I Linearity of power spectral density calculations

[Schwann]

TL;DR Summary

Does PSD computed from concatenated epochs of time series differ from PSD computed from averaged epochs of the same time series?

I have a question related to linearity of power spectral density calculation.

Suppose I have a time series, divided into some epochs. If I compute PSD by Welch's method with a time window equal to the length of an epoch and without any overlap, I obtain this result:

If I calculate the average of my time series over the epochs, obtain the averaged signal, the length of which is equal to the length of one epoch (obviously), and then compute PSD by the same method using this averaged signal, I get a slightly different result:

I thought that these two scenarios could not be different, as PSD from the concatenated epochs is presumably equal to PSD from averaged epochs (in my opinion). However, the results are different.

I am looking for analytical explanation of these discrepancies.

Thank you!

 

[BvU]

Summary:: Does PSD computed from concatenated epochs of time series differ from PSD computed from averaged epochs of the same time series?

No. PSD of the same set of particles is calculated. Time doesn't play a role and you do the same calculation in both cases.

analytical explanation of these discrepancies

You must have a mistake somewhere. Hard to say what without having all the details.

 

[Schwann]

Thank you for your answer. What I meant is not computing PSD based on the same set of time points with different order.

No. PSD of the same set of particles is calculated. Time doesn't play a role and you do the same calculation in both cases.

Whan I meant is the following.

Scenario 1. I have a long time series made of concatenated epochs. Then I compute PSD.

Scenario 2. From the same epochs I compute the average and then compute PSD.

In Scenario 1 we have averaged PSDs from each epoch, because the time window in Welch's method I set as the length of the epoch. In Scenario 2 we have PSD from the averaged signal. It seems that they are not equal, as evident from the plots in my initial question. But I don't understand why...

 

[BvU]

Oops, professional brainwashing over an extended period made me read "particle size distribution" for PSD

To add insult to injury I didn't understand the scenario: I figured PSD versus average of epoch PSDs instead of PSD of epoch average.

An averaged signal ruins a power spectral density: the average of a nonzero signal can be zero in a worst case