“derivative of FFT outputs with respect to time” doesn’t make any sense, so even though the notation used in the paper seems to say just that, I think we can understand it better by reading the text that says “the accelerations are
found by taking the time derivative of the FFT fitted speeds”.
That makes perfect sense: The FFT of the velocity array is used to interpolate between samples, providing velocity as a continuous function of position. That continuous function is then differentiated at the appropriate position (j in the paper) to find the acceleration at every position i. I didn’t read closely enough to find out how i and j are related.
In implementation, every FFT output for frequency f would be multiplied by fi (that is, the frequency times sqrt(-1), not i the position) to produce the FFT of the acceleration function, and then the FFT basis functions would be evaluated in their continuous form (using Math.sin and Math.cos) to produce an acceleration at any desired point.
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