Instead, in the bottom figure, the amount of scaling appears to vary non-linearly, such that it is continuous. This is especially apparent since the figure shows a dotted line representing the spectrum before equalization. You could say that this is just how the figure was drawn, however it is highly consistent across the whole figure; furthermore, it is consistent with other papers and the descriptions in some of the patents seem to suggest that the amplitude discontinuities may be solved. Based on this, I have a hypothesis on how the non-uniform amplitude scaling may have been implemented. The process would be to create a first envelope of the harmonic peaks before transformation; create a second envelope of the harmonic peaks after frequency shifting, but before equalization; and finally scale each bin based on the ration of the first envelope to the second envelope. Although I have not tested it, looking at the figure visually, the proposed procedure would seem to match very closely, assuming a smoothing interpolation. Furthermore, it may also explain why the intermediate stage that shows the spectrum after shifting, but before equalization, is explicitly shown. Such a thing is not actually created by the SPP algorithm as described, but it does make sense to show for the proposed non-uniform amplitude scaling. Finally, in theory, this procedure would not produce any additional amplitude discontinuities, further consistent with the figure.

I will implement this idea as a variant in my existing SPP algorithm, which has already been implemented such that multiple variants (and combinations thereof) can exist in the same procedure (using preprocessor flags). Furthermore, I will consider the possibility that the same procedure is also used for the unwrapped phase envelope.