Circular Time-Shifting Property of the DFT

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We begin with the forward DFT, the twiddle factor W_N equals e to the minus j two pi over N, and the inverse DFT, setting up the right circular shift definition y of n equals x of n minus m, mod N. Starting from y of n, we substitute the inverse DFT for x, expand the exponent into minus k n plus k m, separate the factors, and identify the shifted spectrum, the boxed result Y of k equals W_N to the k m times X of k. For x of n equals one, two, three, four, with N equals four and m equals one, X of k equals ten, negative two plus two j, negative two, negative two minus two j. Shifting right gives y of n equals four, one, two, three, and Y of k equals ten, two plus two j, two, two minus two j, confirming the magnitude stays equal while the phase shifts by two pi k m over N.