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
Match the following :
List - I List - II (a) U(t) (i) \(\dfrac{TZ}{(Z-1)^{2}}\) (b) t (ii) \(\dfrac{Z}{Z-e^{-aT}}\) (c) t2 (iii) \(\dfrac{Z}{Z-1}\) (d) e-at (iv) \(\dfrac{T^{2}Z(Z+1)}{(Z-1)^{3}}\) Codes :
For the following signal :
\(x(n)=\left(\dfrac{1}{2}\right)^{n}u(n)+\left(\dfrac{1}{3}\right)^{n}u(n)\)
Z transforms and ROC has been given in below statements :
(a) \(Z[x(n)]=\dfrac{Z}{Z-\frac{1}{2}}+\dfrac{Z}{Z-\frac{1}{3}}\)
(b) ROC : \(|Z| \gt \dfrac{1}{2}\) and \(|Z| \gt \dfrac{1}{3}\)
(c) \(Z[x(n)]=\dfrac{Z}{Z+\frac{1}{2}}+\dfrac{Z}{Z+\frac{1}{3}}\)
(d) ROC : \(|Z| \lt \dfrac{1}{3}\) and \(|Z| \lt \dfrac{1}{3}\)
Out of the above given statements which are correct ?
The z transform of e −t sampled at 10 Hz will be:
Consider a discrete-time sequencebr
$x[n] = \begin{cases} (0.2)^n, & 0 \le n \le 7 \\ 0, & \text{otherwise} \end{cases}$
The region of convergence of $X(z)$, the z-transform of $x[n]$, consists ofFor a causal discrete-time LTI system with transfer function
$H(z) = \frac{2z^2+3}{(z+\frac{1}{3})(z-\frac{1}{3})}$,
which of the following statements is/are true?
