1. Chapter 2
The z Transform

2. Introduction
A mathematical tool used to the analysis and synthesis of discrete-time control systems: the z transform.
The Laplace transform in continuous-time systems.
With the z transform method, the solutions to linear difference equations become algebraic in nature.
Discrete-Time Signals
The sampled signal: x(0), x(T), x(2T),…, where T is the sampling period.
If the system involves an iterative process carried out by a digital computer, the signal involved is a number sequence: x(0), x(1), x(2),…. – it can be considered as a sampled signal of x(t) when the sampling period T is 1 sec.
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3. The z Transform One-sided z Transform
For most engineering applications the one-sided z transform will have a convenient close-form solution in its region of convergence.
The z transform of any continuous-time function may be written in the series form by inspection.
The inverse transform can be obtained by inspection if X(z) is given in the series form.
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4. z Transforms of Elementary Functions
Unit-Step Function
The series converges if
Unit step sequence
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5. z Transforms of Elementary Functions
Unit-Ramp Function
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6. z Transforms of Elementary Functions
Polynomial Function ak
Exponential Function
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7. z Transforms of Elementary Functions
Sinusoidal Function
See Table 2-1 on pages 29-30
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8. Important Properties and Theorems of the z Transform
Multiplication by a Constant
Linearity of the z Transform
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9. Important Properties and Theorems of the z Transform
Multiplication by ak
Shifting Theorem
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10. Important Properties and Theorems of the z Transform
Shifting Theorem (cont.)
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11. Important Properties and Theorems of the z Transform
Example 2-3
Note that z- 1 represents a delay of 1 sampling period T, regardless of the value of T.
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12. Important Properties and Theorems of the z Transform
Complex Translation Theorem
Initial Value Theorem
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13. Important Properties and Theorems of the z Transform
Final Value Theorem
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14. Important Properties and Theorems of the z Transform
Example 2-9
Determine the final value x() of
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15. The Inverse z Transform
The notation for the inverse z transform is Z-1. The inverse z transform of X(z) yields the corresponding time sequence x(k).
Only the time sequence at the sampling instants is obtained from the inverse z transform.
The inverse z transform yields a time sequence that specifies the values of x(t) only at discrete instants of time.
Many different time functions x(t) can have the same x(kT).
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16. The Inverse z Transform
A z transform table: an obvious method for finding the inverse z transform.
Table 2-2
Four methods for obtaining the inverse z transform:
Direct division method
Computational method
Partial-fraction-expansion method
Inversion integral method
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17. The Inverse z Transform
Poles and Zeros in the z Plane
The locations of the poles and zeros of X(z) determine the characteristics of x(k), the sequence of values or numbers.
We often use a graphical display in the z plane of the locations of the poles and zeros of X(z).
poles at z=-1, z=-2
zeros at z=0, z=-0.5
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18. The Inverse z Transform
Direct Division Method
Example 2-10
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19. The Inverse z Transform
Computational Method
Consider a system
For the Kronecker delta input, X(z) =1
The inverse z transform of G(z) is given by y(0),y(1),y(2),…
Two approaches to obtain the inverse transform:
MATLAB approach
Difference equation approach
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20. The Inverse z Transform
Computational Method (cont.)
MATLAB approach
num=[ ]
den=[ ]
x=[1 zeros(1,40)]
y=filter(num,den,x)
y =
0.0001
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21. The Inverse z Transform
Computational Method (cont.)
MATLAB approach
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22. The Inverse z Transform
Computational Method (cont.)
Difference Equation Approach
for k = -2
for k = -1
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23. The Inverse z Transform
Partial-Fraction-Expansion Method
Each expanded term has a form that may easily be found from commonly available z transform tables.
for simple pole
for multiple pole
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24. The Inverse z Transform
Partial-Fraction-Expansion Method (cont.)
Example 2-14
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25. The Inverse z Transform
Inversion Integral Method
for a simple pole
for a multiple pole
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26. The Inverse z Transform
Inversion Integral Method (cont.)
Example 2-16
Two simple poles: z=z1=1 and z=z2=e-aT
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27. z Transform Method for Solving Difference Equations
The linear time-invariant discrete-time system characterized by the following linear difference equation:
Discrete function
z Transform
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28. z Transform Method for Solving Difference Equations
Example 2-18
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29. Concluding Comments
The basic theory of the z transform method has been presented.
z transform : linear time invariant discrete-time systems
Laplace transform: linear time-invariant continuous-time systems
With the z transform method, linear time-invariant difference equations can be transformed into algebraic equations.
The z transform method allows us to use conventional analysis and design techniques available to analog control systems.
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