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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) = Z

^{0}+ 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

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**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) |

### This post was written by: Sreejith Hrishikeshan

Sreejith Hrishikesan Nair is a M-Tech graduate in Communication Systems. He completed B-tech Degree in Electronics and Communication.He is a person who wants to implement new ideas in the field of Technology.

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