fourier transform – Verify FFT results without equation of waveform

You’re fitting to measured data. The only way that I can see to make sure your fit is good is to analyze the FFT output, and reconstruct the waveform with a function, then see how that matches up to your captured data.

You’re not capturing an integer number of periods. That will cause some spectral bleeding, because the bins don’t match exactly. You probably want to window the captured data before you do the FFT — this will make for slightly more, but cleaner, spectral bleeding.

From the waveform, it’s a fundamental with a strong 5th-harmonic impressed on it — that’s consistent with what I see in your FFT.

You can hypothesize that the above is, indeed, what you’re seeing. With that hypothosis, you can then identify the peaks and interpolate the actual frequency of the sine waves from them (alternately, if it’s rotating or reciprocating machinery, just take the fundamental as being equal to the rotating or reciprocating frequency).

Then if you’re still unsure if you’re getting it right, reconstruct the signal from the phase & amplitude relationship.