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Consider we need the magnitude spectrum of the signal. Signal is recorded in \$N\$ trials to a certain stimuli. Signal varies from each trial because of noise. So, to estimate the actual magnitude spectrum, there are two approaches:

  1. Avg. in temporal domain: Average out trials in temporal domain to obtain signal of less noise and then take its fourier transform to obtain signal's complex frequency content. Visualize magnitude spectrum. The output would be the same, even if we average the fourier transforms(complex), as it is a linear operator.
  2. Avg. in magnitude spectrum domain: Compute fourier transforms of each trial, get magnitude spectrum and average this across trials.

According to me, approach (1) is correct as we are not neglecting any of it. Averaging has to be done in linear domain, which is real and imaginary parts of the complex output (Fourier transform). Whereas the second approach neglects phase spectrum of each trial, i.e. averaging in Polar representation of complex number individually. Many of my friends, even a professor approve the second approach (2). Their argument is that, because the requirement is the magnitude spectrum which is random, because of noise in each trial, so averaging the required output (Fourier transform) over many trials to obtain its estimate is a reasonable step. I am not convinced with it, as spectrum consists of both phase and magnitude and just neglecting one of it to get an estimate individually, by linear approaches, (here Arithmetic mean) is not proper, as the true underlying magnitude spectrum, getting modulated by noise, may be some form of non-linearity (function mapping temporal signal to magnitude spectrum). Thus, averaging magnitude spectrum out won't null out the noise. So, please help by providing opinions of what may be correct and why ?

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Averaging includes the introduction of errors, be it in the temporal OR in the frequency domain.

But let's address your specific concern. My opinion is that if the SNR is good, neglecting the phase distortion created by the noise does not introduce a big error. So IMHO averaging in time or in frequency (signal amplitude) will give good results in both cases if SNR is good, and bad results, also in both cases.

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  • \$\begingroup\$ But, many say, even if SNR is bad, averaging spectrum magnitude is better estimate than averaging complex spectrum and computing magnitude... \$\endgroup\$ – Wupadrasta Santosh Feb 10 '18 at 6:49

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