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Let x(t) be a continuous time periodic signal with fundamental period T = 1 seconds. Let {ak} be the complex Fourier series coefficients of x(t), where k is integer valued. Consider the following statements about x(3t):
I. The complex Fourier series coefficients of x(3t) are {ak} where k is integer valued
II. The complex Fourier series coefficients of x(3t) are {3ak} where k is integer valued
III. The fundamental angular frequency of x(3t) is 6 π rad/s
For the three statements above, which one of the following is correct?
  • a)
    only II and III are true
  • b)
    only I and III are true
  • c)
    only III is true
  • d)
    only I is true
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
Let x(t) be a continuous time periodic signal with fundamental period ...
Fourier series coefficient ak is unaffected by scaling operating. Thus (I) is true and (II) is false.
T = 1sec for x(t) and if it compressed by ‘3’ then the resultant period T = 1/3
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Most Upvoted Answer
Let x(t) be a continuous time periodic signal with fundamental period ...

Explanation:

Statement I:
- When x(t) is replaced by x(3t), the fundamental period becomes T' = 1/3 seconds.
- The complex Fourier series coefficients of x(3t) are {ak} only if the fundamental period remains the same.
- Therefore, Statement I is false.

Statement II:
- When x(t) is replaced by x(3t), the complex Fourier series coefficients change to {3ak} due to the change in fundamental period.
- This is because the coefficients are scaled by the reciprocal of the time scale factor.
- Therefore, Statement II is true.

Statement III:
- The fundamental angular frequency of x(t) is 2π rad/s.
- When x(t) is replaced by x(3t), the fundamental angular frequency becomes 6π rad/s.
- Therefore, Statement III is true.

Therefore, the correct answer is only I and III are true.
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Let x(t) be a continuous time periodic signal with fundamental period T = 1 seconds. Let {ak} be the complex Fourier series coefficients of x(t), where k is integer valued. Consider the following statements about x(3t):I. The complex Fourier series coefficients of x(3t) are {ak} where k is integer valuedII. The complex Fourier series coefficients of x(3t) are {3ak} where k is integer valuedIII. The fundamental angular frequency of x(3t) is 6 πrad/sFor the three statements above, which one of the following is correct?a)only II and III are trueb)only I and III are truec)only III is trued)only I is trueCorrect answer is option 'B'. Can you explain this answer?
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Let x(t) be a continuous time periodic signal with fundamental period T = 1 seconds. Let {ak} be the complex Fourier series coefficients of x(t), where k is integer valued. Consider the following statements about x(3t):I. The complex Fourier series coefficients of x(3t) are {ak} where k is integer valuedII. The complex Fourier series coefficients of x(3t) are {3ak} where k is integer valuedIII. The fundamental angular frequency of x(3t) is 6 πrad/sFor the three statements above, which one of the following is correct?a)only II and III are trueb)only I and III are truec)only III is trued)only I is trueCorrect answer is option 'B'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about Let x(t) be a continuous time periodic signal with fundamental period T = 1 seconds. Let {ak} be the complex Fourier series coefficients of x(t), where k is integer valued. Consider the following statements about x(3t):I. The complex Fourier series coefficients of x(3t) are {ak} where k is integer valuedII. The complex Fourier series coefficients of x(3t) are {3ak} where k is integer valuedIII. The fundamental angular frequency of x(3t) is 6 πrad/sFor the three statements above, which one of the following is correct?a)only II and III are trueb)only I and III are truec)only III is trued)only I is trueCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let x(t) be a continuous time periodic signal with fundamental period T = 1 seconds. Let {ak} be the complex Fourier series coefficients of x(t), where k is integer valued. Consider the following statements about x(3t):I. The complex Fourier series coefficients of x(3t) are {ak} where k is integer valuedII. The complex Fourier series coefficients of x(3t) are {3ak} where k is integer valuedIII. The fundamental angular frequency of x(3t) is 6 πrad/sFor the three statements above, which one of the following is correct?a)only II and III are trueb)only I and III are truec)only III is trued)only I is trueCorrect answer is option 'B'. Can you explain this answer?.
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