How do I average frequency spectra?

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.

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.

• this assumes the signal is "stationary", right? Apr 29, 2013 at 17:51
• 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.
– Peter K.
Apr 29, 2013 at 17:55
• Can you perhaps give some justification and details about the assumptions needed? Or point towards a source? Thanks in advance! Jan 9 at 9:44
• I ask because this seems to be applying Bartlett's method in a case where the original data is already non-contiguous. Jan 9 at 10:52