inverse z transform
TRANSCRIPT
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
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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
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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
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Inverse z- transform example 1
Consider a sequence with z transform of
Where the ROC is as shown
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Inverse z- transform example 1
S
Note that can be rewritten as
To find the constants and we use the following
Similarly
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Inverse z- transform example 1
Note that Now can be rewritten as
The inverse z transform results in the shown below
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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
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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
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Partial fraction example 2
Find the inverse z transform for the sequence given by
If the ROC is as shown
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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
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Partial fraction example 2
Now the function can be rewritten as shown below
Or
Where
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Partial fraction example 2
The constants and can be found as follows
can now be written as
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Partial fraction example 2
Recall that from the z-transform pairs table we have
Therefore is given by
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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
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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
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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
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Inverse z transform by using power series example 3
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Inverse z transform by using power series example 4
Consider the z transform defined by
Find by using long division
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Inverse z transform by using power series example 4
Solution
This series reduces to
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Inverse z transform by using power series example 5
Find the inverse z transform of the sequence defined by
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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
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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
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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
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Z-transform properties table
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Z-transform properties table
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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
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Example 6
From the z-transform table we have
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Example 7
Determine the z-transform of
Solution
By using the time reversal property we have
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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
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Example 8
If we multiply we get the following answer
The inverse z-transform which is the convolution of is given by
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Example 9
Find the inverse z transform for the function defined by
solution
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Example 9
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