I have a time series and apply the FFT to get a spectrum.

Let's assume that my sampling frequency and the length of the time sample are chosen such that I end up with a $\Delta f = 0.1$ Hz.

As this is still rather noisy, I'd like to change $\Delta f$ to 0.5 Hz a posteriori.

How do I combine the fourier components of the adjacent bins? Am I right to just sum them? Or do I take the square root of the sum of squares? Is there a term for what I am trying to achieve, so that I can find some references on my own?

  • $\begingroup$ what is the statistical model of noise ? $\endgroup$ – AlexTP May 22 '17 at 8:49
  • $\begingroup$ I don't have one. the background to question is that i plot the spectrum. i would like it to be more visually pleasing, while still being essentially correct and taking advantage of the fact that I have the data at a higher resolution in principle. $\endgroup$ – fft_newbie May 22 '17 at 9:04
  • $\begingroup$ "essentially correct", however, depends on whether you correctly represent your signal or noise, and that can only be said about something that you have a model for. There's no "universally right" approach here, you need to model things. $\endgroup$ – Marcus Müller May 22 '17 at 9:05
  • $\begingroup$ to be quite frank, I dont know how to get a precise model. The only thing I can say is that it is vibration data, and I'm interested in frequencies below 200 Hz. I assume the signal to be the superposition of sines and some white noise (I guess). Could you give me some keywords regarding to what exactly you mean with a model, and especially on how then to decide how to combine the frequency bins based on that model? $\endgroup$ – fft_newbie May 22 '17 at 12:28

There are different methods for doing what you want. But first i have to tell you power spectrum density (PSD) is real and positive and you have to obtain it from squared FFT of signal or from FFT of signal's auto-correlation (FFT of auto-correlations is always positive). To reduce noise you could apply some damping window on your signal. This method is called modified periodogram. Also you could divide your signal to smaller segments and find PSD of each segment then find average of PSD of each segment. This method is called Bartlet method. Also your suggested method, summation of adjacent bins or applying moving average works too (you must apply it on PSD).

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