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For example, I'm given a sequence x(n) = { 3, 5, 1, 4, -3, 5, -2, -2, 4 } whose 9 point DFT is X[k]. Without computing IDFT, i need to find the sequence y(n) whose 9 point DFT is given by $Y[k] = (-1)^k ~ e^{j \frac{\pi}{3} k}$ . How do I approach this question?

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  • $\begingroup$ Do it in algebra, derive a formula, apply the formula for a thorough analysis. $\endgroup$ – Cedron Dawg Jul 18 '20 at 13:01
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    $\begingroup$ Looking again, what does X[k], or x[n], have to do with Y[k]? $\endgroup$ – Cedron Dawg Jul 18 '20 at 15:48
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    $\begingroup$ @my_knee_Hertz. Cedron is correct. You can't even begin to find y(n) until you know how Y[k] is related to X[k]. $\endgroup$ – Richard Lyons Jul 19 '20 at 22:26
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Hints only:

  1. Think about time-frequency duality of DFT pairs. What is the DFT of sequence $-1^k$?

  2. And then there is a phase term in the DFT, so there should be a shift in discrete time sequence. Use time-delay property of DFT.

These 2 properties should give you the solution.

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