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Which one of the following is the correct statement? The region of convergence of z-transform of x[n] consists of the values of z for which x[n] is:
- a)Absolutely integrable
- b)Absolutely summable
- c)Unity
- d)<1
Correct answer is option 'B'. Can you explain this answer?
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Which one of the following is the correct statement? The region of con...
Explanation: The region of convergence of z-transform of x[n] consists of the values of z for which x[n]r-n is absolutely summable.
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Which one of the following is the correct statement? The region of con...
Region of Convergence (ROC) in Z-transform:
The Region of Convergence (ROC) is a critical concept in the Z-transform domain and determines the values of z for which the Z-transform of a discrete-time signal is well-defined.
Absolutely Summable:
The correct statement is that the region of convergence of the Z-transform of x[n] consists of the values of z for which x[n] is absolutely summable. This means that the series formed by the magnitude of x[n] is convergent. In other words, the sum of the absolute values of x[n] over all n is finite.
Explanation:
The Z-transform of a discrete-time signal x[n] is given by X(z) = Σ(x[n] * z^(-n)). For the Z-transform to exist, the series formed by x[n] * z^(-n) must converge. This is ensured by the condition of absolute summability.
Importance of ROC:
The ROC also determines the stability and causality of the system represented by the Z-transform. Different ROCs correspond to different system behaviors, and understanding the ROC is crucial in analyzing and designing systems in the Z-domain.
Conclusion:
In summary, the region of convergence of the Z-transform of x[n] consists of the values of z for which x[n] is absolutely summable. This condition ensures the convergence of the Z-transform series and is essential for analyzing the properties of discrete-time signals and systems in the Z-domain.
The Region of Convergence (ROC) is a critical concept in the Z-transform domain and determines the values of z for which the Z-transform of a discrete-time signal is well-defined.
Absolutely Summable:
The correct statement is that the region of convergence of the Z-transform of x[n] consists of the values of z for which x[n] is absolutely summable. This means that the series formed by the magnitude of x[n] is convergent. In other words, the sum of the absolute values of x[n] over all n is finite.
Explanation:
The Z-transform of a discrete-time signal x[n] is given by X(z) = Σ(x[n] * z^(-n)). For the Z-transform to exist, the series formed by x[n] * z^(-n) must converge. This is ensured by the condition of absolute summability.
Importance of ROC:
The ROC also determines the stability and causality of the system represented by the Z-transform. Different ROCs correspond to different system behaviors, and understanding the ROC is crucial in analyzing and designing systems in the Z-domain.
Conclusion:
In summary, the region of convergence of the Z-transform of x[n] consists of the values of z for which x[n] is absolutely summable. This condition ensures the convergence of the Z-transform series and is essential for analyzing the properties of discrete-time signals and systems in the Z-domain.
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Which one of the following is the correct statement? The region of convergence of z-transform of x[n] consists of the values of z for which x[n] is:a)Absolutely integrableb)Absolutely summablec)Unityd)<1Correct answer is option 'B'. Can you explain this answer?
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