Linearity of power spectral density calculations

In summary: In the first example, PSD is computed from the concatenated epochs, which means the average will always be nonzero. In the second example, the average is taken over the epochs, so the average will be zero only if all the signals are zero.
  • #1
Schwann
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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:

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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  • #2
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.
 
  • #3
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...
 
  • #4
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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1. What is the concept of linearity in power spectral density calculations?

The concept of linearity in power spectral density (PSD) calculations refers to the property that the PSD of a linear combination of signals is equal to the sum of the individual PSDs of each signal. This means that the PSD of a signal is not affected by linear operations such as scaling, addition, or subtraction.

2. How is linearity of power spectral density calculations useful in signal processing?

The property of linearity in PSD calculations allows for the analysis and manipulation of signals in the frequency domain. This is particularly useful in signal processing tasks such as filtering, noise reduction, and spectral analysis.

3. Can non-linear signals have a linear power spectral density?

No, non-linear signals do not have a linear PSD. Non-linear signals are characterized by a non-linear relationship between the input and output, which means that the PSD cannot be expressed as a linear combination of individual PSDs.

4. Are there any limitations to the linearity of power spectral density calculations?

While the concept of linearity is useful in many signal processing applications, it is important to note that it may not hold true in all cases. For example, if the signals being combined are not stationary or if they have a non-linear relationship, the PSD may not be linear.

5. How can linearity be tested in power spectral density calculations?

Linearity can be tested in PSD calculations by performing a superposition test. This involves comparing the PSD of a signal with the PSD of the same signal after applying a linear operation, such as scaling or addition. If the two PSDs are equal, then the signal exhibits linearity in the frequency domain.

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