The z Transform
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by Dr. Robert Akl
Generalizing the DTFT
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2
Complex Exponential Excitation
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The Transfer Function
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Systems Described by Difference Equations
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5
Direct Form II Realization
Direct Form II realization of a discrete-time system is similar in form to Direct Form II realization of continuous-time systems
A continuous-time system can be realized with integrators, summing junctions and multipliers
A discrete-time system can be realized with delays, summing junctions and multipliers
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6
Direct Form II Realization
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The Inverse z Transform
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Existence of the z Transform
Time Limited Signals
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9
Existence of the z Transform
Right- and Left-Sided Signals
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Existence of the z Transform
Right- and Left-Sided Exponentials
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11
Existence of the z Transform
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Existence of the z Transform
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Existence of the z Transform
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Some Common z Transform Pairs
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z-Transform Properties
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z-Transform Properties
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z-Transform Properties
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z-Transform Properties
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19
The Inverse z TransformSynthetic Division
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20
Synthetic Division
The Inverse z Transform
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Synthetic Division
The Inverse z Transform
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22
Synthetic Division
The Inverse z Transform
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23
Partial Fraction Expansion
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Partial Fraction Expansion
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25
z-Transform Properties
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z-Transform Properties
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27
z-Transform Properties
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z-Transform Properties
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29
z-Transform Properties
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Inverse z Transform Example
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Inverse z Transform Example
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04/19/23
Inverse z Transform Example
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The Unilateral z Transform
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Properties of the Unilateral z Transform
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Solving Difference Equations
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Solving Difference Equations
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37
Pole-Zero Diagrams and Frequency Response
For a stable system, the response to a sinusoid applied attime t = 0 approaches the response to a true sinusoid (appliedfor all time).
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38
Pole-Zero Diagrams and Frequency Response
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Pole-Zero Diagrams and Frequency Response
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40
Transform Method Comparison
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Transform Method Comparison
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Transform Method Comparison
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Transform Method Comparison
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System Response to a Sinusoid
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System Response to a Sinusoid
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System Response to a Sinusoid
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