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Assertion (A): For the rational transfer function H(z) to be causal, stable and causally invertible, both the zeroes and the poles should lie within the unit circle in the z-plane.
Reason (R): For a rational system, ROC is bounded by poles
Reason (R): For a rational system, ROC is bounded by poles
- a)Both A and R are true and R is the correct explanation of A
- b)Both A and R are true bit R is NOT the correct explanation of A
- c)A is true but R is false
- d)A is false but R is true
Correct answer is option 'B'. Can you explain this answer?
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Assertion (A): For the rational transfer function H(z) to be causal, s...
Explanation: For the rational transfer function H (z) to be causal, stable and causally invertible, both the zeroes and the poles should lie within the unit circle in the z-plane. For a rational system, ROC is bounded by poles.
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Assertion (A): For the rational transfer function H(z) to be causal, s...
Assertion (A): For the rational transfer function H(z) to be causal, stable and causally invertible, both the zeroes and the poles should lie within the unit circle in the z-plane.
Reason (R): For a rational system, ROC is bounded by poles.
Explanation:
To understand the assertion and reason given in the question, let's break it down into two parts and analyze each one separately.
Part 1: For the rational transfer function H(z) to be causal, stable and causally invertible, both the zeroes and the poles should lie within the unit circle in the z-plane.
Explanation of Assertion (A):
The statement in Assertion (A) is correct. In order for a rational transfer function, represented by H(z), to be causal, stable, and causally invertible, both the zeroes and the poles of the transfer function should lie within the unit circle in the z-plane.
Causal: A rational transfer function is causal if all the poles of the transfer function lie strictly inside the unit circle. This condition ensures that the output of the system depends only on the present and past inputs, not on future inputs.
Stable: A rational transfer function is stable if all the poles of the transfer function lie strictly inside the unit circle. Stability implies that the output of the system remains bounded for any bounded input.
Causally Invertible: A rational transfer function is causally invertible if all the zeroes of the transfer function lie strictly inside the unit circle. Causal invertibility ensures that the original input can be uniquely determined from the output of the system.
Explanation of Reason (R):
The statement in Reason (R) is also correct. For a rational system, the Region of Convergence (ROC) is bounded by the poles of the transfer function.
ROC: The ROC is the region in the z-plane where the z-transform of a sequence converges. In the case of a rational transfer function, the ROC can be bounded or unbounded.
For a rational system, the ROC is bounded by the poles of the transfer function. This means that the poles determine the boundary of the ROC. If any pole lies outside the unit circle, the ROC will be unbounded and the system will not be causal, stable, or causally invertible.
Conclusion:
Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A). The poles of a rational transfer function determine the boundary of the ROC, and for the transfer function to be causal, stable, and causally invertible, both the zeroes and the poles should lie within the unit circle in the z-plane.
Reason (R): For a rational system, ROC is bounded by poles.
Explanation:
To understand the assertion and reason given in the question, let's break it down into two parts and analyze each one separately.
Part 1: For the rational transfer function H(z) to be causal, stable and causally invertible, both the zeroes and the poles should lie within the unit circle in the z-plane.
Explanation of Assertion (A):
The statement in Assertion (A) is correct. In order for a rational transfer function, represented by H(z), to be causal, stable, and causally invertible, both the zeroes and the poles of the transfer function should lie within the unit circle in the z-plane.
Causal: A rational transfer function is causal if all the poles of the transfer function lie strictly inside the unit circle. This condition ensures that the output of the system depends only on the present and past inputs, not on future inputs.
Stable: A rational transfer function is stable if all the poles of the transfer function lie strictly inside the unit circle. Stability implies that the output of the system remains bounded for any bounded input.
Causally Invertible: A rational transfer function is causally invertible if all the zeroes of the transfer function lie strictly inside the unit circle. Causal invertibility ensures that the original input can be uniquely determined from the output of the system.
Explanation of Reason (R):
The statement in Reason (R) is also correct. For a rational system, the Region of Convergence (ROC) is bounded by the poles of the transfer function.
ROC: The ROC is the region in the z-plane where the z-transform of a sequence converges. In the case of a rational transfer function, the ROC can be bounded or unbounded.
For a rational system, the ROC is bounded by the poles of the transfer function. This means that the poles determine the boundary of the ROC. If any pole lies outside the unit circle, the ROC will be unbounded and the system will not be causal, stable, or causally invertible.
Conclusion:
Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A). The poles of a rational transfer function determine the boundary of the ROC, and for the transfer function to be causal, stable, and causally invertible, both the zeroes and the poles should lie within the unit circle in the z-plane.
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Assertion (A): For the rational transfer function H(z) to be causal, stable and causally invertible, both the zeroes and the poles should lie within the unit circle in the z-plane.Reason (R): For a rational system, ROC is bounded by polesa)Both A and R are true and R is the correct explanation of Ab)Both A and R are true bit R is NOT the correct explanation of Ac)A is true but R is falsed)A is false but R is trueCorrect answer is option 'B'. Can you explain this answer?
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