Example 9.6.  Find the inverse z-transform   .   Use (a) series, (b) table of z-transforms, (c) residues.  

Solution 9.6.

Solution 9.6 (a).  Method of series.  

    Expand   in a series involving powers of .  
    
        
        
The sequence of coefficients in the Laurent series is what we desire, and we see that  

        .  

Solution 9.6 (b).  Writing    we identify    and use line 3 in the Table 9.1 to obtain

        .

Solution 9.6 (c).  Writing    we see that    has a simple pole at  .  Using the Corollary 9.1 for finding the inverse z-transform we obtain  

        
        
Using  the function    and value    in Corollary 9.2 we get  
        
        .   

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