Example 9.4.  The z-transform of the exponential sequence   [Graphics:Images/ZTransformIntroMod_gr_116.gif]   is   [Graphics:Images/ZTransformIntroMod_gr_117.gif].  

[Graphics:Images/ZTransformIntroMod_gr_119.gif]

Explore Solution 9.4.

[Graphics:../Images/ZTransformIntroMod_gr_120.gif]

[Graphics:../Images/ZTransformIntroMod_gr_121.gif]

 

 

 

We can explore the situation when  [Graphics:../Images/ZTransformIntroMod_gr_122.gif]  and  [Graphics:../Images/ZTransformIntroMod_gr_123.gif].

[Graphics:../Images/ZTransformIntroMod_gr_124.gif]

[Graphics:../Images/ZTransformIntroMod_gr_125.gif]

[Graphics:../Images/ZTransformIntroMod_gr_126.gif]

 

 

 

[Graphics:../Images/ZTransformIntroMod_gr_127.gif]

[Graphics:../Images/ZTransformIntroMod_gr_128.gif]

[Graphics:../Images/ZTransformIntroMod_gr_129.gif]

 

 

 

We are done.

Aside.  We can find the inverse of  X[z]  using residues.

[Graphics:../Images/ZTransformIntroMod_gr_130.gif]

[Graphics:../Images/ZTransformIntroMod_gr_131.gif]

 

 

 

[Graphics:../Images/ZTransformIntroMod_gr_132.gif]

[Graphics:../Images/ZTransformIntroMod_gr_133.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) 2006 John H. Mathews, Russell W. Howell