown in this form The coefficients can be found by deriving number of times as shown   14

15. Inverse z transform by using power series expansion From the definition of the z-transform we can write the z-transform as This is known as L aurent series From this series we can find the sequence as illustrated by the next example   15

16. Inverse z transform by using power series example 3 Find the inverse z-transform for the sequence defined by Solution Note the sequence can be expanded as If we compare with the Laurent series we can extract as follows   16

17. Inverse z transform by using power series example 3 17

18. Inverse z transform by using power series example 4 Consider the z transform defined by Find by using long division   18

19. Inverse z transform by using power series example 4 Solution This series reduces to   19

20. Inverse z transform by using power series example 5 Find the inverse z transform of the sequence defined by   20

21. Inverse z transform by using power series example 5 Solution Because the region of convergence, the sequence is a left-sided The solution can be obtained by long division as indicated 21

22. Z-transform properties The z-transform has many useful properties similar to Fourier transform properties These properties can be used to find the inverse z-transform for certain complex z functions as it will be demonstrated in the examples These properties are Linearity 22

23. z transform properties Time shifting Multiplication by an exponential Differentiation of Conjugation of a complex sequence Time reversal Convolution of a sequence Initial value theorem These properties are summarized in the table shown in the next two slides   23

24. Z-transform properties table 24

25. Z-transform properties table 25

26. Example 6 Determine the z-transform and the ROC for the sequence Solution We can divide into two different functions Now can be rewritten as   26

27. Example 6 From the z-transform table we have 27

28. Example 7 Determine the z-transform of Solution By using the time reversal property we have   28

29. Example 8 Compute the convolution of the following two sequences using the z transform Solution Note that the z transform of each of the previous sequences is given by and   29

30. Example 8 If we multiply we get the following answer The inverse z-transform which is the convolution of is given by   30

31. Example 9 Find the inverse z transform for the function defined by solution   31

32. Example 9 32