jjgoings / pade

do the Fourier transform using the method of Padé approximants

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pade's Issues

Sign of sine problem

Interestingly, scipy.fftpack predicts a negative sign for the three peaks of the discrete Fourier transform of the function sin(2t) + sin(4t) + sin(8t). Currently the Padé code predicts positive signs. The reason for this is in lines 68 and 69 - np.poly1d contracts the a coefficients with the wrong-order phase terms (Eq. 29 of Bruner et al). np.poly1d(np.flip(a)) and likewise for b fixes the issue. This also fixes imaginary-component sign issues with several other functions I've tested.

Why the sign of the peaks of the FT of a simple sum of sine functions would be negative is beyond me, and I'd like to understand it better. Another example of this is sin(t) + cos(2t) + sin(3t) + cos(4t). You get alternating imaginary and real peaks at ω = {1,2,3,4}, as expected, but again the imaginary peaks (corresponding to the sine) are all negative. Padé also has a lot of difficulty with this example, even with damping.