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X(k) is the Discrete Fourier Transform of a 6-point real sequence x(n).
If X(0) = 9 + j0, X(2) = 2 + j2, X(3) = 3- j0, X(5) =1- j1, x(0) is
  • a)
    3
  • b)
    9
  • c)
    15
  • d)
    18
Correct answer is option 'A'. Can you explain this answer?
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X(k) is the Discrete Fourier Transform of a 6-point real sequence x(n)...
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X(k) is the Discrete Fourier Transform of a 6-point real sequence x(n)...
Given:
- X(k) is the Discrete Fourier Transform (DFT) of a 6-point real sequence x(n).
- X(0) = 9 + j0
- X(2) = 2 + j2
- X(3) = 3 - j0
- X(5) = 1 - j1

To Find:
The value of x(0)

Solution:
The Discrete Fourier Transform (DFT) of a real sequence has the following properties:
- X(0) is always real.
- X(N-k) = X*(k) for k = 1 to N-1, where * represents the complex conjugate.

Using the above properties:
- X(0) = 9 + j0 (given)
- X(6-0) = X(6-0) = X(6) = X*(0) = 9 - j0 (complex conjugate property)
- X(6) is always real.

Therefore, we can conclude that X(6) = 9 - j0 is also real, which means the imaginary component is zero. Hence, x(0) = X(6) = 9.

Conclusion:
The value of x(0) is 9. Therefore, the correct answer is option 'A'.
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X(k) is the Discrete Fourier Transform of a 6-point real sequence x(n).If X(0) = 9 + j0, X(2) = 2 + j2, X(3) = 3- j0, X(5) =1- j1, x(0) isa)3b)9c)15d)18Correct answer is option 'A'. Can you explain this answer?
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