interestingly, it's still off by a constant phase, and i'm not sure why yet it's also interesting that the way the sinusoids sum its details are equal to its larger shape. i imagine there's a trigonometric identity for that. (Pdb) data = np.random.random(4) (Pdb) quaddata = np.concatenate([data]*4) (Pdb) p data array([0.16894141, 0.65128647, 0.69280144, 0.71307183]) (Pdb) p abs(micro_ift(micro_ft(quaddata, 4), 4, upsample=True)) array([0.16894141, 0.65128647, 0.69280144, 0.71307183, 0.16894141, 0.65128647, 0.69280144, 0.71307183, 0.16894141, 0.65128647, 0.69280144, 0.71307183, 0.16894141, 0.65128647, 0.69280144, 0.71307183]) (Pdb) p abs(micro_ift(micro_ft(quaddata, 4), 4, upsample=True))[::4] array([0.16894141, 0.16894141, 0.16894141, 0.16894141])