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Assertion (A): The z-transform of the output of the sampler is given by the series.
Reason (R): The relationship is the result of the application of z = e-sT, where T stands for the time gap between the samples.
Reason (R): The relationship is the result of the application of z = e-sT, where T stands for the time gap between the samples.
- a)Both A and R are true and R is correct explanation of A
- b)Both A and R are true but R is not correct explanation of A
- c)A is true but R is false
- d)A is false but R is true
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
Assertion (A): The z-transform of the output of the sampler is given b...
Explanation: T is termed as the time of the sampling instant and z transform is always defined for the instant of the sampling event and this can be as desired by the user.
Most Upvoted Answer
Assertion (A): The z-transform of the output of the sampler is given b...
Assertion (A): The z-transform of the output of the sampler is given by the series.
Reason (R): The relationship is the result of the application of z = e^(-sT), where T stands for the time gap between the samples.
Explanation:
The z-transform is a mathematical transformation that is used to analyze and process discrete-time signals. It is the discrete-time counterpart of the Laplace transform, which is used for continuous-time signals.
The z-transform of a discrete-time signal x[n] is defined as the infinite sum:
X(z) = Σ(x[n] * z^(-n)), where n ranges from -∞ to ∞.
The z-transform can be used to analyze the frequency content and behavior of discrete-time signals. It is particularly useful for analyzing linear time-invariant (LTI) systems, which are systems whose behavior does not change over time.
Assertion (A): The z-transform of the output of the sampler is given by the series.
This statement is true. The output of a sampler is a discrete-time signal, and the z-transform is used to analyze discrete-time signals. Therefore, the z-transform of the output of the sampler can be represented as a series.
Reason (R): The relationship is the result of the application of z = e^(-sT), where T stands for the time gap between the samples.
This statement is false. The relationship between z and s is given by z = e^(sT), where T is the sampling period. This relationship allows us to convert a continuous-time transfer function into a discrete-time transfer function using the z-transform. The z-transform is then used to analyze the discrete-time system.
Therefore, the correct answer is option C: A is true but R is false. The z-transform of the output of the sampler is given by a series, but the relationship between z and s is z = e^(sT), not z = e^(-sT).
Reason (R): The relationship is the result of the application of z = e^(-sT), where T stands for the time gap between the samples.
Explanation:
The z-transform is a mathematical transformation that is used to analyze and process discrete-time signals. It is the discrete-time counterpart of the Laplace transform, which is used for continuous-time signals.
The z-transform of a discrete-time signal x[n] is defined as the infinite sum:
X(z) = Σ(x[n] * z^(-n)), where n ranges from -∞ to ∞.
The z-transform can be used to analyze the frequency content and behavior of discrete-time signals. It is particularly useful for analyzing linear time-invariant (LTI) systems, which are systems whose behavior does not change over time.
Assertion (A): The z-transform of the output of the sampler is given by the series.
This statement is true. The output of a sampler is a discrete-time signal, and the z-transform is used to analyze discrete-time signals. Therefore, the z-transform of the output of the sampler can be represented as a series.
Reason (R): The relationship is the result of the application of z = e^(-sT), where T stands for the time gap between the samples.
This statement is false. The relationship between z and s is given by z = e^(sT), where T is the sampling period. This relationship allows us to convert a continuous-time transfer function into a discrete-time transfer function using the z-transform. The z-transform is then used to analyze the discrete-time system.
Therefore, the correct answer is option C: A is true but R is false. The z-transform of the output of the sampler is given by a series, but the relationship between z and s is z = e^(sT), not z = e^(-sT).
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