Reading a scrambled DFT list: separate the signs from the subscripts

To invert a DFT list like Y[k] = {X4, -X5, X0, -X1, X2, -X3}, treat it as two independent changes. Alternating signs mean (-1)^k, which is a circular shift of x[n] by N/2 (N even). The position c of X0 means X[k-c], which multiplies by e^(+j2*pi*c*n/N). Result: y[n] = e^(j2*pi*c*n/N) x[<n-N/2>_N], with the ramp applied after the shift. Sign rule: x moving right gives a minus ramp on X, X moving right gives a plus ramp on x, and moving left flips either. Confirm by building Y from np.fft.fft(x) and comparing np.fft.ifft(Y) with the formula.