You could also look at where along the unit circle the roots are located, the angle roughly correspond to the frequency.
You are right. However, in practice, it is very hard to associate the angle with the dominant frequency. In that paper, Case 2 (Fig 6&Table 2), only the real root exceeds the 0.2 threshold criterion. The dominant frequency, however, is definitely not 0. Other roots are all far away from the unit circle (the distances are 0.3+).
The mean would just go into the zero mode of the Fourier transform, wouldn’t it?
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To conclude my case, I’d like to post here my approach of oscillation detection. The oscillation index is defined as
r = \frac{\sqrt{\frac{1}{N-1}\sum_{i=1}^{N-1}(\epsilon_{i+1} - \epsilon_i)^2}}{\frac{1}{N-1}\sum_{i=1}^{N-1}\epsilon_i}
When r>1.0 for a number of consecutive iterations, then the oscillation is confirmed. Below is an example of this index.
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