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Fourier series in complex form

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Fourier series in trigonometry form

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Here, I'm measuring the same signal (an impulse force - or at least as close as possible to an impulse, represented by a knocking force). The work I'm doing is parameter study (i.e.: studying the effect of different bandwidths and numbers of FFT lines).

This is on the same number of FFT lines but different bandwidth

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And this is the same bandwidth and different FFT lines

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The main conclusion I can draw is that changing one parameter (while keeping the other is the same) would change the amplitude of the frequency spectrum. I can explain this via math above (smaller frequency resolution leads to longer measuring time, and thus lower Fourier coefficients).

But what does it mean in a practical and realistic manner? As in for actual sampling process, how would changing the parameters (measuring time, sampling points...) affect the frequency spectrum of the same signal?

Edit: The smaller frequency resolution would eventually lead to lower amplitude in frequency spectrum is considered as analytical math by my instructor, while the actual system works with numerical math. So a more "physical" (or "what actually happens") answer is needed.

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Are you using the correct FFT form for spectral analysis? You need to divide by N when mapping time->frequency domain with an N-point FFT for a power spectrum. If you want power spectral density you have to factor in the bin bandwidth and window effective noise bandwidths.

The amplitude of a spectral line represents the power of a single signal frequency, power spectral density reflects power per unit bandwidth, which is different.

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  • $\begingroup$ Technically speaking, the software I use is a "blackbox" of FFT. I just have to set the parameters I need, here include the bandwidth and number of FFT lines (respectively, 1/2.56 times the sampling frequency and the number of sampled point in time domain). The graphs above are straight up FFT result of the same force signal under different conditions. $\endgroup$
    – ComradeH
    Feb 5 at 18:48

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