Assertion A Analysis

The given sequence is $X[n] = \left(\frac{1}{2}\right)^n u(n)$.

The Z-transform of a causal exponential sequence $a^n u[n]$ is given by $\frac{1}{1-az^{-1}}$ with a Region of Convergence (ROC) of $|z| > |a|$.

For $a = \frac{1}{2}$, the Z-transform is $X(z) = \frac{1}{1-\frac{1}{2}z^{-1}}$ and the ROC is $|z| > \frac{1}{2}$. This matches the assertion.

System Stability Condition

A discrete-time Linear Time-Invariant (LTI) system is considered stable if and only if the ROC of its impulse response includes the unit circle, which is the circle defined by $|z| = 1$.

Assertion A and Stability

In Assertion A, the ROC of the Z-transform $X(z)$ is given as $|z| > \frac{1}{2}$.

To check for stability, we determine if this ROC includes the unit circle ($|z|=1$). Since any value of $z$ such that $|z|=1$ also satisfies $|z| > \frac{1}{2}$ (because $1$ is greater than $\frac{1}{2}$), the unit circle is indeed included within the ROC.

Therefore, the system described in Assertion A is stable.

Assertion A is correct.

Reason R Analysis

Reason R states that for a stable system, the ROC must include the unit circle.

This statement accurately describes the necessary and sufficient condition for the stability of a causal LTI system in the Z-domain.

Reason R is correct.

Relationship between Assertion A and Reason R

Assertion A correctly identifies a system as stable based on its Z-transform and ROC.

Reason R provides the underlying principle that justifies this conclusion: the inclusion of the unit circle in the ROC is the criterion for stability.

Since the ROC in Assertion A ($|z| > \frac{1}{2}$) fulfills the condition stated in Reason R, Reason R serves as a correct explanation for why Assertion A is true.

Conclusion

Both Assertion A and Reason R are accurate statements, and Reason R correctly explains Assertion A.