Determine a sequence whose N point DFT is given without computing inverse

Question

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?

DSP Rookie · Answer

Hints only:

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

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.

Determine a sequence whose N point DFT is given without computing inverse

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