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Note: This question has been marked as a duplicate of What do the X and Y axis stand for in the Fourier transform domain?. While the answer will likely have an overlap, so the question should be linked, I see the context and background as sufficiently different to warrant a separate question for anyone who is having a similar issue to what I'm having; and it has been noted that some additional information may be useful in answering this question.

I'm trying to find the periods of time series and was directed toward Fourier analysis. I am having a hard time understanding what is going on in ref/Fourier > Applications > Frequency Identification. Would anyone be able to explain, or direct me to an explanation somewhere? Thank you!

EDIT: Here is some sample data. It is a list representing a 55-year time series with one measurement per year, for a total of 56 points. What I have been doing is subtracting off the seventh-order polynomial regression and running the absolute value of the fourier on both the raw data and the data with the overall trend removed.

{0.,0.,0.,0.00118064,0.,0.00437956,0.,0.,0.0035461,0.00107875,0.00715015,0.00501505,0.0040568,0.00294985,0.00514933,0.00609137,0.0047619,0.00100806,0.00196464,0.000882613,0.00187793,0.00282486,0.00172563,0.000807754,0.00318218,0.00233645,0.0015528,0.00149365,0.000776398,0.000688705,0.00198151,0.00236593,0.00225904,0.00149589,0.,0.00332889,0.00179748,0.00126743,0.00289352,0.00218103,0.00265675,0.00104275,0.00509495,0.00292887,0.00161616,0.00327869,0.00413793,0.00223928,0.00502067,0.00434075,0.00392465,0.00469373,0.00595745,0.00521147,0.00567752,0.00293255}

Note, for the Fourier graphs below, the data have not been padded to the right; should I do that?

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marked as duplicate by Daniel Lichtblau, Jens, Michael E2, happy fish, m_goldberg May 15 '17 at 22:30

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    $\begingroup$ related $\endgroup$ – andre314 May 14 '17 at 19:50
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    $\begingroup$ I have put some basic information on Fourier here. Let me know if this helps of if you need more. $\endgroup$ – Hugh May 14 '17 at 21:02
  • $\begingroup$ @Hugh: Thank you, that's some very helpful information; though one thing I'm actually still not clear on is, what do each of the axes represent with standard Fourier in Mathematica? $\endgroup$ – user3769181 May 15 '17 at 22:13
  • $\begingroup$ @andre: That looks like exactly what I'm looking for. Is it necessary to pad to the right? And I'm not clear on how you converted the position of the peak to the period. Thank you! $\endgroup$ – user3769181 May 15 '17 at 22:14
  • $\begingroup$ I have tried to add some extra information to the article here. However, I think you need to describe your difficulty more precisely for me to help you. Try to give us an example of your problem. $\endgroup$ – Hugh May 16 '17 at 11:30

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