The z Transform M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

The z Transform

M. J. Roberts - All Rights Reserved. Edited

by Dr. Robert Akl

Generalizing the DTFT

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

2

Complex Exponential Excitation


3

The Transfer Function


4

Systems Described by Difference Equations


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


6

Direct Form II Realization


7

The Inverse z Transform


8

Existence of the z Transform

Time Limited Signals


9


Right- and Left-Sided Signals


10


Right- and Left-Sided Exponentials


11



12



13



14

Some Common z Transform Pairs


15

z-Transform Properties


16



17



18



19

The Inverse z TransformSynthetic Division


20

Synthetic Division



21

Synthetic Division



22

Synthetic Division



23

Partial Fraction Expansion


24

Partial Fraction Expansion


25



26



27



28



29



30

Inverse z Transform Example


31



32

04/19/23



33

The Unilateral z Transform


34

Properties of the Unilateral z Transform


35

Solving Difference Equations


36

Solving Difference Equations


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).


38



39



40

Transform Method Comparison


41



42



43



44

System Response to a Sinusoid


45



46



47

Documents

The z Transform M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl