Signals and Systems

6.2 REGION OF CONVERGENCE (ROC)

The region of convergence are the values of z for which the z-transform converges. Equation (6.1) shows that z-transform is an infinite power series which is not always convergent for all values of z. Therefore, the region of convergence should be mentioned along with the z-transformation.

Example 6.1   Find the z-transform of the following sequences

(i) x₁(n) = {1, 2, 3, 4, 5, 6, 7} and (ii) x₂(n) = {1, 2, 3, 4, 5, 6, 7}

↑                   

Solution

1. For the given sequence x₁(n) = {1, 2, 3, 4, 5, 6, 7}, we can write x₁(0) = 1, x₁(1) = 2, x₁(2) = 3, x₁(3) = 4, x₁(4) = 5, x₁(5) = 6 and x₁(6) = 7.

   The z-transform of the sequence x₁(n) is given by

    

    

   In the present case n = 0 to n = 6.

    

    

   X₁(z) has finite ...