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GATE EE 2020 Official Paper

Option 4 : |z| > 1/2

CT 1: Ratio and Proportion

2672

10 Questions
16 Marks
30 Mins

**Concept:**

**Z transform:**

The **Z transform** of x(t) is, denoted by X(z), is defined as:

\(X\left( z \right) = \mathop \sum \limits_{n = - \infty }^\infty x\left( n \right){z^{ - n}}\)

\({a^n}u\left( n \right)\mathop \to \limits^{ZT} \frac{1}{{1 - a{z^{ - 1}}}}\); ROC:|z|>|a|

**Shifting property:**

If \(x\left( n \right)\mathop \to \limits^{ZT} x\left( z \right)\) ; ROC: R

Then \(x\left( {n - {n_0}} \right)\mathop \to \limits^{ZT} {z^{ - {n_0}}}x\left( z \right)\); ROC: R

The time-shifting will not affect ROC.

**Calculation:**

Given that, \(x\left( n \right) = {\left( {\frac{1}{2}} \right)^n}1\left[ n \right]\)

Z Transform (ZT) of x(n) will be,

\(x\left( z \right) = \frac{z}{{z - \frac{1}{2}}} = \frac{1}{{1 - \frac{1}{2}{z^{ - 1}}}}\) ; ROC: \(\left| z \right| > \frac{1}{2}\)

And, \(z\left( {x\left( {n - k} \right)} \right) = \;\frac{{{z^{ - k}}}}{{1 - \frac{1}{2}{z^{ - 1}}}}\) ; ROC: \(\left| z \right| > \frac{1}{2}\)