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inverse z transform

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Zlatan Ahmadovic
Zlatan Ahmadovic

The document discusses various methods for finding the inverse z-transform, including inspection of z-transform pairs, partial fraction expansion, and power series expansion. It provides examples of using each method to find the inverse z-transform of given z-functions. It also discusses properties of the z-transform, such as time shifting and convolution, that can help in solving inverse problems. Sample problems demonstrate applying the techniques to compute inverse z-transforms and use properties to solve for sequences.

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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
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
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
Inverse z- transform example 1 Consider a sequence [] with z transform of = 1 1 1 4 р1 1 1 2 р1 > 1 2 Where the ROC is as shown 4
Inverse z- transform example 1 S＄ Note that () can be rewritten as X = 1 1 1 4 р1 1 1 2 р1 = 1 1 1 4 р1 + 2 1 1 2 р1 To find the constants 1 and 2 we use the following 1 = 1 1 4 р1 | р1=4 = 1 Similarly 2 = 1 1 2 р1 | р1=2 = 2 5
Inverse z- transform example 1 Note that () Now can be rewritten as X = 1 1 1 4 р1 + 2 1 1 2 р1 The inverse z transform results in the [] shown below = 1 4 + 2 1 2 6
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
Partial fraction with - The long division converts the () function in the following form = =0 巨 р + =1 1 р1 Where is the numerator order, is the denominator order, are constants of the partial fraction and are the roots 8
Partial fraction example 2 Find the inverse z transform for the sequence given by If the ROC is as shown 9
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
Partial fraction example 2 Now the function () can be rewritten as shown below Or Where 0 = 2 11
Partial fraction example 2 The constants 1 and 2 can be found as follows () can now be written as 12
Partial fraction example 2 Recall that from the z-transform pairs table we have Therefore [] is given by 13
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
Inverse z transform by using power series expansion From the definition of the z-transform we can write the z-transform as = = []р = + 2 2 + 1 1 + 0 + 2 р1 + 2 р2 + This is known as Laurent series From this series we can find the sequence [] as illustrated by the next example 15
Inverse z transform by using power series example 3 Find the inverse z-transform for the sequence defined by = 2 1 1 2 р1 1 + р1 1 р1 Solution Note the sequence [] can be expanded as = 2 1 2 1 + 1 2 р1 If we compare with the Laurent series we can extract [] as follows 16
Inverse z transform by using power series example 3 17
Inverse z transform by using power series example 4 Consider the z transform defined by = 1 1р1 > Find [] by using long division 18
Inverse z transform by using power series example 4 Solution This series reduces to = [] 19
Inverse z transform by using power series example 5 Find the inverse z transform of the sequence defined by = 1 1 р1 < 20
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
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
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
Z-transform properties table 24
Z-transform properties table 25
Example 6 Determine the z-transform and the ROC for the sequence = 3(2) 4(3) Solution We can divide [] into two different functions 1 = (2) 2 = (3) Now [] can be rewritten as = 31 42 26
Example 6 From the z-transform table we have 27
Example 7 Determine the z-transform of [] = [] Solution By using the time reversal property we have 28
Example 8 Compute the convolution of the following two sequences using the z transform 1 = 1, 2, 1 2 = 1, 0 5 0 も Solution Note that the z transform of each of the previous sequences is given by 1 = 1 2р1 + р2 and 2 = 1 + р1 + р2 + р3 + р4 + р5 29
Example 8 If we multiply 1()2() we get the following answer The inverse z-transform which is the convolution of = 1 2 is given by 30
Example 9 Find the inverse z transform for the function defined by = log 1 + р1 > solution 31
Example 9 32

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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 = 1 1 1 4 р1 1 1 2 р1 > 1 2 Where the ROC is as shown 4
  • 5. Inverse z- transform example 1 S＄ Note that () can be rewritten as X = 1 1 1 4 р1 1 1 2 р1 = 1 1 1 4 р1 + 2 1 1 2 р1 To find the constants 1 and 2 we use the following 1 = 1 1 4 р1 | р1=4 = 1 Similarly 2 = 1 1 2 р1 | р1=2 = 2 5
  • 6. Inverse z- transform example 1 Note that () Now can be rewritten as X = 1 1 1 4 р1 + 2 1 1 2 р1 The inverse z transform results in the [] shown below = 1 4 + 2 1 2 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 = =0 巨 р + =1 1 р1 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 0 = 2 11
  • 12. Partial fraction example 2 The constants 1 and 2 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 = = []р = + 2 2 + 1 1 + 0 + 2 р1 + 2 р2 + This is known as Laurent 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 = 2 1 1 2 р1 1 + р1 1 р1 Solution Note the sequence [] can be expanded as = 2 1 2 1 + 1 2 р1 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 = 1 1р1 > 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 = 1 1 р1 < 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
  • 26. Example 6 Determine the z-transform and the ROC for the sequence = 3(2) 4(3) Solution We can divide [] into two different functions 1 = (2) 2 = (3) Now [] can be rewritten as = 31 42 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 1 = 1, 2, 1 2 = 1, 0 5 0 も Solution Note that the z transform of each of the previous sequences is given by 1 = 1 2р1 + р2 and 2 = 1 + р1 + р2 + р3 + р4 + р5 29
  • 30. Example 8 If we multiply 1()2() we get the following answer The inverse z-transform which is the convolution of = 1 2 is given by 30
  • 31. Example 9 Find the inverse z transform for the function defined by = log 1 + р1 > solution 31