What is the inverse z transform for z/(z^2-2z+2).

Question

Inverse z transform of Z/(Z^2-2Z+2) ?

Answer 1

The inverse z transform can be written as (1/ 2 pi i) times a contour integral in the complex plane.

Because of the term z² -2z +2 in the denominator,   there are two poles in the complex plane:   z = 1 + i  and z = 1 -i

 

The integrand of the contour integral is   zⁿ /[( z -(1+i) ) (z - (1 - i))]  [ 1/ 2 pi i ]

 

Caauch's theorem can be used to evaluate this contour integral.

 

The result is (1 + i)ⁿ /(2 i)  + (1 - i)ⁿ  / (- 2i)

 

Using the fact that  1 + i =  2^1/2 exp( i pi/4)    this can be rewritten as

 

2^n/2  2 cos( n pi/4)