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Signals and Systems by Smarajit Ghosh

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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 ...

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