Linearity of power spectral density calculations

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Discussion Overview

The discussion revolves around the linearity of power spectral density (PSD) calculations, specifically comparing results obtained from concatenated epochs of a time series versus those obtained from an averaged signal derived from the same epochs. Participants explore the implications of using Welch's method in these two scenarios and seek analytical explanations for observed discrepancies in the results.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions whether the PSD computed from concatenated epochs differs from that computed from averaged epochs, suggesting that they should yield the same result.
  • Another participant asserts that the same set of particles is being analyzed in both cases, implying that time does not influence the outcome of the PSD calculation.
  • A participant clarifies that the scenarios involve different approaches: one computes PSD from concatenated epochs while the other computes it from the average of those epochs, leading to different results.
  • Concerns are raised about the potential for the averaged signal to affect the PSD, with one participant noting that averaging a nonzero signal could result in a zero signal, which would impact the PSD calculation.

Areas of Agreement / Disagreement

Participants express differing views on whether the PSD results from the two scenarios should be equivalent. There is no consensus on the reasons for the discrepancies observed in the results.

Contextual Notes

Participants acknowledge the complexity of the scenarios and the potential for misunderstandings regarding the definitions and implications of the calculations involved. Specific details about the calculations and conditions under which the discrepancies arise remain unresolved.

Schwann
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TL;DR
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:

1594982808504.png


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:

1594982921650.png


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!
 
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Schwann said:
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.
 
Thank you for your answer. What I meant is not computing PSD based on the same set of time points with different order.
BvU said:
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.
1594998152344.png


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

1594998270007.png


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

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
 
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