Version 1.0, August 31, 2001, Copyright, Hugh Jack 1993-2001

2.9 z-TRANSFORMS

 

· For a discrete-time signal , the two-sided z-transform is defined by . The one-sided z-transform is defined by . In both cases, the z-transform is a polynomial in the complex variable .

 

· The inverse z-transform is obtained by contour integration in the complex plane . This is usually avoided by partial fraction inversion techniques, similar to the Laplace transform.

 

· Along with a z-transform we associate its region of convergence (or ROC). These are the values of for which is bounded (i.e., of finite magnitude).

 

· Some common z-transforms are shown below.

  1. Common z-transforms

Signal

z-Transform

ROC

 

1

All

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

· The z-transform also has various properties that are useful. The table below lists properties for the two-sided z-transform. The one-sided z-transform properties can be derived from the ones below by considering the signal instead of simply .

  1. Two-sided z-Transform Properties

Property

Time Domain

z-Domain

ROC

Notation



 



 



 

Linearity

 

 

At least the intersection of and

Time Shifting

 

 

That of , except if and if

z-Domain Scaling

 

 

 

Time Reversal

 

 

 

z-Domain
Differentiation

 

 

 

Convolution

 

 

At least the intersection of and

Multiplication

 

 

At least

Initial value theorem

causal