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Which of the following is a Dirichlet condition with respect to the signal x(t)?
- a)x(t) has a finite number of discontinuities in any period
- b)x(t) has finite number of maxima and minima during any period
- c)x(t) is absolutely integrable in any period
- d)All of the mentioned
Correct answer is option 'D'. Can you explain this answer?
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Which of the following is a Dirichlet condition with respect to the si...
Explanation: For any signal x(t) to be represented as Fourier series, it should satisfy the Dirichlet conditions which are x(t) has a finite number of discontinuities in any period, x(t) has finite number of maxima and minima during any period and x(t) is absolutely integrable in any period.
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Which of the following is a Dirichlet condition with respect to the si...
Dirichlet Conditions:
Dirichlet conditions are a set of conditions that determine whether a periodic signal can be represented by its Fourier series. These conditions were formulated by the mathematician Johann Dirichlet and are used to ensure that the Fourier series of a signal converges to the original signal.
There are four main Dirichlet conditions:
1. Periodicity: The signal must be periodic, meaning it repeats itself over a specific time interval.
2. Finite Number of Discontinuities: The signal should have a finite number of discontinuities within any period. This means that there cannot be an infinite number of abrupt changes or jumps in the signal.
3. Finite Number of Extrema: The signal should have a finite number of maxima and minima within any period. This ensures that the signal does not oscillate infinitely or have infinitely sharp peaks.
4. Absolute Integrability: The signal must be absolutely integrable within any period. This means that the integral of the absolute value of the signal over any period should be finite.
Answer Explanation:
In the given options, the option (d) "All of the mentioned" is the correct answer because it satisfies all the Dirichlet conditions. Let's analyze each option to understand why:
a) x(t) has a finite number of discontinuities in any period: This condition is satisfied as it ensures that the signal does not have an infinite number of abrupt changes or jumps.
b) x(t) has a finite number of maxima and minima during any period: This condition is satisfied as it ensures that the signal does not oscillate infinitely or have infinitely sharp peaks.
c) x(t) is absolutely integrable in any period: This condition is satisfied as it ensures that the integral of the absolute value of the signal over any period is finite.
Therefore, option (d) satisfies all the Dirichlet conditions and is the correct answer.
Dirichlet conditions are a set of conditions that determine whether a periodic signal can be represented by its Fourier series. These conditions were formulated by the mathematician Johann Dirichlet and are used to ensure that the Fourier series of a signal converges to the original signal.
There are four main Dirichlet conditions:
1. Periodicity: The signal must be periodic, meaning it repeats itself over a specific time interval.
2. Finite Number of Discontinuities: The signal should have a finite number of discontinuities within any period. This means that there cannot be an infinite number of abrupt changes or jumps in the signal.
3. Finite Number of Extrema: The signal should have a finite number of maxima and minima within any period. This ensures that the signal does not oscillate infinitely or have infinitely sharp peaks.
4. Absolute Integrability: The signal must be absolutely integrable within any period. This means that the integral of the absolute value of the signal over any period should be finite.
Answer Explanation:
In the given options, the option (d) "All of the mentioned" is the correct answer because it satisfies all the Dirichlet conditions. Let's analyze each option to understand why:
a) x(t) has a finite number of discontinuities in any period: This condition is satisfied as it ensures that the signal does not have an infinite number of abrupt changes or jumps.
b) x(t) has a finite number of maxima and minima during any period: This condition is satisfied as it ensures that the signal does not oscillate infinitely or have infinitely sharp peaks.
c) x(t) is absolutely integrable in any period: This condition is satisfied as it ensures that the integral of the absolute value of the signal over any period is finite.
Therefore, option (d) satisfies all the Dirichlet conditions and is the correct answer.
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Which of the following is a Dirichlet condition with respect to the signal x(t)?a)x(t) has a finite number of discontinuities in any periodb)x(t) has finite number of maxima and minima during any periodc)x(t) is absolutely integrable in any periodd)All of the mentionedCorrect answer is option 'D'. Can you explain this answer?
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