Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R Assertion A: For a given sequence $X[n]=(\frac{1}{2})^n u(n)$ and its Z-transform $X(z)=\frac{1}{1-\frac{1}{2}z^{-1}}, \quad |z|>\frac{1}{2}$ is a stable system. Reason R: For a stable system ROC (Region of Convergence) must include the unit circle. In the light of the above statements, choose the most appropriate answer from the options given below
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.
Important Questions from Z Transform
The z transform of the following real exponential sequence
x(n) = {a n ;n >= 0} , {= 0 ; n < 0} and a > 0 is given by
The causal signal with z-transform z 2(z - a) -2 is
(u[n] is the unit step signal)
The ROC of a system is the
The similarity between the Fourier transform and the z-transform is that
The z-transform of a causal periodic signal can be determined from the knowledge of the z-transform of its:
