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# Lecture 2: Properties of the Z-Transform

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Schedule :: Intro :: Inverse :: Properties :: Solution :: Transfer Functions :: System Realization :: Bilateral :: Frequency Response :: Rubber Sheet Geometry :: FIR Filters

In the descriptions of these properties, take

Delay property

This is very analogous to the differentiation property of Laplace transforms, and will similarly allow us to solve differential equations.

So is the delay operator. (As is the differentiation operator.) Also
Note the difference between these two!

This property is used to introduce the initial conditions when we use transforms to solve difference equations.

Similar to the last property,

Convolution
Like the convolution property for Laplace transforms, the convolution property for Z-transforms is very important for systems analysis and design. In words: The transform of the convolution is the product of the transforms. This holds for both Laplace and Z-transforms.

If and then

where denotes convolution (in this case, discrete-time convolution).

Multiplication by

Multiplication by

Initial Value theorem
For a causal ,

Final Value theorem
If has no poles outside the unit circle (i.e. it is stable),
Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, May 17). Lecture 2: Properties of the Z-Transform. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Signals_and_Systems/node3_1.html. This work is licensed under a Creative Commons License