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

Where the ROC is as shown

4

Inverse z- transform example 1

S

Note that can be rewritten as

To find the constants and we use the following

Similarly

5

Inverse z- transform example 1

Note that Now can be rewritten as

The inverse z transform results in the shown below

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

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

11

Partial fraction example 2

The constants and 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

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

Solution

Note the sequence can be expanded as

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

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

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

Solution

We can divide into two different functions

Now can be rewritten as

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

Solution

Note that the z transform of each of the previous sequences is given by and

29

Example 8

If we multiply we get the following answer

The inverse z-transform which is the convolution of is given by

30

Example 9

Find the inverse z transform for the function defined by

solution

31

Example 9

32