How did letter d.) become letter e.)? How do we perform this averaging process? Its clear the averaging the signal causes some noise to be reduced.
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If you are in a position to acquire $N$ snapshots of data, $s_n(t)$, for $n=0\ldots N-1$ then you can just do:
$$ S_n(k) = |\mathrm{fft}(s_n(t))|^2\\ \hat{S}(k) = \frac{1}{N} \sum_{n=0}^{N-1} S_n(k) $$
and a certain amount of averaging will reduce the apparent noise on the spectrum.
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1$\begingroup$ this assumes the signal is "stationary", right? $\endgroup$– endolithApr 29, 2013 at 17:51
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2$\begingroup$ Right: any time you're using the spectrum, you're assuming stationarity. When you're averaging multiple spectra as here, you're assuming stationarity over a longer period. $\endgroup$– Peter K. ♦Apr 29, 2013 at 17:55
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$\begingroup$ Can you perhaps give some justification and details about the assumptions needed? Or point towards a source? Thanks in advance! $\endgroup$ Jan 9 at 9:44
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$\begingroup$ I ask because this seems to be applying Bartlett's method in a case where the original data is already non-contiguous. $\endgroup$ Jan 9 at 10:52
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$\begingroup$ @AdomasBaliuka Please ask a new question if this answer doesn't help you. Feel free to reference this question and answer in your question, to set context for what you're asking. $\endgroup$– Peter K. ♦Jan 9 at 13:20