Correct Answer - Option 4 : u(t) = z / (z - 1)
Concept:
The z-transform of a discrete-time signal x(n) is defined as follows:
\(X\left( z \right) = \mathop \sum \limits_{n = - \infty }^\infty x\left( n \right){z^{ - n}}\)
Or,
\(x\left( n \right)\mathop \leftrightarrow \limits^{\;z\;} X\left( z \right)\)
ROC (Region of Convergence) defines the set of all values of z for which X(z) attains a finite value.
ROC is the set of values of z for which the sequence x(n) z-n is absolutely summable, ie.,
\(\mathop \sum \limits_{n = - \infty }^\infty \left| {x\left( n \right){z^{ - n}}} \right| < \infty \)
Analysis:
x(n) = u(n)
The Z transform is given as:
\(X(z) = \;\mathop \sum \limits_{n = 0 }^{ \infty} {\left( 1 \right)}{z^{ - n}}\)
X(z) = 1 + z-1 + z-2 + z-3 + .......
\(X(z)=[ \frac{1}{1-z^{-1}}]\)
\(X(z)=\frac{z}{z-1}\) with \(0 < \left| z \right| < 1\)
