What is derivative of FFT outputs with respect to time?

Question

I am quite new Digital Signal Processing.I am trying to implement anti-cogging algortihm for my permanent magnet synchronous motor (=PMSM) control algortihm. I follow this documentation

I collected velocity data according to the rotation angle. I have 1024 bytes long sampling buffer. Each sample is stored when my rotation encoder gave a pulse, so each index actually means mechanical rotation angle. Each stored value is the velocity at related angle.

I transformed the collected samples to frequency domain with FFT. It's not clear to me what is my Fs (=sampling frequency). When I examined the frequency spectrum, dominant harmonics are related multiplies of 14 - 27 index numbers which is number of slot(27) and number of pp(14).

But the last step of the suggested method, Acceleration Based Waveform Analysis, need calculated derivative of FFT outputs with respect to time. Outputs are in frequency domain, so how could I calculate derivative of FFT outputs with respect to time and why does the method need this calculation?

Answer 1

This answer is obsolete. It tells only how to calculate the Fourier transform of a derivative if the Fourier transform of the original function is known. The actual problem is revealed in comments and the question is fixed.

A great part of the problem is how to get the angular acceleration as a function of rotation angle when the measurements are made by recording the angular velocity i.e. the rotation speed as a function of rotation angle. A sample of rotation speed has been stored every time the motor has rotated a certain angle increment. The sample queue contains 1024 samples (no scaling info is included).

The final goal is to make movements smooth in a robot by implementing an anti-gogging system. Usual robot motors with simplest possible drive pulses generate non-uniform torque which make movements twitching. The torque smoothing will be tried by using PWM to generate more complex drive pulses which compensate twitches

The torque variations cannot be properly measured from motor current and voltage. because they are caused by structural non-uniformities such as reluctance variation due rotation

The useless original answer:

You obviously want the spectrum of the acceleration and you already have the spectrum of the velocity. I do not know what exact notation you use, but let's name to Vk the complex valued sample in the spectrum. k is the index and k=0 means the DC component.

Let's assume your physical sampling period is T, so your adjacent points Vk and V(k+1) have physical frequency difference 1/T.

The spectrum of the acceleration can be calculated as follows:

Ak = (Vk) * j * 2Pi *k/T

where j is the imaginary unit.

This is taken from the general properties of the Fourier transform - how to calculate the transform of the time derivative of a function. The original transform should be multiplied by

j* 2Pi * f

where f is the frequency.

I guess you have searched the formula for Ak from writings of DFT. It's not there. Discrete time signals do not have such thing as time derivative. This is a special case. Now the signal was assumed to be a chain of samples of continuous time signal which has the derivative.