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I have a long signal (1 day sampled at 500 Hz), which results in many frequencies if I calculate the amplitude spectrum (see e.g. here). However, I am only interested in the "big picture", let's say only frequencies of natural numbers (i.e. 1, 2, ..., 250 Hz).

Should I simply remove all other frequencies? Should I do something like a binning? Or is there even a better solution?

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Apply a sliding window function of 500 samples length to the signal, for example a Hamming window. Choose a suitable overlap between windows e.g. 75%. Calculate the spectrum of each window and then average over all windows. Then the frequency bins lie exactly at the natural numbers as you wanted.

But beware: each bin does not only represent one exact frequency, but the energy in a frequency band, in this case $[f-0.5,f+0.5]$Hz

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