Frequency domain integration and differentiation

I'm learning about fourier analysis and am experimenting with integration and differentiation in the frequency domain.

For test cases in the time domain, I generate 3 single frequency data sets: acceleration, velocity and displacement.


Next, I do an FFT of the time domain velocity data,
then integrate in the frequency domain,
then inverse FFT back to time domain and
compare with the test case displacement data.


Next, I do an FFT of the time domain velocity data,
then differentiate in the frequency domain,
then inverse FFT back to time domain and
compare with the test case acceleration data.


Both test cases work, but with a lag (lead?) in time. Since my test data is periodic, I can do a circular shift to get an exact result.

However, the time lag/lead is frequency dependent.

How do I handle an arbitrary signal?

For a 512 sample FFT, do I generate 256 time signals,
apply a separate circular shift for each frequency,
then add all 256 time signals together?

Or have I missed something?

Thanks, RIH5342
.

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