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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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    $\begingroup$ this assumes the signal is "stationary", right? $\endgroup$ – endolith Apr 29 '13 at 17:51
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    $\begingroup$ 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 '13 at 17:55

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