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Z Transform of U(n)

  • Find the Z Transform and ROC of U(n) ?

    The Z transform of a discrete time signal x(n) is given by,






    Here given x(n) = u(n)

    Therefore,






    We know that U(n) = 1; n ≥ 0
                                 = 0; n < 0
    Therefore,






    X(z) = Z0 + Z-1 +Z-2 + Z-3 + Z-4 +………….
            = 1 + Z-1 +Z-2 + Z-3 + Z-4 +………….

    It is clear that the infinite series is a Geometric Progression (GP)

    The sum of the GP is given by

    Sum = First Term / (1 – Common Ratio)

    The common ratio (r) is given by

    r = second term/first term
      = third term/second term

    So, r = Z-1 /1 = Z-2/ Z-1 = Z-1

    Hence, the sum of the series is given by





    ROC of U(n)


    The ROC of U(n) is given by

    |r| < 1

    |Z-1| < 1

    |1/Z| < 1

    |Z| > |1|
    ROC of U(n)



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